Discussion Paper No.372

Graham-type Trade Model under theCondition of Full Employment: Ricardian
Trade Model with Link Commodities
Hideo Sato
（Tohoku University, Professor Emeritus）
August 2017

TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND

MANAGEMENT TOHOKU UNIVERSITY

27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANGraham-type Trade Model under the Condition of Full Employment:
Ricardian Trade Model with Link Commodities
Hideo Sato

Abstract
This paper provides a Graham-type trade model, which is a multi-country
multi-commodity Ricardian (one-factor) trade model and attaches great importance tocommodities produced in common in more than one country, or link commodities: e.g.
automobiles among Japan, USA, and Germany, IT products among Japan, Korea, andChina, and beef among Brazil, Australia, and USA. The link commodities determine
the relative wage rates of all the countries producing the same link commodities,
thereby determining the relative prices of all commodities produced in these countries.
Among the pattern of the international division of labor (IDL pattern) formed in themulti-country multi-commodity trade model, the linkage type IDL patterns linking all
the countries directly and indirectly is overwhelmingly many, the limbo type IDL
patterns having more than one disconnection accounts for a small part, especially oneof them, the perfect specialization patterns (PSPs) having no link commodities areuncommon. When small changes in demand occur, quantity adjustments without pricechanges are conducted among countries linked directly and indirectly (the Grahamcase), and price adjustments among countries disconnected (the Mill case).
Conventional trade theories have focused their attention on the PSP and have notincorporated the link commodities into their models. The link commodities, therefore
the quantity adjustments or the Graham case matters. We need to free ourselves fromundue emphasis on the PSPs.
Keywords: link commodities; multi-country multi-commodity; quantity adjustments;
F. D. Graham
Acknowledgment: I would like to thank Kazuhiro Kurose (Tohoku University) andYoshinori Shiozawa (Osaka City University, Professor Emeritus) for their usefulsuggestions and helpful comments.
1Graham-type Trade Model under the Condition of Full Employment:
Ricardian Trade Model with Link Commodities
Hideo Sato

1 Introduction
Conventional trade theories have not sufficiently incorporated two important stylized
facts, existence of intermediate goods traded internationally and existence ofcommodities produced in common in more than one country (we refer to these ascommodities with comparative average in plain terms or link commodities in technicalterms), into their models. According to WTO (2013), share of intermediate goods
accounted for 51-55% of world non-fuel export in the years 2000-2011. In addition,
among a huge number of items traded practically, we can see not a few commodities
with comparative average, e.g. automobiles among Japan, USA, and Germany, IT
products among Japan, Korea, and China, beef among Brazil, Australia, and USA, andso on. Nevertheless, most trade models have ignored these facts, especially the latter
fact.
However, new stream of research into trade theory is changing this kind of situation.
Shiozawa (2007, 2017a) and Oka (2017) presented a new trade model withintermediate goods and link commodities, and Sato (2017) showed the outline of a
Graham-type trade model, which is a multi-country multi-commodity Ricardian(one-factor) trade model and attaches great importance to link commodities.
This paper provides the details of a Graham-type trade model. Although the modelhas two versions, a full employment version and an unemployment version, here, wedeal with only the former. The remainder of this paper is organized as follows. Section 2
describes the essence of Graham’s relatively unknown theory of international values,
because our model is based on Graham’s model and a modified version of the original.
Section 3 explains the Graham-type trade model under the condition of full employment.
Section 4 is concluding remarks.
Furthermore, we add three appendixes. First, we deal with the three country threecommodity numerical example in Jones (1961), which clarified what pattern is the mostefficient if the perfect international specializations is formed. We show that it is2uncommon for the perfect specializations to be practically formed. Second, we
introduce the pioneering H. von Mangoldt’s numerical trade model and compare it with
Graham’s model. The third is a supplement of subsubsection 3.2.4.

2 Graham’s theory of international values
Frank D. Graham (1890-1949), a mainstream US economist, researched international
values from the 1920s and published his major book The Theory of International
Values in 19481, the year before his accidental death. His research, however, has notbeen praised within mainstream economics and has been almost forgotten. The reason
is that while the origin of mainstream trade theory is J. S. Mill’s theory of reciprocal
demand (Mill, 1848), Graham criticized Mill’s theory thoroughly 2 . Also,
non-mainstream economists, especially Marxian economists, who are critical of Mill’stheory, have refused to accept Graham’s theory as excellent and have ignored it
entirely. The reason is that, while the labor theory of value is the most important
foundation for Marxian economics, Graham regarded the theory as a stumbling blockand refused it. However, Graham’s theory was decisively important for the new trade
theory. We explain the features of his theory in some detail.
(1) Graham was the first to present the existence of an equilibrium solution in a
multi-country multi-commodity trade model.
1

His related works are Graham (1923, 1932, 1948). The following explanation mainly
relies on Graham (1948).
2
Soon after the publication of Graham (1948), many book reviews and papers relating
to Graham’s theory were written, and most of them supported Mill’s theory. Elliot(1950) insisted that Graham’s two-country multi-commodity cases can be translated
into offer curves by using the conception of A. Marshall’s “representative bales”(Marshall, 1923, p.157). Metzler (1950) mentioned that Elliot was right in two-country
cases, but not in three-or-more country cases, and Whitin (1953) wrote that Elliot wasalso right in three-or-more country cases. Before these comments, Viner (1937) had
criticized Graham, supporting Mill and Marshall. Melvin (1969) also interpreted
Graham’s model on the base of offer curves. Almost attempts to understand Graham’s
trade theory within a framework of conventional trade theories did not succeed,
because the theory transcended conventional theories.
3We can sum up the fundamental structure of Graham’s model as follows.
1. There are many countries and many commodities.
2. There are no intermediate goods and no profits. All commodities are for
consumption.
3. For each country, constant opportunity costs, economic scales, and demand
structures are given.
4. Full employment and trade equilibrium (or national expenditure equals national
income in each country) are fulfilled.
5. There are no transport costs and no trade barriers.
Under these assumptions, the patterns of the international division of labor (hereafter,
IDL patterns), international values, and each country’s volumes of production, export,
import, and consumption are determined uniquely.
Graham explains the above, while providing no mathematical treatment3, by using
many numerical examples. In earlier trade theories, although there was an example thatan equilibrium solution was derived in a two-country multi-commodity case4, in a
multi-country multi-commodity case, some possible IDL patterns were only shown at
best5. Indeed, Graham was the first to present the existence of an equilibrium solution
in a multi-country multi-commodity (four-country three-commodity or 10-country
10-commodity) case6.
(2) To explain domestic values and international values by the same logic, Graham
3

McKenzie (1954a) presented a mathematical treatment for Graham’s model and
McKenzie (1954b) proved the existence and uniqueness of equilibrium in a
multi-country multi-commodity one-factor trade model with some properties, of which
subset is, McKenzie claimed, Graham’s model.
4
See von Mangoldt (1975) and Appendix 2.
5
See Section 4, Chapter 8 of Viner (1937).
6
Graham calculated equilibrium solutions of his numerical examples through atedious process of trial and error. Although he requested mathematicians of great
repute to provide a mathematical formula solving for the equilibrium of his numericalexamples, they were unable to furnish any such formulas (Graham, 1948, p. 95).
According to McKenzie (1999), one of them, von Neumann replied that no analytic
solution was possible (p. 5). In fact, as shown later, there is no way to solve theproblem at one stroke, and several complicated processes are needed in order to obtain
the equilibrium solutions practically. As far as I know, it is Noguchi (1990) to provide
an algorithm for solving for the first time. However, Noguchi’s approach is limited to
the linkage type (see 3.1.1) and considerably differs from our approach.
4expresses production techniques of commodities not by labor costs (inputtedlabor), but by opportunity costs.
According to Graham, each country’s production techniques differ in every sector.
While the labor theory of value expresses the difference in these techniques by using
the difference in labor input coefficients, he expresses it by using the difference in theopportunity cost of each commodity. Concretely, he designates a specific commodity
as a benchmark commodity (the opportunity cost of this commodity is one) and
expresses the production techniques of other commodities by the number of units
producible by giving up production of one unit of the benchmark commodity. The
opportunity costs are essentially constant7, as distinct from those of neoclassical trade
theory which are increasing. Graham describes the reason for using opportunity costs
as follows:
When we think in terms of opportunity cost it can be conclusively demonstrated
that Ricardo, Mill, and the neo-classicists, were wholly wrong in supposing thatthe same rule which regulates the relative value of commodities in one country
does not regulate the relative value of the commodities exchanged between two
or more countries. (Graham 1948, p. 333)
We also explain the other two given conditions. The economic scale of each country
is expressed by the production volumes of the benchmark commodity which is realized
when each country specializes in the commodity. Although full employment is
supposed, the volumes of production factors and absolute productivity levels are notshown. Therefore, differentials in per capita income or wage rates among countries are
not argued in the theory of international values directly and are treated as anotherproblem 8 . The demand structures of each country are given by the expenditure
coefficients (proportions of amounts expended on each commodity in total
expenditure). The sum of the coefficients is one (i.e., all income is expended for
7

Graham refers to the case of variable opportunity costs too, and indicates that the
number of commodities produced in common in more than one country would growunder the increasing opportunity costs (Graham 1948, pp. 146-151).
8
Graham is not indifferent to the problem. For example, he writes that national
prosperity (per capita income or wage rate) is a function of two variables, per capita
physical productivity and the terms of trade, and the former is more important (p. 50,
pp. 212-213, p. 233). He also refers to money wages (p. 261, p. 307).
5

consumption) in every country.
(3) International values are determined by the opportunity costs in each country andlink commodities.
International values or the world relative prices of commodities are determined notby reciprocal national demand, but by the opportunity costs in each country just likedomestic values. What is important in this determination is the existence ofcommodities produced in common in more than one country, termed link commodities(p. 254, p. 332)9. This link commodities link the opportunity costs of countries that
produce the same link commodities, meaning that the relative prices of all thecommodities produced in these countries are determined uniquely. In principle, every
country has at least one link commodity, suggesting numerous link commodities in theworld at large. As a result, a body of link commodities links the opportunity costs of all
countries and thus determines the international values of all the commodities in theworld. The link commodities are, in turn, determined by the interaction among theopportunity costs, economic scales, and demand structures in each country. According
to Graham, the link commodities were the missing link of the classical theory of value.
(4) In the face of changes in demand, international values are highly stable.
International values, formed once, are highly stable in the face of changes indemand. Such changes are adjusted through changes in production volumes andexport-import volumes without price changes. If drastic changes in demand occur,
prices might change slightly. In this case, the price changes are necessarily
accompanied by changes in the IDL pattern. Newly formed international values arealso based on the linkage of the opportunity costs in each country.
However, depending on the three given conditions of 3 in the above (1), the linkage
of opportunity costs might be disconnected. Graham calls such a state of disconnection
limbo, and regards this state as highly improbable10. In the limbo case, a small change
9Graham also called them the common products (p. 69, p. 257) or the common
commodities (pp. 253-4). Considering the concept of the terms, however, “linkcommodities” are the most suitable.
10
According to Graham, in limbo cases, small changes in demand cause changes incommodity prices and also commodity prices are unstable and indeterminate (p. 35).
Although Graham almost ignored the limbo cases, as pointed out by McKenzie(1954a) and shown in subsection 3.2, the possibility of limbo cases should not be6

in demand brings about an immediate change in international values. Exemplified byusing a two-country two-commodity model, the limbo case is a situation in which eachcountry specializes in a commodity with a comparative advantage, an ordinary caseused in textbooks. According to him, however, a situation in which one country
produces two commodities and the other country produces either commodity with acomparative advantage has a far higher probability. Then, international values are
determined by the opportunity cost of the former country and reciprocal demand plays
no part.

3 A Graham-type model under the condition of full employment
3.1 Derivation of the equilibrium solution in the M-country
N-commodity model
3.1.1 Model Setting and Definition of Terms
At first, we set a general Graham-type model that assumes full employment as follows.
1. There are M countries and N commodities. Here, M and N are integers more than 2
and N is larger than M.
2. There are no intermediate goods and no profits. All commodities are for
consumption.
3. Full employment and trade equilibrium (or national expenditure equals national
income in each country) are fulfilled.
4. There are no transport costs and no trade barriers.
5. There are no international movements of labor and domestic wage rates are equalin all sectors.
6. For each country, production techniques expressed by constant labor input
coefficients, volumes of usable labor, and demand structures expressed by
expenditure coefficients are given. Although we don’t absolutely need theinformation about the labor input coefficients of some sectors (e. g. the carignored.
7

industries in developing countries or the crude oil extraction industries in
non-oil-producing countries) in which a probability of having a comparativeadvantage is almost zero, we give all sector’s data for convenience of explanation
and assume that the degrees of comparative advantage between two countries
selected arbitrarily differ in every sector. The sum of the expenditure coefficients is
one in every country.
In the full employment version of the Graham-type model, under these assumptions,the IDL patterns, international values or the world relative prices, wage rates in eachcountry, and each country’s volumes of production, export, import, and consumption
are determined uniquely. Before explaining logic of the determination and a way toderive the equilibrium solution, we will definite some terms.
At first, the equilibrium solution consists of identification of the IDL pattern, eachcommodity’s equilibrium production volumes in each country, each country’s
equilibrium wage rates, and the equilibrium international values. By obtaining theequilibrium solution, we can calculate the each commodity’s volumes of export, import
and consumption in each country.
Given the international division of labor, some sectors in each country continue theproduction activity and other sectors cease it. We call the former active points and thelatter non-active points. The IDL patterns have to be reasonable. Here, “reasonable”
means a situation in which both the “production costs of active points = commodity
prices” and “production costs of non-active points > commodity prices” are fulfilled.
Since the purpose of economic models is to understand the real world, let us watch
the actual situation of the international division of labor. Then, we can see a scene that
there does not exist in ordinary trade models, that is, a fact that there are commodities
produced in common in more than one country: Cars produced in Japan, USA., andGermany, IT products in China, Korea, and Japan, textile products in some developing
countries, and so on. We call such commodities link commodities after Graham. The
link commodities determine the relative wage rates of all the countries producing thesame link commodities, thereby determining the relative prices of all commodities
produced in these countries. As a same commodity has an identical price, the relativelabor productivities (inverses of the labor input coefficients) of the link commodities8

are precisely the relative wage rates 11 , and the relative prices are obtained bymultiplying the relative wage rates by constant labor input coefficients.
When countries have the same link commodities, we say, these countries are directly
linked. Countries, however, can also be indirectly linked. Suppose that there arecountries A, B, and C, and that A and B produce a link commodity in common, further
that B and C produce another link commodity in common. In this case, all the three
countries are linked: A and C are not directly but indirectly (by the medium of B)
linked. Hereafter, the term “link” means not only “link directly” but also “linkindirectly” and we use the term “linkage” to express “the state of being linked”.
The IDL patterns can be classified into two types. One is when all the countries arelinked through link commodities. We refer to this as the linkage type. In this type, therelative wage rates of all countries and the relative prices of all commodities can be
expressed by labor input coefficients according to each IDL pattern. In other words,
once the patterns are determined, all the relative wage rates and commodity prices(hereafter, the wage rates/prices) are determined by the patterns themselves, or there isa one-to-one correspondence between the IDL patterns and the wage rates/prices.
The second type is called the limbo type. In this type, the linkage of countries is not
perfect, and one or more disconnections of the linkage occur. Therefore, determining
all the wage rates/prices only by the IDL patterns is not possible. Theoretically, thedisconnection can occur in the range from 1 to M–1. When there are M–1
disconnections, perfect specialization patterns (hereafter, PSPs) are formed, whichhave no link commodities and are ordinary cases in a textbook explanation of a
two-country two-commodity trade model: each country specializes in a commoditywith a comparative advantage. An ordinary case in a two-country two-commodity
model is an extreme case in a multi-country multi-commodity model. We need to pay
attention that, except this extreme case, there are link commodities also in the limbo
type and that the link commodities perform the above-mentioned functions.

11

Suppose that countries A and B produce a same link commodity (e.g. commodity 1).
Then, the commodity’s price (p1) is expressed as the product of wage rates (wA, wB)
and labor input coefficients (aA1, aB1), or p1=wA*aA1=wB*aB1. Therefore, wB/wA=
aA1/aB1.
9

Figure 1 illustrates these two types in a six-country case. Six countries (expressed byx) are all linked in the linkage type, whereas in the limbo type, the linkage is
disconnected in two places and six countries are divided into three groups, within
which more than one country is linked unless the groups consist of a single country.
linkage typelimbo type with two disconnectionsx-----x-----x-----x-----x-----x

x-----xx-----x-----xxFig. 1 An example of the two types of the IDL patterns

3.1.2 Derivation of the equilibrium solution in the linkage typeHere, we explain a way to derive equilibrium solution in the linkage type practically.
The process has three phases as follows: searching for and identifying reasonable IDL
patterns, setting up simultaneous equations according to each IDL pattern and solving
them mathematically, and selecting an economically meaningful solution set. We
describe details in order.
First, we have to search for and identify reasonable IDL patterns classified into thelinkage type. Whether an IDL pattern is reasonable is determined only by the labor
input coefficients and judgement of reasonableness is, though laborious, simple. As, inthe linkage type, all the wage rates/prices are already known according to the IDL
patterns, we only have to compare the commodity prices with the production costs of
non-active points, which are obtained by multiplying their labor input coefficients by
the each country’s wage rate. The number of reasonable patterns of the linkage type is(M+N–2)!/{(M–1)!(N–1)!} in an M-country N-commodity case12. If M and N are large,
because of a large number of IDL patterns to be judged13, it is difficult even to identify
the patterns. Including the rest of the process, the support of computer program would
be needed in order to calculate actually.
12

Based on Shiozawa (2012, p. 50), the number of reasonable IDL patterns with ldisconnections is (M+N–l–2)!/{(M–l–1)!(N–l–1)!l!}. By substituting zero for l in thisexpression, the number of linkage type IDL patterns is obtained since the linkage type
has no disconnection.
13
The number is {M^(N-1)}{N^(M-1)} according to Shiozawa’s direct suggestion.
10

Second, for all the reasonable patterns, we identify the wage rates/prices andformulate simultaneous equations, which are composed of M equations expressing full
employment conditions of each country and N equations expressing supply-demandbalance for each commodity. However, independent equations are M+N–1, because
one of N equations is not independent owing to Walras’ law. Unknowns are production
volumes in each active point, the number of which is M+N–1 in the linkage type
(McKenzie, 1954a, p. 175). Why M+N–1? There must be N active points to ensure that
all commodities are produced. In this situation, the countries are entirely unconnected.
Therefore, M–1 additional active points are needed to link all of them. Thus, as theequations and unknowns are equal in number, we can solve all the sets of equations
mathematically. However, whether the solutions obtained mathematically aremeaningful economically is another problem, leading to the next process.
Third, since the production volumes must be positive economically, we have toselect a set having all positive solutions from the (M+N–2)!/{(M–1)!(N–1)!} sets ofsolutions. If there is such a set, the solutions of this set are the equilibrium solutions
required. The IDL pattern, production volumes, and the wage rates/prices are
determined. The consumption volumes and export-import volumes in each country arealso able to be calculated easily.
We now give an example of the above-mentioned in the case of a three-county
four-commodity. There are the three countries of A, B, and C and the four commoditiesof 1, 2, 3, and 4. We define aij, bij, Li, pj, and wi as commodity j’s labor input coefficientin country i, commodity j’s expenditure coefficient in country i, volumes of usable
labor in country i, commodity j’s price, and wage rate of country i, respectively. The
numéraire is commodity 1. Commodity j’s production volumes in country i areexpressed by xij. Consumption volumes are expressed as wiLibij/pj and export-import
volumes are difference between production volumes and consumption volumes in eachcountry. In a three-county four-commodity case, the production volumes of six active
points are unknowns and those of non-active points are zero. For example, in the IDL
pattern that country A produces commodities 1 and 2, country B commodities 2 and 3,
and country C commodities 3 and 4, the wage rates/prices and simultaneous equations
are expressed as follows. Although we have to rewrite these in the case of otherpatterns, this is easy and would be sufficient for exemplification.
11

Prices and wage rates:
p1=1
p2=aA2/aA1
p3=(aB3/aB2)*p2=(aB3/aB2)*(aA2/aA1)
p4=(aC4/aC3)*p3=(aC4/aC3)*(aB3/aB2)*(aA2/aA1)
wA=1/aA1
wB=(aA2/aB2)*wA=aA2/(aB2*aA1)
wC=(aB3/aC3)*wB=(aB3*aA2)/(aC3*aB2*aA1)
Conditions of full employment:
aA1*xA1+aA2*xA2=LA
aB2*xB2+aB3*xB3=LB
aC3*xC3+aC4*xC4=LC
Conditions of supply-demand balance (only three of the four are independent):
xA1*p1=wA*LA*bA1+wB*LB*bB1+wC*LC*bC1xA2*p2+xB2*p2=wA*LA*bA2+wB*LB*bB2+wC*LC*bC2
xB3*p3+xC3*p3=wA*LA*bA3+wB*LB*bB3+wC*LC*bC3
xC4*p4=wA*LA*bA4+wB*LB*bB4+wC*LC*bC4
As the wage rates/prices are expressed only by the labor input coefficients, we canconfirm that, once the IDL pattern is determined, the wage rates/prices are determined
only by the condition of production techniques. Although there are no explicit
expressions, by multiplying the both sides of full employment expressions by wage
rates, and also by deforming the left-hand sides adequately: by replacing products oflabor input coefficients and wage rates with commodity prices, and furthermore bymultiplying the right-hand sides by the summation expressions of expenditure
coefficients or bi1+bi2+bi3+bi4 (=1), we can also confirm that national income(summation of production volumes multiplied by prices) equals national expenditure(summation of amounts expended on each commodity) 14 or trade equilibrium is
14

For example, about country A: p1*xA1+p2*xA2=wA*LA*(bA1+bA2+bA3+bA4).
12

fulfilled. If the solutions of the above equations are all positive, these are theequilibrium solutions. When the solutions include zero or negative production volumes,
we have to solve other sets of equations.
3.1.3 Derivation of the equilibrium solution in the limbo type
If there is no set that all the solutions are positive, it means that one or moredisconnections of the linkage occur. Then, we must expand the search range to find theequilibrium solution to the IDL patterns of the limbo type. The process is as follows.
First, the same as the linkage type, we identify the reasonable patterns of the limbo
type. The number of the patterns are very large: ∑(M+N–l–2)!/{(M–l–1)!(N–l–1)!l!}(l=1, 2, …, M–1) 15 . Also in the case of the limbo type, whether the pattern isreasonable is determined only by the labor input coefficients. However, because not all
the wage rates/prices are determined according to the IDL patterns, we have to adopt a
different way from the linkage type. There are two methods.
We explain those in the case of the IDL patterns with l disconnections. In this case,countries are divided into l+1 groups and the IDL has to be reasonable within eachgroup and also among groups. Reasonableness of the IDL within each group is able tobe checked easily because the relative wage rates are determined by linkage. For the
IDL among groups to be reasonable, there is a condition. It is that relative wage rates
between countries belonging to different groups have to be within a specific range.
This wage rates constraint condition has to be met between all the combinations of two
out of the l+1 groups, or there are l+1C2 constraint conditions relating wage rates. If
wage rates satisfying all these conditions can exist under an IDL pattern, the IDL
pattern is judged as reasonable (we show examples in subsubsection 3.1.4). Contrary, if
these conditions are contradictory each other, the IDL pattern is judged as notreasonable. For convenience of explanation, we describe this method accompanied by
identification of the range of wage rates as the judging method 1.
The other method called the judging method 2 uses the identified IDL patterns oflinkage type. If, while holding the condition that all the commodities are produced andall the countries produce at least one commodity, we remove one active point of a
15

See fn. 12. All the patterns from with one disconnection to with M–1 disconnections
are summed.
13

linkage type IDL pattern, one disconnection occurs and a limbo type IDL pattern with
one disconnection is derived. Further, by adding the same operation to this newly
obtained pattern, we can obtain an IDL pattern with two disconnections. By repeating
the same operation up to M–1 disconnections, we can identify all the limbo type IDL
patterns (we show an example in subsubsection 3.1.4).
Next, after identifying the patterns, we formulate equations and solve them. Whenthe number of disconnections is l, l+1 country groups are formed. While the wage
rates/prices within each country group are determined by the IDL pattern itself, thosebetween groups are not determined only by the pattern. To determine all the wage
rates/prices, we have to add the wage rate of a country in each group which does notproduce a numéraire commodity as the unknown 16 . The number of additional
unknowns is l. On the other side, the number of active points diminishes by the numberof disconnections in the limbo type (McKenzie, 1954a: see the judging method 2 inorder to understand intuitively). Eventually, regardless of the number of disconnections,the total unknowns are still M+N–1, and we can solve all the sets of equations
mathematically.
Finally, we have to select a set of solutions that fulfills the following two conditions:
all the solutions are positive and the obtained wage rates are within the adequate range
in the case of using the judging method 1 or the solution set passes a competitive test in
the case of the judging method 2. This test is to check whether non-active points arecompetitive by comparing the production costs of non-active points with thecommodity prices. As the entire wage rates/prices are already obtained, the test is
simple. If at least one non-active point is competitive, the set is disqualified. Of course,
the verification of the wage rates’ range and the competitive test are equivalent. In any
case, only one set satisfies these two conditions and this set is the equilibrium solution.
In the pattern that country A produces commodities 1 and 2, country B commodities
3 and 4, and country C commodity 4 only, the wage rates/prices and simultaneous
equations are expressed as below. As some of wage rates are unknowns in the limbo
type, we express them as xi (i=A, B, C). Here, the production volumes of five active
points and country B’s wage rate (xB) are unknowns.
16

Although it is possible to add not wage rates but commodity prices as unknowns,
we select wage rates in view of tractability.
14

Prices and wage rates:
p1=1
p2=aA2/aA1
p3=aB3*xB
p4=aB4*xB
wA=1/aA1
wB=xB
wC=(aB4/aC4)*xB
Conditions of full employment:
aA1*xA1+aA2*xA2=LA
aB3*xB3+aB4*xB4=LB
aC4*xC4=LC
Conditions of supply-demand balance (only three of the four are independent):
xA1*p1=wA*LA*bA1+xB*LB*bB1+(aB4/aC4)*xB*LC*bC1xA2*p2=wA*LA*bA2+xB*LB*bB2+(aB4/aC4)*xB*LC*bC2
xB3*p3=wA*LA*bA3+xB*LB*bB3+(aB4/aC4)*xB*LC*bC3
xB4*p4+xC4*p4=wA*LA*bA4+xB*LB*bB4+(aB4/aC4)*xB*LC*bC4
If the solution set obtained fulfills the above-mentioned two conditions, this is theequilibrium solution. Otherwise, we have to calculate about other IDL patterns.
In the limbo type, a small change in demand may cause a change in the wage
rates/prices. By solving the above equations for xB
xB =LA/aA1*(bA3+bA4)/{aB4/aC4*LC*(bC1+bC2)+LB*(bB1+bB2)}
This expression shows the following17: expenditure coefficients of all countries are
involved in determination of the wage rates of countries B and C (it should be notedthat although there is not e.g. bA1 or bA2 in the above expression, a change in these may
brings about a change in bA3 or bA4); if country A increases (decreases) expenditure
coefficients of commodities 3 and 4 produced in countries B and C, the wage rates of
17

Because this expression is effective as long as the IDL pattern does not change, thefollowing explanation assumes that there is no change in the pattern.
15

countries B and C increase (decrease); conversely, if countries B and C increase(decrease) expenditure coefficients of commodities 1 and 2 produced in country A,
wage rates of both countries decrease (increase). The reason is simple. In this IDL
pattern, since quantity adjustment does not work between group of commodities 1 and
2 and group of commodities 3 and 4, national income and commodity prices, therefore
wage rates have to change in order to absorb change in expenditure coefficients withfull employment and trade equilibrium kept18.

3.2 Analysis of the model by using the three-country four-commodity
numerical example
3.2.1 Identification of the reasonable IDL patterns
We set a three-country four-commodity numerical example and analyze characteristicsof the Graham-type model by observing changes in the equilibrium values arising fromchanges in the given conditions. Labor input coefficients are given as Table 1. Here,country A symbolizes developed country, country B emerging country, and country C
developing country. Units of the commodities are chosen in such a manner that all thelabor input coefficients of country A are one and the commodities are numbered inorder of diminishing country A’s comparative advantage between countries A and B19.
Labor input coefficients of country C are given arbitrarily. The numéraire iscommodity 1.

18

We will give a supplementary explanation. Suppose that bA3 increases a little, bA1
deceases a little and others do not change. Then, if we define commodity j’s
consumption volumes in country i as cij, cA1 (=wALAbA1/p1) decrease, cA2 (=wALAbA2/p2)
is unchanged, and cB1, cB2, cC1, and cC2 moves in the same direction with wB and wC. In
order to fulfill country A’s full employment, cB1, cB2, cC1, and cC2, therefore wB and wC
must increase.
19
Although, without this process, the number of the IDL patterns to be investigatedreaches 432, the number diminishes to 112 owing to this process.
16

Tab. 1 Labor input coefficients
Comm.1 Comm.2

Comm.3 Comm.4

Country A

1

1

1

1

Country B

5

4

3

2

Country C

60

25

30

7

From this table, ten linkage type IDL patterns are determined as bellow. First
parentheses show the IDL patterns and e.g. A123 means that country A produces
commodities 1, 2, and 3, second parentheses show commodity prices in order from
commodity 1 to 4, and third wage rates from country A to C.
Linkage type IDL patterns
4+1+1 type (a country produces 4 commodities and the other two 1 each)1) (A1234; B4; C4) (1; 1; 1; 1) (1; 1/2; 1/7)
2) (A1; B1234; C4) (1; 4/5; 3/5; 2/5) (1; 1/5; 2/35)
3) (A1; B1; C1234) (1; 25/60; 1/2; 7/60) (1; 1/5; 1/60)
3+2+1 type (a country produces 3 commodities, another 2, and the other 1)4) (A123; B34; C4) (1; 1; 1; 2/3) (1; 1/3; 2/21)
5) (A123; B3; C24) (1; 1; 1; 7/25) (1; 1/3; 1/25)
6) (A1; B123; C24) (1; 4/5; 3/5; 28/125) (1; 1/5; 4/125)
7) (A12; B234; C4) (1; 1; 3/4; 1/2) (1; 1/4; 1/14)
8) (A13; B3; C234) (1; 5/6; 1; 7/30) (1; 1/3; 1/30)
9) (A1; B13; C234) (1; 1/2; 3/5; 7/50) (1; 1/5; 1/50)
2+2+2 (all the countries produce 2 commodities each)10) (A12; B23; C24) (1; 1; 3/4; 7/25) (1; 1/4; 1/25)
Next are the limbo type IDL patterns. The IDL patterns and wage rates (including
their range) are shown below (commodity prices are omitted). In the patterns with two
disconnections, country C needs to fulfill two wage rates constraints.
Limbo type IDL patterns with one disconnection11) (A123; B4; C4) (1; 1/3-1/2; 2wB/7)

12) (A12; B34; C4) (1; 1/4-1/3; 2wB/7)

13) (A1; B234; C4) (1; 1/5-1/4; 2wB/7)

14) (A1; B23; C24) (1; 1/5-1/4; 4wB/25)

1715) (A1; B3; C234) (1; 1/5-1/3; wB/10)

16) (A12; B3; C24) (1; 1/4-1/3; 1/25)

17) (A13; B3; C24) (1; 1/3; 1/30-1/25)

18) (A1; B123; C4) (1; 1/5; 4/125-2/35)

19) (A1; B1; C234) (1; 1/5; 1/60-1/50)

20) (A123; B3; C4) (1; 1/3; 1/25-2/21)

21) (A1; B13; C24) (1; 1/5; 1/50-4/125)

22) (A12; B23; C4) (1; 1/4; 1/25-1/14)

Limbo type IDL patterns with two disconnections (PSPs)
23) (A12; B3; C4) (1; 1/4-1/3; 1/25-1/7, wB/10-2wB/7)
24) (A1; B23; C4) (1; 1/5-1/4; 1/60-1/7, 4wB/25-2wB/7)
25) (A1; B3; C24) (1; 1/5-1/3; 1/25-1/7, 4wB/25-wB/10)
From this list, we can confirm that there is one (1+1C2) wage rates constraint in thelimbo type patterns with one disconnection and three (2+1C2) with two disconnections.
We also give concrete examples of the judging method 2. We can derive e.g. thepattern 11) by removing A4 from the pattern 1). In the same way, 11) or 12) is derived
by removing B3 or A3 from 4), 23) by removing B4 from 12), and so on. Furthermore,by paying attention to the wage rate of 11) which is derived from 1) and 4), we can seethat it lies between those of 1) and 4). Such a relation of derivation and wage rates
suggests that e.g. 1) and 4) adjoin each other and 11) forms the boundary between 1)
and 4) on the world production frontier (hereafter, WPF). According to Shiozawa(2007), the WPF of multi-country multi-commodity has a shape of convex polytope
which is covered by facets, the number of which is 10 in the case of three-country
four-commodity, and each facet represents each IDL pattern of the linkage type, andjoints of the facets represent the IDL patterns of the limbo type. In two-dimensionalgraphs of two-country (or multi-country) two-commodity, lines correspond to thelinkage type patterns and vertexes the limbo type patterns, and in three-dimensionalgraphs of three-country four-commodity, surfaces correspond to the linkage typepatterns and ridgelines the limbo type patterns20.
On closer examination of these patterns, we can see that there are many cases thatsome active points do not follow the grades of comparative advantage between two
countries. For example, although country B’s commodity 4 is the lowest comparative
20

See e.g. McKenzie (1954b, p. 151) about the two-dimensional graphs and Figure 2
in 3. 2. 5 about the three-dimensional graphs.
18

advantage between countries B and C, country B produces only commodity 4 together
with country C in the patterns 1) and 11). In 3) and 19), country B, together withcountry A, produces only commodity 1 in which country B has the lowest comparativeadvantage between countries A and B. Similarly, in 5) , 8), 15), 16), 17), 20), 23), and25), country B produces only commodity 3 in which country B does not have thehighest comparative advantage between countries A and B, and also between countries
B and C. Simple relationships between two countries disappear behind complicatedrelationships among three countries. This indicates that research into multi-country
multi-commodity models have a intrinsic significance of its own in the same way that
simplicity

andclearnessof

two-country

two-commodity

andtwo-country

multi-commodity models have a important meaning.
3.2.2 Wage rates and the IDL patterns
As country A’s wage rate is always 1, we pick up only wage rates of countries B and C
and compile them into Table 2 by limiting to the linkage type. Wage rates (WR in thetable) are arranged in the decreasing order, and also the number of active points (NAP)
and the IDL patterns (Patterns) are shown.
Tab. 2 Wage rates, the number of active points, and the IDL patterns
Country
B
Country
C

WR

1/2

1/3

1/3

1/3

1/4

1/4

1/5

1/5

1/5

1/5

NAP

1

1

1

2

2

3

1

2

3

4

Patterns

1)

5)

8)

4)

10)

7)

3)

9)

6)

2)

WR

1/7

2/21

1/14

2/35

1/25

1/25

1/30

4/125

1/50

1/60

NAP

1

1

1

1

2

2

3

2

3

4

Patterns

1)

4)

7)

2)

5)

10)

8)

6)

9)

3)

There are very large wage rates differentials according to the IDL patterns. Generally,
the fewer the number of active points is, the more advantageous the wage rate is.
However, it is not absolutely so. The NAP of country B varies from 1 to 4 in four
patterns with the same WR of 1/5, which is the most disadvantageous for country B.
Also in the case of country C, there is one exception. Such a state never occurs in
two-country multi-commodity. This is a peculiar phenomenon to multi-country
multi-commodity.
19

The range of the wage rates differentials reaches out from the minimum tomaximum of the productivity differentials of individual sectors between all the
combinations of two out of the three countries, namely from 1/2 to 1/5 between
countries A and B, from 1/7 to 1/60 between countries A and C, and from 2/7 to 5/60
between countries B and C, which can be easily calculated from Table 2. This is amatter of course, considering that wage rates differentials are productivity differentials
of link commodities. This is merely another expression of the fact that all thecommodities have a possibility to be link commodities at a point of time when only
production techniques are given, or before volumes of usable labor and demand
structures are given.
3.2.3 Determination of Wage rates, commodity prices, and production volumes
Here, by giving volumes of labor and expenditure coefficients, we determine the wage
rates/ prices and production volumes. Assume that country A has 1000 labor, country B
1000, and country C 3000 and all the four expenditure coefficients in each country are
0.25. When we calculate production volumes of active points about all the patterns
from 1) to 10), only 4) has all positive solutions, and others have one or two negative
solutions. Under these conditions, the IDL pattern 4) is formed and production
volumes and consumption volumes are as Table 3 (numerical values are rounded:hereafter the same shall apply), and differentials between both volumes areexport-import volumes.
Tab. 3 Production volumes and consumption volumes (1)Pattern 4)
Country A
Country B
Country C

Production volumes

Consumption volumes

Comm.1 Comm.2 Comm.3 Comm.4 Comm.1 Comm.2
404.8

404.8

190.5
214.3

Comm.3 Comm.4

250.0

250.0

250.0

375.0

178.6

83.3

83.3

83.3

125.0

428.6

71.4

71.4

71.4

107.1

We change expenditure coefficients: for all countries, those of commodities 1 and 2
from 0.25 to 0.3 and those of commodities 3 and 4 from 0.25 to 0.2. Also at this time,
the pattern having all positive solutions is only 4). Therefore, the wage rates/prices do
not change at all. Production volumes and consumption volumes are as Table 4.
20

Tab. 4 Production volumes and consumption volumes (2)Pattern 4)
Country A

Production volumes

Consumption volumes

Comm.1 Comm.2 Comm.3 Comm.4 Comm.1 Comm.2
485.7

485.7

Country B

28.6
295.2

Country C

Comm.3 Comm.4

300.0

300.0

200.0

300.0

57.1

100.0

100.0

66.7

100.0

428.6

85.7

85.7

57.1

85.7

Changes in demand are adjusted by changes in production volumes andexport-import volumes without price changes. What Graham indicated occurs.
3.2.4 Distribution of labor among countries and the IDL patterns
In this subsubsection, we investigate the effects of distribution of labor among
countries on the IDL patterns. We Start from Table 4 and reduce country A’s labor one
by one from initial value 1000, remaining the others unchanged. Thereby, country A’s
production volumes diminish with diminishing labor. When the labor reaches 928,
production volumes of commodity 3 become negative, and there are no linkage type
IDL patterns having all positive solutions. As we can easily predict, only the limbo
type IDL pattern 12) satisfies the above-mentioned two conditions (see 3.1.3).
Let us continue to reduce country A’s labor. The pattern 12) is held for a certain time,and, during the time, the wage rates of countries B and C continue to decrease together
with prices of commodities 3 and 4. When country A’s labor reaches 696, country B’s
production costs of commodity 2 come to 0.9994 and fall below the commodity price(=1). Country B’s industry of commodity 2 starts production activity and the IDL
pattern 7) is formed. For reference, we show an example of the pattern 12) in Table 5.
Tab. 5 Production volumes and consumption volumes (3)Pattern 4)
Country A
Country B
Country C

Production volumes

Consumption volumes

Comm.1 Comm.2 Comm.3 Comm.4 Comm.1 Comm.2
400.0

400.0
309.5

Comm.3 Comm.4

240.0

240.0

185.7

278.6

35.7

86.2

86.2

66.7

100.0

428.6

73.8

73.8

57.1

85.7

This is the case that country A’s labor is 800. The wage rates are (1; 0.29; 0.08), and21

commodity prices (1; 1; 0.86; 0.57). In harmony with that the pattern 12) is derived
from 4) and 7), the wage rates/prices of 12) are also between those of 4) and 7).
Thus, changes in distribution of labor among counties change the IDL patterns. All
the transitions of the patterns which are induced by changes in country A’s labor are as
follows.
Country A’s labor

the IDL patterns

wage rates

Over 3714

1): (A1234; B4; C4)(1; 1/2; 1/7)

3714–2477

11): (A123; B4; C4)(1; 1/3-1/2; 2wB/7)

2476–9294): (A123; B34; C4)(1; 1/3; 2/21)

928–69712): (A12; B34; C4)(1; 1/4-1/3; 2wB/7)

696–6087): (A12; B234; C4)(1; 1/4; 1/14)

607–23022): (A12; B23; C4)(1; 1/4; 1/25-1/14)

229–15910): (A12; B23; C24)

(1; 1/4; 1/25)

158–12714): (A1; B23; C24)

(1; 1/5-1/4; 4wB/25)

Under 127

6): (A1; B123; C24)

(1; 1/5; 4/125)

From the observation above, we can indicate four points. First, in the situation thatlabor input coefficients and expenditure coefficients are fixed, changes in distributionof labor among countries change the IDL patterns and therefore the wage rates/prices.
The cause is changes in shapes of the WPF. While changes in composition of demand
volumes are very small because of the fixed expenditure coefficients21, shapes of the
WPF change largely. As a result, demand points lie on the different facets (or the IDL
patterns) of the WPF (see Appendix 3).
Second, the more country A’s relative labor volumes to both countries B and C are,the higher the both countries’ wage rates are. Although, there are some exceptions as
stated above (see Table 2), ceteris paribus, the smaller labor volumes are, the morefavorable wage rates are in general. Third, in the transition process of the IDL patterns,
the linkage type and the limbo type appear alternately. Fourth, this transition processprogresses along the patterns resembling mutually without a jump. This means that21

By expressing commodity j’s demand volumes in country i as dij, we obtain dij = bij
wijLij/pj. Because wij and pj change with changes in IDL patterns, dij changes a little.
22changes in the wage rates/prices caused by changes in the IDL patterns are not radical
but gradual unless changes in the given conditions are radical.
3.2.5 Changes in demand and the IDL patterns
Now, we observe the movement of equilibrium solutions which is induced by changes
in expenditure coefficients under the condition that the WPF is fixed. Labor input
coefficients are the same as Table 1, country A has 1000 labor, country B 1000, andcountry C 3000. Each country’s expenditure coefficients are equal for eachcommodity: those of commodities 2 and 3 are fixed at 0.25, and those of commodities
1 and 4 change 0.01 by 0.01from 0.01 to 0.49. Then, the transition of the IDL patterns
is as follows. Only expenditure coefficients of commodity 1 are shown and those ofcommodity 4 (0.5 – expenditure coefficients of commodity 1) are omitted.
Expenditure coefficientsthe IDL patterns

wage rates

0.011): (A1234; B4; C4)(1; 1/2; 1/7)

0.02–0.1111): (A123; B4; C4)(1; 1/3-1/2; 2wB/7)

0.12–0.32

4): (A123; B34; C4)(1; 1/3; 2/21)

0.33–0.41

20): (A123; B3; C4)(1; 1/3; 1/25-2/21)

0.42–0.49

5): (A123; B3; C24)

(1; 1/3; 1/25)

Changes in expenditure coefficients also change the IDL patterns. In the transition of
the patterns, the linkage type and the limbo type appear alternately, and the transition
progresses along the patterns resembling mutually. Let us pay attention to the
movement of country C’s wage rates. The wage rates continue to decreaseintermittently with the decline in the expenditure coefficients of commodity 4, in
which country C has the highest comparative advantage. In general, the larger the
expenditure coefficients of commodities having higher comparative advantage become,the more favorable the wage rates become.
Especially in the limbo type, there is another interesting point about the relation
between demand volumes and the wage rates/prices. We explain this point by using a
two-country three-commodity example. Labor input coefficients and volumes of labor
are as Table 6, and each country’s expenditure coefficients are equal for each23commodity: those of commodity 3 is fixed at 0.2, and those of commodity 1 increases
0.01 by 0.01from 0.01 to 0.79, and those of commodity 2 decreases 0.01 by 0.01from
0.79 to 0.01.
Tab. 6 Labor input coefficients and volumes of labor
Labor input coefficients
Comm.1 Comm.2 Comm.3

Volumes of laborCountry A

20

40

30

400

Country B

50

20

20

600

Fig. 2 World production frontier and the locus of production point
The WPF and the locus of production points are drawn in Figure 222. The threedomains 1, 2, and 3 are the linkage type IDL patterns, and the two ridgelines are thelimbo type IDL patterns, contents of which are shown in each parenthesis. Numericalvalues are the possible production volumes of each commodity at each point. The
dotted line is the locus of production points, and starts from the point of right end, via
22This example and the figure are based on Shiozawa (2014).
24

domain 3, ridgeline, domain 2, ridgeline, and domain 1, and reaches the point of leftend. It should be noted that the movement of the expenditure coefficients is converted
into the movement of the physical quantities. The reason why, in spite of fixed
expenditure coefficients, commodity 3’s volumes change on each domain and ridgeline,is that the wage rates/prices change according to the IDL patterns23.
At first, we confirm the relative wage rates. When the wage rate of country A is 1,
that of country B is 2 on the domain3, 2→1.5 on the first ridgeline, 1.5 on the domain 2,
1.5→0.4 on the next ridgeline, and 0.4 on the domain3. While, on each domain,
production points move in proportion to expenditure coefficients without change in thewage rates/prices, on each ridgeline, those move with accompanied by change in thewage rates/prices. The transitions of production points from the ridgelines to thedomains occur at the time when country B’s non-active points (of commodity 3 firstly,and next, of commodity 1) become competitive in the deterioration process of country
B’s wage rate.
We explain in more detail about a state occurring on the ridgelines. On the firstridgeline, quantity adjustments without price changes are conducted between
commodities 1 and 3. On the other hand, production volumes of commodity 2 of which
expenditure coefficients decrease do not change, and its price changes. On the second
ridgeline, a similar thing occurs. Thus, an important characteristic of Graham-type
trade model that prices and quantities move independently is valid not only in thelinkage type but also in the limbo type.
3.2.6 Changes in production techniques and the wage rates/prices
Here, we describe the effects of changes in production techniques on the wage
rates/prices. Suppose that the IDL pattern 10) (A12; B23; C24) is formed. Commodity
2 is the common link commodity among three countries, wage rates are (1; 1/4; 1/25),
and commodity prices (1; 1; 3/4; 7/25).
What happens when the labor productivity of commodity 4 in country C doubles(the labor input coefficient halves)?

Nothing happens except that the price ofcommodity 4 halves. Country C’s real wage rate certainly increases due to the decline
23

See fn. 21.
25

in price of commodity 4, but this is common with the other countries in which laborproductivity does not increase at all. This is not so good results for country C. The
fruits of increases in labor productivity of commodities produced in only home country
leak into foreign countries24.
On the contrary, increases in labor productivity of the link commodities raise thehome countries’ wage rates. Increases in wage rates, in turn, raise production costs ofcommodities of which increases in labor productivity are lower than those of the linkcommodities. Suppose that the labor productivity of commodity 2 in country C
increases and consequently country C’s wage rate increases. Then, there can be twoconsequences. One is the case that the commodity 4 made in country C holds
competitiveness in spite of price increases and commodity terms of trade of bothcountries A and B deteriorate. The other is the case that the commodity 4 made incountry C loses competitiveness and the IDL pattern changes. Unlike changes in
distributions of labor among countries and demand structures, changes in production
techniques always change commodity prices. If changes in production technique occur
in plural sectors of plural countries, and thereby the structures of comparativeadvantage change widely, the IDL patterns and the wage rates/prices also change
widely.
3.2.7 Probability of the IDL patterns: Graham case versus Mill caseIn the case of three-country four-commodity, there are 10 linkage type IDL patterns, 12
limbo type IDL patterns with one disconnection, and 3 limbo type IDL patterns withtwo disconnections (PSPs). Which patterns do have high frequency of appearance? As
described before, it is clear that which patterns are formed is determined by theinteraction among production techniques, distribution of labor, and demand structures.
However, it is very difficult to calculate a probability of the appearance in athree-country four-commodity case. Therefore, we try it in a simple two-country
three-commodity case (countries A and B, commodities 1, 2, and 3).
Labor input coefficients of country A are standardized as one and those of country B
are expressed a1, a2, and a3 (a1 > a2 > a3 > 0) in order from commodity 1 to 3. Volumes
24

Such a thing was indicated in Pasinetti (1981) and Lewis (1969).
26

of labor in country A are one, and those of country B LB (> 0). Expenditure coefficientsof both countries are each 1/3 for all the commodities and the numéraire is commodity
1. Then, country A’s wage rate is always one, and there exist five reasonable IDL
patterns: (A123; B3), (A12; B3), (A12; B23), (A1; B23), and (A1; B123). Of course,
expenditure coefficients also have an effect on the probability of the IDL patterns. If an
expenditure coefficient of a specific commodity is larger, the probability of the patternwhich both countries produce the specific commodity is larger (see fn. 25). Here, bysetting the coefficients uniform, we remove expenditure coefficients bias.
At first, let us examine the conditions that the pattern (A123; B3) is formed. In thispattern, commodity prices are (1; 1; 1) and the wage rates are (1; 1/a3). By expressing
production volumes of active points as xij (i=A, B; j=1, 2, 3), conditions of full
employment and supply-demand balance are as follows.
xA1+xA2+xA3=1
a3*xB3=LB
xA1=1/3+(1/3)*LB/a3
xA2=1/3+(1/3)*LB/a3
xA3+ xB3=1/3+(1/3)*LB/a3
When all the solutions are positive, this pattern is formed. It is obvious that xA1, xA2,
and xB3 are positive, but xA3 is not so. By solving the above equations for xA3, we
obtain xA3=1/3-(2/3)*LB/a3, accordingly, xA3 > 0 ⇔ LB < a3/2. Therefore, if LB < a3/2,
the pattern (A123; B3) is formed.
Similarly, we examine the pattern (A12; B23). In this case, commodity prices are (1;1; a3/a2), wage rates are (1; 1/a2), and the conditions are as follows.
xA1+xA2=1
a2*xB2+a3*xB3=LB
xA1=1/3+(1/3)*LB/a2
xA2+ xB2=1/3+(1/3)*LB/a2
a3/a2* xB3=1/3+(1/3)*LB/a2

27

It is obvious that xA1 and xB3 are positive, but xA2 and xB2 are not so. By solving theequations, we obtain xA2=2/3-(1/3)*LB/a2 and xB2=(2/3)*LB/a2-1/3. From the both
expressions, we obtain xA2 and xB2 > 0 ⇔ a2/2 < LB < 2a2. Though we omit details, theconditions forming the pattern (A1; B123) is 2a1 < LB.
Next is the limbo type pattern (A12; B3). The wage rate of country B (wB) is added
to unknowns and the equations are as follows.
xA1+xA2=1
a3*xB3=LB
xA1=1/3+(1/3)*wB*LB
xA2=1/3+(1/3)*wB*LB
a3*wB*xB3=1/3+(1/3)*wB*LB
In the limbo type, in addition to the condition that all the solutions are positive, wB
has to be within a specific range, namely, 1/a2 < wB < 1/a3. On the other hand, bysolving the equations for wB, we obtain wB =1/(2LB). From the two expressions, we canunderstand that the conditions forming the pattern (A12; B3) is a3/2 < LB < a2/2.
Likewise, we can derive 2a2 < LB < 2a1 for the pattern (A1; B23).
We can compile the above results into Figure 3.
(A123; B3)(A12; B3)

(A12; B23)

(A1; B23)(A1; B123)

LB
0

a3/2

a2/2

2a2

2a1

∞

Fig. 3 The IDL patterns and differentials of labor productivity
This figure shows that we can calculate the probability of the IDL patterns bydefining the range of LB and values of a1, a2, and a3 appropriately. If we assume thatthe range of LB is from 0 to 10, a1 is 4, a2 3, and a3 2, the probability of the limbo typeis 25%. If the range of LB is from 1 to 8, within which both countries do not fail to gainfrom trading, the probability is roughly 36 %. Note that LB is the relative labor volumes28

of country B and a j (j=1, 2, 3) are the labor productivity differentials (LPD) 25. Figure
3 also shows that the greater the LPD’s differences between sectors are, the higher the
probability of the limbo type is. With the increases in the number of countries andcommodities, the probability of the limbo type becomes lower, because the LPD’sdifferences between sectors are smaller.
The above is the two-country three-commodity case. By using our three-country
four-commodity example, we will try a simulation. Labor input coefficients are thesame as Table 1 and all the expenditure coefficients are 1/4. We generate 400 sets ofrandom numbers which are composed of 3 positive integers fallen in the range of 100 –
30000 (within which all the 25 IDL patterns can be formed), regard the 3 integers as
distributions of labor among countries, and identify the patterns formed under the
distributions. The result is as follows: the linkage type is 317 sets (79%), the limbo
type with one disconnection 82sets (21%), and the limbo type with two disconnections(PSPs) 1set.
When attention is focused on individual IDL patterns, the patterns which country A
produces three or more commodities are many (315 sets) regardless of the type, andcontrary, the patterns which country C produces three or more commodities are null.
This stems from the assumption of labor input coefficients. Owing to the country C’s
production techniques assumed to be very low level in comparison with both countries
A and B, country C must have very large labor volumes to produce three or morecommodities. Consequently, the patterns which country C produces only one
commodity are exclusive, namely, account for 394 out of 400 sets. To itemize thepatterns, the linkage type is 312 sets and the limbo type 82 sets. The patterns whichcountry C produces only one commodity are not so many especially in the linkage type25

If we define the labor volumes and labor input coefficients of both countries as Li
and aij (i=A, B; j=1, 2, 3), the result of calculation is replaced as follows: LB → LB/LA,
a1 → aB1/aA1, a2 → aB2/aA2, a3 → aB3/aA3. Also, we show the result in the case ofgeneralized expenditure coefficients. Suppose as follows: while labor volumes andlabor input coefficients are the same as this subsubsection, expenditure coefficients
expressed by bj (j=1, 2, 3) are different for each commodity, but are equal between
both countries. Then, four boundary points of Figure 3 are replaced in order from left
side as follows: a3*b3/(1–b3), a2*b3/(1–b3), a2*(1–b1)/b1, and a1*(1–b1)/b1. Therefore, if
e.g. b2 is larger (b1 and b3 are smaller), the probability of the pattern (A12; B23)
increases.
29

(4/10). Rather, such patterns are more in the number and also larger in the ratio in thelimbo type (8/15).
Based on the above examination, we can conclude that probability of the linkage
type is considerably higher than that of the limbo type, however, that of the limbo type
is not so low that we can ignore it, and that of the PSPs are extremely low. We call the
aspect of the quantity adjustment of supply-demand balance the Graham case and the
aspect of the price adjustment the Mill case. Then, the linkage type is exclusively the
Graham case, the limbo type other than PSPs is coexistence of the Graham case andthe Mill case, and PSPs are exclusively the Mill case. Therefore, in the adjustment of
supply-demand balance in the world trade, given the condition of constant production
costs, the Graham case overwhelms Mill case.
3.2.8 Summary of the results obtained through observation
We have observed the relation of equilibrium values and the given conditions. The
results are summarized as follows.
1. The IDL patterns, which determined only by the condition of production
techniques, are many and the relative wage rates differ according to the patterns.
The width of the wage rates differentials extends from the minimum to the
maximum of the productivity differentials of individual sectors.
2. In multi-country multi-commodity trade models, because simple relationships
between two countries withdraw behind complicated relationships among many
countries, there are many cases that some active points in the reasonable IDL
patterns do not follow the grades of comparative advantage between two countries
selected arbitrarily.
3. By giving labor volumes and demand structures in addition to productive
techniques, a specific IDL pattern is determined. If it is the linkage type, the wage
rates/prices are uniquely determined by the pattern itself.
4. In the linkage type, the wage rates/prices are highly stable in the face of changes
in demand, and the quantity adjustment without changes in wage rates/prices (theGraham case) is conducted.

30

5. According to the three givens, disconnections of the linkage occur and thelimbo-type IDL patterns are formed. The frequency is certainly low but not so lowthat we can ignore it.
6. Also in the limbo case, the linkages and, therefore, the Graham case exist. The
price adjustment responding to changes in demand (the Mill case) is limited inplaces where there are the disconnections.
7. Generally, with some exceptions, the smaller labor volumes are and the larger
expenditure coefficient of commodities having higher comparative advantage
become, the more favorable the wage rates are.
8. Productivity increase in only commodities with comparative advantage does notraise wage rates. To raise them, productivity increase in link commodities isneeded.

4 Concluding remarks
A keyword of the Graham-type trade model is link commodities. The possibility thatthe link commodities exist was indicated by some economist other than Graham andearlier than Graham (see Appendix 2). None of them, however, thought that the linkcommodities are important for trade theory. There were several reasons. First, they
could not cast aside a stereotype that the price adjustments or the Mill case matterbecause Mill’s prestige was great. Second, since there were no detailed world trade
statistics, they did not think that the link commodities existed in real trading world. For
example, Edgeworth (1894) wrote that the standard commodity (the link commodity in
our terms) was a hypothetical commodity, and though existent in fact, would probably
be insignificant in magnitude (p. 634). Although, with a development of world trade
statistics after the World War II, the existence of commodities produced in common in
more than one country has been recognized, they have not been treated as the linkcommodities, but have been discussed within a framework of intra-industry trade ordifferentiated products (Grubel, 1967; Krugman, 1980).
Third, it is not sufficiently understood that the number of the link commodities isonly a small part of the number of trade items. As previously stated, the number of the31link commodities is, in theoretical trade models, at most “the number of countries
minus one”. Present world trade statistics classify trade items on the base of the HS(Harmonized Commodity Description and Cording System) code. The number of thecommodity groups is approximately 1200, based on 4-digit HS code (headings), androughly 5000 on 6-digit HS code (subheadings). We may be able to group trade items
that “absolute advantages are approximately equal and elasticities of substitution
relatively high” (Chipman, 1965, p. 500) into one category. Similarly, we can groupcountries that income per capita and economic structure are approximately equal into
one category. Considered comprehensively, contrary to Chipman, we can say that there
are overwhelmingly more commodities than countries. Suppose a three-country
thirty-commodity model. Then, the number of the link commodities is at most two or
one fifteenth. Although the link commodities may be not so noticeable in real tradingworld, they carry out their functions adequately and sufficiently.
The link commodities, therefore the quantity adjustments or the Graham casematters. We need to free ourselves from undue emphasis on the PSPs.
In this paper, we provided the full employment version of the Graham-type model.
In the real world, however, underemployment is a normal state. Needless to say, it isdesirable that the assumptions of models reflect reality, and in fact, the Graham-typemodel which places high priority on quantity adjustments over price adjustments isessentially compatible with an underemployment in the following meaning: themovement of productive resources among domestic sectors, which requires a long timein the real, is indispensable for quantity adjustments under the condition of full
employment, while changes in the operating rate and employment rate are sufficientfor those under the condition of underemployment. We will provide theunderemployment version of the Graham-type trade model in our next paper.

32

Appendix 1: On Jones’ perfect specialization numerical example
Jones (1961) has provided a Ricardian three-country three commodity numericalexample and showed which pattern is efficient among PSPs. The answer is the patternwhich minimizes the product of labor input coefficients of active points. This is valid
in a general case of M-country M-commodity. These models are the complete oppositeof the Graham-type model in the sense that there are no link commodities. Table A1-1
is the Jones’ numerical example: country names, commodity names, and arrangements
are changed.
Table A1-1 Jones’ numerical example
Labor input coefficients
Comm.1 Comm.2 Comm.3
Country2

3

10Country

4

5

10Country3

7

10The labor input coefficients of the effective PSP’s active points is printed in boldface.
The product of these is 90, and smaller than the other combinations. This is surely oneof the reasonable IDL patterns. In addition to this, the six linkage type patterns and thesix limbo type patterns with one disconnection are reasonable. As already mentioned,
the total 13 reasonable IDL patterns are determined only by production techniques.
However, it is the distributions of labor and the demand structures to identify a specificpattern out of the 13 patterns. As mentioned later soon, when these two conditions are
given arbitrary, the linkage type has the highest probability, the limbo type with one
disconnection is next to it, and the PSP is uncommon. However, since the probabilityof the PSP is not null, we can provide examples of the PSP. Here, we provide an
example under the conditions that full employment and trade equilibrium are fulfilled.
What is the condition that the PSP is formed? It is that the active points arecompetitive. Concretely,
3wC < 2wA and 3wC < 4wB(commodity 1 in country C)

3wA < 5wB and 3wA < 7wC(commodity 2 in country A)

10wB < 10wA and 10wB < 10wC

(commodity 3 in country B)
33

By simplifying,
3wA/5 < wB < wC < 2wA/3
If we can give the distributions of labor and the demand structures so as to meet thiscondition, the PSP is formed. Although combinations of the both to meet the condition
are countless, it is considerably difficult to identify them practically. There is an idea tosimplify. Assume that all the expenditure coefficients are 1/3. Then, conditions ofsupply-demand balance are expressed as follows.
xC1*p1=wA*LA*(1/3)+wB*LB*(1/3)+wC*LC*(1/3)
xA2*p2=wA*LA*(1/3)+wB*LB*(1/3)+wC*LC*(1/3)
xB3*p3=wA*LA*(1/3)+wB*LB*(1/3)+wC*LC*(1/3)
By replacing production volumes of these expressions with “labor volumes divided bylabor input coefficients” and commodity prices with “labor input coefficients
multiplied by wage rates” and by arranging adequately, the following is obtained.
2LC*wC=wA*LA+wB*LB
2LA*wA= wB*LB+wC*LC
2LB*wB=wA*LA+wC*LC
By subtracting second from first expression and deforming it, we obtain wA/wC=LC/LA.
We can understand that the inverse of the relative wage rates are equal to the relativelabor volumes. Therefore, the above inequalities of wage rates are replaced with thefollow inequalities of labor volumes.
3LA/2 < LC < LB < 5LA/3
When country A’s labor is e.g. 600, those of countries B and C must be in the rangeof 900 – 1000 and “country C’s labor < country B’s labor”. Under the other distributionof labor, the PSP with full employment and trade equilibrium is not realized. If thesetwo conditions are not important, only the fulfillment of the above inequalities of wage
rates is the condition realizing the PSP, which may be accompanied by unemployment
and (or) trade imbalance. We give an example of the PSP satisfying two conditions.
Labor input coefficients are the same as Table A1-1, labor volumes are 600, 990, and
960 in order from country A to C, expenditure coefficients are all 1/3, and thenuméraire is commodity 1. Production volumes, consumption volumes, and wage rates
are as Table A1-2, and commodity prices are 1, 8/5, and 320/99 in order from
commodity 1 to 3.
34

Tab. A1-2 An example of the perfect specialization
Production volumes

Consumption volumes

Comm.1 Comm.2 Comm.3

Comm.1 Comm.2 Comm.3

Country A

200

Country B
Country C

99
320

Wage rates

320/3

200/3

33

8/15

320/3

200/3

33

32/99

320/3

200/3

33

1/3

Thus, we can bring all the IDL patterns into existence by manipulating values oflabor volumes and expenditure coefficients. However, the frequency of the IDL
patterns is not equal. We will try a simulation. Labor input coefficients are the same as
Table A1-1 and all the expenditure coefficients are 1/3. We generate 200 sets ofrandom numbers which are composed of 3 positive integers fallen in the range of 100 –10000 (within which all the 13 IDL patterns can be formed), regard the 3 integers as
distributions of labor among countries, and identify the patterns formed under the
distributions. The result is as follows: the linkage type is 167 sets, the limbo type with
one disconnection 32sets, and the PSPs only 1 set. The frequency of the PSPs is verylow.
Under other labor input coefficients, the frequency certainly differs, but theprobability of the PSPs decreases with the increase of the number of countries, because
the number of the wage rates’ constraints increases. It would be far more productive toresearch the IDL patterns including the link commodities than the PSPs.

35

Appendix 2: On von Mangoldt’s theory of international values
Hans von Mangoldt (1824-68), a German Economist, presented two-country threecommodity and two-country five-commodity numerical examples, which are almost
identical to the Graham-type model, in his book Grundriß der Volkswirthschaftslehre
published in 1863. These examples are dealt with in Edgeworth (1894), thereby
becoming widely known. In 1975 when a long time had passed after that, Appendix II(pp. 185-224) of the book, in which the numerical examples were included, wastranslated into English and appeared as von Mangoldt (1975) in Journal ofInternational Economics (according to editor, the first English translation). Althoughvon Mangoldt’s numerical examples are a little complicated, F. Y. Edgeworth gave aclear explanation about the two-country three-commodity case. Here, we explain thetwo-country five commodity case with simplifying.
Tow countries I and II produce five commodities A, B, C, D, and E. Both countries
have a given volume of productive resources (although terms of productive force,
productive power, and productive factor are used, we can also regard them as labor
volumes), by which all the five commodities can be produced at constant costs. Realproduction costs of each commodity and pre-trade consumption volumes of bothcountries are given as Table A2-1. Although the both countries’ volumes of productive
resources do not showed clearly, we can calculate them from Table A2-1: 1900 incountry I and 1200 in country II.
Tab. A2-1 Real production costs and pre-trade consumption volumes
Real production costs

Consumption volumes

A

B

C

D

E

A

B

C

D

E

Country I

4

7

6

8

5

100

60

60

40

80

Country II

5

9

3

7

4

50

40

60

30

50

With a start of trade, both countries’ demands increase. Mangoldt make a peculiar
assumption that demands increase only in commodities of which relative prices decline,
and demand volumes are given in such a way that full employment is fulfilled. A result
is as Table A2-2.

36

Tab. A2-2 Production volumes and demand volumes under trade (1)Production volumes
A
Country I

B

171+3/7 118+38/49

Country II

00

Demand volumes under trade

C

D

E

A

B

C

D

E

0

47+6/7

0

100

60

105

40

87.5

60

30

50

165 22+1/7 137.5 71+3/7 58+38/49

Commodity D is the link commodity (terms of “standard commodity” are used) andis exported from country I to country II. Commodity prices are 1/2, 7/8, 3/7, 1(numéraire), 4/7 in order from commodity A to E. Full employment and tradeequilibrium are fulfilled.
Furthermore, Mangoldt presents additional three cases by changing suppositions. In
all cases, country I’s pre-trade consumption volumes of commodity E increase (from80 to 115+3/7, 128, and 144 respectively), therefore, country I’s volumes of productive
resources increase (from 1900 to 2077+1/7, 2140, and 2220 respectively) and there is
no change in other pre-trade situations than this. Then, equilibrium solutions (IDL
patterns, production volumes, and commodity prices) are obtained as follows. Also inthese cases, full employment and trade equilibrium are fulfilled.
Additional case 1: limbo type (IABD; IICE), prices (1/2, 7/8, 3/7, 1, 4/7)
Production volumes
A
Country I

B

C

D

E

A

B

C

0

70

0

100

60

105 40 126.25

165

0171+3/7 118+38/49

Country II

00

Demand volumes under trade176.25 71+3/7 58+38/49

D

60

E

30

50

Additional case 2: limbo type (IABD; IICE), prices (1/2, 7/8, 75/164, 1, 100/164)
Production volumes

Demand volumes under trade

A

B

C

D

E

A

B

C

D

E

Country I

176+9/41

122+206/287

072.012

0

100

60

98.4

40

131.2

Country II

0

0

158.4

0

181.2

76+9/41

62+206/287

60

30.01250

Additional case 3: linkage type (IABDE; IICE), prices (1/2, 7/8, 15/32, 1, 5/8)
Production volumes
A
Country I
Country II

B

178+1/8 124+2/7
0

0Demand volumes under trade

C

D

E

A

B

C

D

E

0

72+13/16

11

100

60

96

40

144

156

0

183 78+1/8 64+2/7 60 32+13/16

37

50

Mangoldt’s theory of international values differs from that of Graham in two points,
except for the number of countries. Firstly, Mangoldt gives demand conditions in termsof physical quantities instead of expenditure coefficients. Secondary, while Graham
ignores the limbo type, Mangoldt treats equally the linkage type and the limbo type.
It is really astonishing that, as early as 1863, these numerical examples were
presented. Mangoldt’s this splendid idea, however, was not developed into an
innovation of trade theory not only by himself but also by Edgeworth and Viner who
evaluated Mangoldt highly. The primary reason would be that they considered theexistence of the link commodities, the standard commodities in their terms, a mere
device for explaining international values or a special case which is rare in the realworld. For Mangoldt, it was no matter whether the link commodities exist or not. He
gave the above numerical examples as exemplification that international values isdetermined by the equation of international demand and supply, as symbolized in thetitle of the English translation paper, On the Equation of International Demand.
J. S. Chipman surveyed trade theory in the mid 1960s, and evaluated Mill highly,
Mangoldt favorably, and Graham severely (Chipman, 1965). Difference of attitude
toward Mill, therefore toward mainstream trade theory26, we think, brought about
difference of evaluation by Chipman. How did Graham himself evaluate Mangoldt?
Although Graham had praised Mangoldt in Graham (1932), he changed the evaluation
later, and eventually said “Mangoldt missed the right road only by a hair but a wrong
turning eventually led him to far astray” (Graham, 1948, p. 69)27.

26

Gomes (1990) has designated A. A. Cournot and von Mangoldt as two forerunnersof neoclassical trade theory, and has referred to Graham as a critic of neoclassical trade
theory.
27
Some economists other than Mangoldt: e.g. Sidgewick (1887, pp. 205-209),
Edgeworth (1894, pp. 619-621, 630-634), Nicholson (1897, pp. 301-310), and Bastable(1903, p. 43), also referred to the possibility of existence of the link commodities
relatively early. However, none of them realized the commodities’ innovative
significance for trade theory. See also Shiozawa (2017b).
38

Appendix 3: Distribution of labor among countries and the IDL patterns:
Explanation by using two-country two-commodity
In the subsubsection 3.2.4, we described the reason that changes in distribution of labor
among countries change the IDL patterns. However, the explanation may be hard tounderstand. We will illustrate it by using two-country two-commodity case. Labor
input coefficients and expenditure coefficients are set as Table A3-1 and fixed. Whilecountry B’s labor are fixed at 60, country A’s labor change from 40 to 20 and further to
10.
Tab. A3-1 Labor input coefficients and expenditure coefficients
Labor input coefficients Expenditure coefficients
Comm.1

Comm.2

Comm.1

Comm.2

Country A

1

1

0.5

0.5

Country B

4

2

0.5

0.5

Then, world production frontiers and demand points are illustrated as Figure A3-1.

Fig. A3-1 World production frontiers and the demand points
World production frontiers are cases that country A’s labor is 40, 20, and 10 in orderfrom outer side, and points on each frontier are demand points. Of six triangles shown39

by using auxiliary lines, the lower three are country B’s production frontiers with thesame size, and the upper three are country A’s production frontiers with different sizes.
The reason why compositions of demand volumes, which are expressed by inclinationsof the straight lines (not drawn) connecting demand points and the origin, change alittle despite of fixed expenditure coefficients is because the wage rates/prices change
according to the IDL patterns (see fn. 21). As shown in Figure A3-1, owing to the largechange of the production frontier’s shape, demand points lie on different facets or avertex. This means that the IDL patterns change from (A12; B2) to (A1; B2), further to(A1; B12) in order from outer side.

40

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