Discussion Paper No.382

A Graham-type Trade Model with Keynesian
Unemployment:
Simultaneous Determination of International Values,
Wage Rates, and Employment Quantities

Hideo Sato
（Tohoku University, Professor Emeritus）
January 2018

TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND

MANAGEMENT TOHOKU UNIVERSITY

27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANA Graham-type Trade Model with Keynesian Unemployment:
Simultaneous Determination of International Values, Wage Rates, and Employment Quantities

Hideo Sato (Tohoku University, Professor Emeritus)

Summary:This paper provides a Graham-type trade model with Keynesian unemployment. In the realworld, the existence of unemployment, the greater part of which is involuntary, is a normal
state. Nevertheless, many of the conventional trade models assume full employment.
Although there is a literature which focuses on the relations between international tradeand unemployment, the unemployment dealt with in the literature is voluntary in the almost
cases. In our model, involuntary unemployment or Keynesian unemployment occurs.
Though labor endowments are given, full employment is not presupposed. Rather,
employment quantities are determined simultaneously with international values and wage
rates.

Keyword:Multi-country multi-commodity; Link commodities; Keynesian unemployment,
Quantity adjustment, F. D. Graham
Acknowledgement:
Japanese drafts of this paper were presented at the Seminar on Institutional Economic
Dynamics (Kyoto University: 30th January, 2016), the Special Seminar on Society andEconomy (Tohoku University: 22th June, 2017), the 14th Workshop on the Theory ofInternational Values (Senshu University: 8th September, 2017), the 76th Annual
Meeting of the Japan Society of International Economics (Nihon University: 21st
October, 2017), and the 7th Annual Meeting of the Keynesian Society Japan (Chuo
University: 3rd December, 2017) and were significantly improved due to many usefulcomments from the participants. We thank all the participants. Also we would like tothank Kazuhiro Kurose (Tohoku University) for his useful suggestions and helpfulcomments.

1

Contents with partial summary:
1 Introduction
2 Framework of the Graham-type trade model and definition of terms
The Graham-type trade model is a multi-country multi-commodity Ricardian trade model
attaching great importance to link commodities. This has two versions: full employment
and underemployment versions. Three given conditions (production techniques, laborendowments, and demand structure) determine equilibrium solutions (patterns of theinternational division of labor [IDL], international values, wage rates, and production
volumes in the full employment version: employment quantities in addition to the above
four in the underemployment version). Although the equilibrium solution is unique in theformer version, there is a probability of multiple equilibria in the latter version.

3 Setting of the model and steps to derive the equilibrium solutions
There is no way to obtain the equilibrium solutions at one stroke, so that we have to follow
a little complicated procedure, which comprises of four steps. In step 1, we search for andidentify reasonable IDL patterns. In step 2, according to the each identified reasonable IDL
pattern, we set up simultaneous equations and solve them mathematically. In step 3, weselect one or more solution sets that fulfill one or two conditions. The selected sets are thecandidates of the equilibrium solutions. In step 4, we inspect whether the selected sets
fulfill the labor quantity constraints (LQCs) and unemployment rates constraints (URCs).
The solution sets that pass through the inspection are the equilibrium solutions.

4 Simultaneous equations in a 3-country 4-commodity case
5 Probability of multiple equilibria: An analytical explanation using a 2-country
3-commodity example
6 Reason for the probability: An explanation using 2-country 2-commodity
numerical examples
Because each country’s employment quantities are variable in the underemployment
version, world production volumes equal to the given world demand may be attained byvarious combinations of each country’s production volumes. However, there is the casethat the combination is only one, and therefore, the equilibrium solution is unique.

7 Effect of changes in demand on employment and wage rates
In all the linkage type and a part of the limbo type, changes in demand bring about quantity
adjustments without price changes. Only the domestic demand increase increases domestic
employment and foreign demand increase never contributes to increase in domestic
employment. In the limbo type, changes in demand may cause changes in relative wage
rates. However, the direction of the change is the exact opposite of conventional trademodels (e.g. Mill’s theory of reciprocal demand) presupposing full employment: therelative wage rates of countries which increase domestic demand for commodities
produced in foreign countries don’t decline, but rise.

2

8 Excessive demand and unusual shortage of demand
According to demand conditions, there might be the case that the candidate of theequilibrium solution is only one and the candidate does not fulfill the LQCs or URCs, andtherefore, there is no equilibrium solution unless the conditions change. In this case, somekind of demand adjustment is inevitable because an excessive demand or an unusual
shortage of demand occurs.

9 Mitigation of trade equilibrium conditions: Trade imbalance caseIn the trade imbalance case, countries which continue to increase a trade surplus cancontinue to increase domestic employment without increases in domestic demand. In thereal world, however, this is possible only in a short time. In a long time, domestic
employment increases are realized only by domestic demand increases.

10 Discription using 3-country 4-commodity numerical examples
10.1 Identification of the IDL patterns
In 3-country 4-commodity numerical examples, there are the 25 reasonable IDL patterns.
Ten of them are the linkage type, 12 are the limbo type with one disconnection, and 3 are
the limbo type with two disconnections, or perfect specialization patterns.

10.2 Derivation of the equilibrium solutions
In one of our cases, 7 of 25 are candidates of the equilibrium solutions and are they are
narrowed down to 1 or 3 equilibrium solutions by the LQCs and URCs. In another case, thecandidate is only one. If there still remain multiple equilibria after screening by the LQCs
and URCs, the model itself is not able to determine which solution is realized finally.
Various reasons outside the reach of the model, e.g. path dependency, accident, and so on,
determine this.

10.3 Numerical simulation of multiple equilibria
Through a numerical simulation using a 3-country 4-commodity example, we can knowthat the analytical results obtained in Section 5 would be also valid in the multi-country
multi commodity case. It should be noted that the transition of combinations of multiple
equilibria and the change of the realized IDL pattern are very different things. Even though
the combinations change, the IDL pattern once realized stabilizes owing to pathdependency as long as the LQCs and URCs are satisfied.

11 Concluding remarks
References3

1 Introduction
In Sato (2017b), we provided a full employment version of the Graham-type trademodel, which is a multi-country multi-commodity Ricardian trade model and attaches
great importance to link commodities. In the present paper, we provide anunderemployment version of the model.
Many of the conventional trade models assume full employment: e.g. HeckscherOhlin-Samuelson model (Samuelson, 1948, 1949), Dornbusch-Fisher-Samuelsonmodel (Dornbusch et al. 1977), Krugman model laying the foundation of the New
Trade Theory (Krugman, 1980), Melitz model forming the basis of the New New
Trade Theory (Melitz, 2003). In the real world, however, the existence of
unemployment, the greater part of which is involuntary, is a normal state. Therefore,
there is also a literature which focuses on the relations between international trade andunemployment. The unemployment dealt with in the literature, in the almost cases,
occurs due to the existence of imperfect labor markets (higher wage rates than market
clearing level, mismatches in employment, divided labor markets, labor market
frictions, and so on 1 ). On the other hand, there are almost no trade models
incorporating unemployment that occurs due to a shortage of effective demand2.
In our model, involuntary unemployment or Keynesian unemployment occurs 3.
Although labor endowments are given in the model, full employment is notpresupposed. Conversely, employment quantities are determined simultaneously withinternational values and wage rates.

1

See Davidson and Matusz (2004, 2010), Dutt et al. (2009), Helpman and Itskhoki(2010), and Belenkiy and Riker (2015). Also see p. 67 of Choi and Harrigan eds.
(2003) and p.115 of Feenstra (2016).
2
Chapter 7 of Dosi et al. (1990) criticized Dornbusch-Fisher-Samuelson model forassuming full employment and extended the model to an underemployment case. Thismodel, however, had only 4 conditional expressions for 7 endogenous variables, andtherefore, did not have appropriate construction as a model.
3
An early attempt at incorporating involuntary unemployment into a trade model was
Haberler (1950). However, as indicated by Tabuchi (2017, pp. 274-5), thisunemployment was voluntary unemployment according to the definition of J. M.
Keynes (Keynes, 1936: see Chapter 2 and 3).
4

The remainder of this paper is constructed as follows. Section 2 shows theframework of the Graham-type trade model and gives a definition of terms. In Section
3, a Graham-type trade model with unemployment is set and the way to derive
equilibrium solutions of the model is presented. Section 4 exemplifies simultaneous
equations in a 3-country 4-commodity case. In Section 5, by using 2-country
3-commodity, the probability of multiple equilibria is explained, and in Section 6, by
using 2-country 2-commodity, the reason for the probability. In Section 7, weinvestigate the effect of changes in demand on employment and wage rates. In Section
8, we describe the possibility of excessive demand and unusual shortage of demand. In
Section 9, by mitigating the condition of trade equilibrium, we examine the case of
trade imbalance. In Section 10, 3-country 4-commodity numerical examples are set
and the equilibrium solutions are derived practically. Moreover, a numerical simulationof multiple equilibria is conducted. Section 11 is concluding remarks.

2 Framework of the Graham-type trade model and definition of terms
In the Graham-type trade model 4 , there are three given conditions: production
techniques expressed by constant labor input coefficients, labor endowments, anddemand structure expressed by expenditure coefficients in the full employment version
and by physical units in the underemployment version. By these given conditions, (1)patterns of the international division of labor (IDL), (2) international values or world
relative prices, (3) wage rates in each country, (4) production volumes, therefore, (5)employment quantities in each country are determined. Of course, in the full
employment version, employment quantities are equal labor endowments. We definecollectively the four of above (1) to (4) in the full employment version and the five ofabove (1) to (5) in the underemployment version as equilibrium solutions. Other terms
are defined as follows.
Given the IDL, some sectors in each country continue their production activities andothers cease them. We call the former active points and the latter non-active points.
4

The name comes from F. D. Graham’s theory of international values (Graham, 1923,
1932, 1948). See Sato (2017a, 2017b). Graham’s original theory presupposes full
employment.
5

The IDL patterns have to be reasonable. Here, “reasonable” means a situation that next
two matters are fulfilled: production costs of active points equal commodity prices, andproduction costs of non-active points are higher than commodity prices.
In the Graham-type trade model, the most important keyword is link commodities
that are commodities produced in common in more than one country5 : e.g. cars
produced in Japan, USA, and Germany, IT products in China, Korea, and Japan, andbeef in Brazil, Australia, and USA. The link commodities determine the relative wage
rates of the countries producing the same link commodities, thereby determining therelative prices of all commodities produced in these countries and the relative
production costs of non-active points in these countries. As a same commodity has an
identical price, the relative labor productivities (inverses of the labor input coefficients)of the link commodities are precisely the relative wage rates, and the relative prices
and the relative production costs are obtained by multiplying the relative wage rates byconstant labor input coefficients6.
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,
moreover, 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 (via B) linked.
The term link means not only “link directly” but also “link indirectly” and we use theterm 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,
5

Graham (1948, p. 254 and p. 332).
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 = wAaA1 = wBaB1. Therefore, wB/wA =aA1/aB1. Further, suppose that country A produces commodity 2 and country B
commodity 3. Then, because p2 = wAaA2 and p3 = wBaB3, the relative price p3/p2 =(wB/wA)(aB3/aA2) = (aA1/aB1)(aB3/aA2). Furthermore, for example, because production
costs of commodity 4 in country A (CA4) are wAaA4 and those in country B (CB4) are
wBaB4, the relative production cost CB4/CA4 = (wB/wA)(aB4/aA4) = (aA1/aB1) (aB4/aA4).
6

6

once the IDL 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 (seefn. 6).
The second IDL 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 7. Therefore, todetermine all the wage rates/prices only by the IDL patterns is not possible.
Theoretically, the disconnection can occur in the range from 1 to M–1. When there are
M–1 disconnections, perfect specialization patterns are formed, which have no linkcommodities. We need to pay attention that, except perfect specialization patterns,
there are link commodities also in the limbo type and that the link commodities
perform the above-mentioned functions.
Figure 1 illustrates these two types in a five-country case. Five countries (expressedby x) are all linked in the linkage type, whereas in the limbo type, the linkage is
disconnected in 1 or 4 places and five countries are divided into 2 or 5 groups, within
which more than one country is linked unless the groups consist of a single country.
Figure 1: An example of the two types of the IDL patterns

The linkage typeThe limbo type with 1 disconnection

x-----x-----x-----x-----x

x-----xx-----x-----x

The limbo type with 4 disconnections (perfect specialization)
x

x

x

x

x

At last of this session, we show the framework of the Graham-type trade model in
Figure 2. The left three items are the given conditions. Production techniques
determine the reasonable IDL patterns by only themselves, or independently of othertwo conditions. The reasonable IDL patterns and the labor endowments in each country
combined to construct the world production frontier (WPF) or the world production
7

Graham called such a state of disconnection “limbo” (Graham, 1948, p. 35).
7

possibility set. Demand structure sets the location of demand points on the WPF in thefull employment version or inside the WPF in the underemployment version. The
production points are determined in accord with the demand points, and thereby
equilibrium solutions are determined. Although the equilibrium solution is unique inthe full employment version, in the underemployment version, there is a probability ofmultiple equilibria.
Figure 2: Framework of the Graham-type trade model

Production techniques

Determination of the reasonable IDL patterns

Labor endowments

Construction of world production frontier (WPF)

Demand structure

Setting of the demand points on or inside the WPF

Full employment version
Underemployment version
Determination of the equilibrium solutions
Unique equilibrium
Probability of multiple equilibria

3 Setting of the model and steps to derive the equilibrium solutions
At first, we set a Graham-type trade model with unemployment as follows.
1. There are M countries and N commodities. Here, M and N are integers morethan 2 and N is larger than M.
2. There are no intermediate goods and no profits. All commodities are for
consumption.

83. Trade equilibrium (national expenditure equals national income) in eachcountry is fulfilled8.
4. There are no transport costs and no trade barriers.
5. There are no international movements of labor and domestic wage rates areequal in all sectors.
6. For each country, production techniques expressed by constant labor input
coefficients are given. Although we don’t absolutely need the information about
the labor input coefficients of some sectors (e.g. the car industries in developing
countries or the crude oil extraction industries in non-oil-producing countries)in which a probability of having a comparative advantage is almost zero, we
give all sector’s data for convenience of explanation and assume that thedegrees of comparative advantage between two countries selected arbitrarily
differ in every sector.
7. Although labor endowments in each country are given, full employment is not a
precondition and unemployment can exist. Too high unemployment rates,
however, destabilize societies, and therefore, are not sustainable. Then, we set
the upper limit of unemployment rates. Each country has to be subject to thelabor quantity constraints (LQCs) and the unemployment rate constraints(URCs).
8. Demand conditions in each country are given in terms of physical units.
Next, we explain how to obtain the equilibrium solutions. There is no way to obtainthe solutions at one stroke, so that we have to follow a little complicated procedure,
which comprises of four steps. In step 1, we search for and identify reasonable IDL
patterns. In step 2, according to the each identified reasonable IDL pattern, we set up
simultaneous equations and solve them mathematically. In step 3, we select one ormore solution sets that fulfill the conditions. In step 4, we inspect whether the selected
solution sets fulfill the LQCs and URCs or not. We describe the details of each step
below.

8This condition is possible to be mitigated (see Section 9).
9Step 1: Searching for and identifying reasonable IDL patterns
First, we have to search for and identify reasonable IDL patterns. Whether an IDL
pattern is reasonable or not is determined only by the labor input coefficients. The way
to identify is common between the full employment version and the underemployment
version, but there is a difference between the linkage type and the limbo type.
Therefore, we explain it separately.
In the linkage type, as the entire wage rates/prices and production costs of the
non-active points are already known according to the IDL patterns, we only have tocompare the commodity prices with the production costs (see fn. 6).
The number of reasonable patterns of the linkage type is (M+N–2)!/{(M–1)!(N–1)!}
in an M-country N-commodity case9. If M and N are large, because of a large numberof IDL patterns to be judged10, it is difficult even to identify the patterns. Including therest of the process, the support of computer program would be needed in order tocalculate actually11.
In the limbo type, 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 (l: an
integer and 0 ≤ l ≤ M-1)12. In this case, countries are divided into l+1 groups and the
IDL has to be reasonable within each group and also among groups. Reasonableness of
the IDL within each group is able to be checked easily because the relative wage rates
are determined by linkage. For the IDL among groups to be reasonable, there is acondition. It is that relative wage rates between countries belonging to different groups
have to be within a specific range. This wage rates constraint has to be met between all
the combinations of two out of the l+1 groups, or there are l+1C2 constraints relating
9

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!}. Since the linkage type has no
disconnection, by substituting zero for l in this expression, we obtain the number oflinkage type IDL patterns.
10
The number is {M^ (N-1)}{N^ (M-1)} according to Shiozawa’s direct suggestion.
11
We uploaded the programs to identify the reasonable IDL patterns in the 3-country
4-commodity case to the web site https://www.researchgate.net/profile/Hideo_Sato2.
See “grahamprogramm0” and “grahamprogramm1”. These are possible to download.
12
The value of l is greater than or equal to one in the limbo-type and zero in thelinkage type.
10

wage rates. If wage rates satisfying all these conditions can exist under an IDL pattern,
the IDL pattern is judged as reasonable. Contrary, if these conditions are contradictoryeach other, the IDL pattern is judged as not reasonable. For convenience of explanation,
we describe this method accompanied by identification of the range of wage rates as
the judging method 1.
We explain this method by using a numerical example in Jones (1961). Table 1 isthe example: country names, commodity names, and arrangements are changed.
Table 1: Jones’ numerical example
Labor input coefficients
Comm.1 Comm.2 Comm.3
Country A
2
10
3Country B
4
5
10Country C
710
3

We examine which IDL pattern is reasonable among the limbo type with two
disconnections or perfect specialization patterns. The number of these patterns is 6. At
first, we examine the pattern printed in boldface. Conditions that this pattern isreasonable are as follows (wi: wage rate of country i).
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)

By deforming and rearranging these, we obtain three (2+1C2 = 3) wage rates constraints:
3wA/5 < wB < wA, 3wA/7 < wC < 2wA/3, and wB < wC < 4wB/3. Although the range that
fulfills all the three constraints is very narrow, such a range certainly exists. Therefore,
this IDL pattern is reasonable.
Next, we examine the pattern that country A produces commodity 1, country B
commodity 2, and country C commodity 3. Conditions making this pattern reasonable
are as follows.
2wA < 4wB and 2wA < 3wC(commodity 1 in country A)

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

10wC < 10wA and 10wC < 10wB(commodity 3 in country C)

11By deforming and rearranging these, we obtain three constraints: wA/2 < wB < 3wA/5,
2wA/3 < wC < wA, and 5wB/7 < wC < wB. The range fulfilling all the three inequalities
does not exist13. Therefore, this pattern is not reasonable. Other four patterns are thesame.
This method is very laborious. There is a far easier method, which uses theidentified linkage type IDL patterns: we call it the judging method 2. If, while holding
the condition that all the commodities are produced and all the countries produce atleast one commodity, we remove one active point of a 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 Section 8).
Because all the patterns from with one disconnection to with M–1 disconnections
are summed, the number of the limbo type IDL patterns are very large:∑(M+N–l–2)!/{(M–l–1)!(N–l–1)!l!} (l = 1, 2, …, M–1).
Step 2: Setting up and solving simultaneous equations
We explain this step in the case that the number of the disconnections is l. When thevalue of l is zero, the IDL patterns are the linkage type. Otherwise, the IDL patterns are
the limbo type. In the second step, for all the reasonable IDL patterns14, we have to set
up simultaneous equations and solve them.
The number of the active points matters. It is known that the number is M+N–1–l(McKenzie, 1954, p. 175). Why M+N–1–l? Let us suppose a perfect specialization
pattern that all the commodities are produced. In this situation, the number of theactive points is N and there is no linkage among countries, or the number of the
disconnections is M –1. To link any two countries under this situation, we need to add
one active point. Then, the number of active points is N+1, that of the linkage is 1, andthat of the disconnections is M–1–1. In this way, active points increase by one every
13

We obtain wC < 3wA/5 from the first and third inequalities. However, this contradicts
2wA/3 < wC in the second inequality.
14
The total number is ∑(M+N–l–2)!/{(M–l–1)!(N–l–1)!l!} (l = 0, 1, 2, …, M–1).
12

time when the linkages increase (or the disconnections decrease) by one. Thus, when
all the countries are linked and there is no disconnection, the active points are M+N–1.
If there are more active points than M+N–1–l, one or more multiple linkages occur
between some countries, and multiple relative wage rates occur between the same twocountries. Therefore, the number of the active points has to be exactly M+N–1–l.
Let us consider the composition of the simultaneous equations. Firstly, equations
expressing commodity prices, which take the form of “pj = wiaij”, exist by the numberof active points (namely M+N–1–l), where pj, wi, and aij (> 0) denote price ofcommodity j, wage rate in country i, and labor input coefficients of commodity j incountry i respectively. Secondly, there are N equations expressing supply-demandbalance for each commodity, which take the form “Σxij = Σdij (i= A, B, …, M)”, where
xij and dij (> 0) denote production volumes and demand volumes of commodity j incountry i. However, because one of N equations is not independent owing to Walras’
Law, independent equations are N–1. Thirdly, there exist N equations expressing tradeequilibrium (national expenditure = national income) in each country, which take theform of “Σpjxij = Σpjdij (j = 1, 2, …, N)”. The total number of the independentequations is 2M+2N–2–l.
Next, let us count the number of unknowns. Commodity prices except numéraire are
N–1, wage rate in each country M, and production volumes of active points M+N–1–l.
The total unknowns are 2M+2N–2–l. Thus, we can solve all the equation setsmathematically.
Step 3: Selecting solution sets that fulfill the conditions
Not all the obtained mathematical solutions are meaningful economically, leading to
the next step. We have to select solution sets fulfilling following conditions from the“∑(M+N–l–2)!/{(M–l–1)!(N–l–1)!l!}(l=0, 1, 2, …, M–1)” solution sets. One of theconditions is that all the production volumes and wage rates are positive. Anothercondition is that, in the limbo type, the obtained wage rates are within the adequate
range in the case of the judging method 1, or that the solution set passes acompetitiveness test in the case of the judging method 2. This test is to check whether
non-active points are competitive or not by comparing the production costs of13

non-active points with the commodity prices. As the entire wage rates/prices are
already obtained, the test is simple. If at least one non-active point is competitive, theset is disqualified. Of course, verifying the range of wage rates and the competitivenesstest are equivalent.
Although the solution set that fulfills the conditions is one only in the full
employment version, in the underemployment version, there can be multiple.
Step 4: Inspecting whether selected solution sets fulfill the LQCs and URCs
The solution sets selected in step 3 are not equilibrium solutions but candidates forthem. We have to calculate employment quantities in each country from the obtained
production volumes and the labor input coefficients to inspect whether the candidates
fulfill the LQCs and URCs. The candidates that pass the inspection are the equilibrium
solutions, which can be multiple or unique.

4 Simultaneous equations in a 3-country 4-commodity case
We show the simultaneous equations in a 3-country 4-commodity case. There are thethree countries of A, B, and C and the four commodities of 1, 2, 3, and 4. In addition topreviously mentioned pj, wi, aij, dij, and xij, We define Li (> 0) and αi (0 ≤ αi <1) as laborendowments and the upper limit of unemployment rate in country i respectively. The
numéraire is commodity 1. Consumption volumes equal demand volumes andexport-import volumes are differences between production volumes and consumption
volumes in each country. We provide each one example of the linkage type and thelimbo type. Also, we provide the LQCs and URCs, which are used in the step 4.
At first, as an example of the linkage type, we take the IDL pattern that country A
produces commodities 1 and 2, country B commodities 2 and 3, and country C
commodities 3 and 4.
Price equations (p1 = 1)
p1 = wAaA1
p2 = wAaA2

p2 = wBaB214

p3 = wBaB3

p3 = wCaC3

p4 = wCaC4
Conditions of supply-demand balance (only three of the four are independent)
xA1 = dA1+dB1+dC1xA2+xB2 = dA2+dB2+dC2xB3 +xC3= dA3+dB3+dC3xC4 = dA4+dB4+dC4Conditions of trade equilibrium
p1xA1+p2xA2 = p1dA1+p2dA2+p3dA3+p4dA4
p2xB2+p3xB3 = p1dB1+p2dB2+p3dB3+p4dB4
p3xC3+p4xC4 = p1dC1+p2dC2+p3dC3+p4dC4Labor quantity constraints (LQCs) and unemployment rate constraints (URCs)(1–αA)LA ≤ aA1xA1+aA2xA2 ≤ LA
(1–αB)LB ≤ aB2xB2+aB3xB3 ≤ LB
(1–αC)LC ≤ aC3xC3+aC4xC4 ≤ LC
There are the total 12 independent equations (6 price equations, 3 independentequations of supply-demand balance, and 3 equations of trade equilibrium) and the 12
unknowns (3 commodity prices except numéraire, 3 wage rates, and 6 production
volumes). Although we have to rewrite these in the case of other patterns, in any case,
we can obtain the mathematical solutions.
As to the limbo type, we suppose that country A produces commodities 1 and 2,
country B commodities 3 and 4, and country C commodity 4 only. Equations are as
follows.
Price equations (p1 = 1)
p1 = wAaA1
p2 = wAaA2
p3 = wBaB3
p4 = wBaB4

p4 = wCaC4

Conditions of supply-demand balance (only three of the four are independent)

15

xA1 = dA1+dB1+dC1
xA2 = dA2+dB2+dC2xB3 = dA3+dB3+dC3xB4+xC4 = dA4+dB4+dC4Conditions of trade equilibrium
p1xA1+p2xA2 = p1dA1+p2dA2+p3dA3+p4dA4
p3xB3+p4xB4 = p1dB1+p2dB2+p3dB3+p4dB4
p4xC4 = p1dC1+p2dC2+p3dC3+p4dC4Labor quantity constraints (LQCs) and unemployment rate constraints (URCs)(1–αA)LA ≤ aA1xA1+aA2xA2 ≤ LA
(1–αB)LB ≤ aB3xB3+aB4xB4 ≤ LB
(1–αC)LC ≤ aC4xC4 ≤ LC
Independent equations decrease by one from 12 to 11. However, the unknowns also
decrease by one to be 11. Therefore, as well as the linkage type, we can solve themmathematically.
Between both types, there is one difference, though. In the linkage type, the wage
rates/prices are obtained only from the price equations regardless of other conditions
and are expressed only by the labor input coefficients like below.
wA = 1/aA1
wB = (aA2/aB2)wA = aA2/(aB2aA1)
wC = (aB3/aC3)wB = (aB3aA2)/(aC3aB2aA1)
p2 = aA2/aA1
p3 = (aB3aA2)/(aB2aA1)
p4 = (aC4aB3aA2)/(aC3aB2aA1)
In the limbo type, some of the wage rates/prices are not obtain unless otherconditions are taken into consideration. Easiness (in the linkage type) and difficulty (inthe limbo type) in judging reasonableness of IDL patterns in the first step stem fromthis difference.

16

5 Probability of multiple equilibria: An analytical explanation using a
2-country 3-commodity example
Whether equilibrium is unique or multiple depends on the given conditions. We
explain this by using a 2-country 3-commodity (countries A and B: commodities 1, 2,
and 3) example. Here, we don’t take the LQCs and URCs into consideration because
the purpose of this session is to show probability of multiple equilibria. Labor input
coefficients (aij) and demand volumes for each commodity (dij) are given by using thesame notation as the previous section. The numbers of commodities are assigned inorder of diminishing country A’s comparative advantage: therefore, aB3/aA3 < aB2/aA2 <aB1/aA1. Then, there are five reasonable IDL patterns, namely, (A123; B3), (A12; B3),
(A12; B23), (A1; B23), and (A1; B123).
For each IDL pattern, let us set up simultaneous equations and solve them. If
obtained solution sets fulfill the above-mentioned conditions (all the production
volumes [xij] of active points being positive in linkage type, and, in addition to that,wage rates being within an adequate range in limbo type), the IDL patterns arereasonable. In the following, we derive conditions making each IDL pattern reasonable
as a relational expression between aij and dij. The numéraire is commodity 1. Threeparentheses after hash marks (#) show the IDL patterns, commodity prices (1; 2; 3),
and wage rates (A; B) in order. Among five equations, the former three are theconditions of supply-demand balance and the next two are the conditions of tradebalance.
At first, we examine about the pattern (A123; B3).
# (A123; B3), (1; aA2/aA1; aA3/aA1), (1/aA1; [aA3/aA1]/aB3)
xA1 = dA1+dB1xA2 = dA2+dB2xA3+xB3 = dA3+dB3xA1+xA2(aA2/aA1)+xA3(aA3/aA1) = dA1+dA2(aA2/aA1)+dA3(aA3/aA1)
xB3(aA3/aA1) = dB1+dB2(aA2/aA1)+dB3(aA3/aA1)
While it is obvious that production volumes xA1, xA2, and xB3 are positive, it is notcertain whether xA3 is positive. Then, let us solve the third and fifth equations for xA3.

17

xA3 = dA3+dB3–xB3= dA3+dB3–{dB1+dB2(aA2/aA1)+dB3(aA3/aA1)}/(aA3/aA1)= dA3–{dB1+dB2(aA2/aA1)}/(aA3/aA1)= dA3–(aA1dB1+dB2aA2)/aA3
Therefore,
xA3 > 0  aA3dA3 > aA1dB1+aA2dB2.
The latter inequality is the condition that the pattern (A123; B3) is reasonable.
Next, we examine the pattern (A12; B3). As this IDL pattern is limbo type, not all
the commodity prices and wage rates are determined. By expressing country B’s wage
rate as wB, commodity prices, wage rates, and equations are shown as follows.
# (A12; B3), (1; aA2/aA1; aB3wB), (1/aA1; wB)
xA1 = dA1+dB1xA2 = dA2+dB2xB3 = dA3+dB3xA1+xA2(aA2/aA1) = dA1+dA2(aA2/aA1)+dA3aB3wB
xB3aB3wB = dB1+dB2(aA2/aA1)+dB3aB3wB
It is self-evident that all the production volumes are positive. However, we have to
identify the range of wB that make the pattern (A12; B3) reasonable. Because
production costs of commodity 2 in country B have to be higher than commodity 2’s
price and production costs of commodity 3 in country A have to be higher thancommodity 3’s price, the following inequalities have to be satisfied.
aB2wB > aA2/aA1 and aA3/aA1 > aB3wB
By simplifying, the following wage rates constraints are obtained.
aA3/(aA1aB3) > wB > aA2/(aA1aB2)
On the other hand, by solving the above first, second, and forth equations for wB,
dA1+dB1+(dA2+dB2)(aA2/aA1) = dA1+dA2(aA2/aA1)+dA3aB3wB
dB1+dB2(aA2/aA1) = dA3aB3wB
wB = {dB1+dB2(aA2/aA1)}/(dA3aB3)
By substituting this expression for wB in the wage rates constraints,aA3/(aA1aB3) > {dB1+dB2(aA2/aA1)}/(dA3aB3) > aA2/(aA1aB2)
aA3/aB3 > (dB1aA1+dB2aA2)/(dA3aB3) > aA2/aB218

aA3dA3 > dB1aA1+dB2aA2 > (aA2/aB2)dA3aB3This is the condition making the pattern (A12・B3) reasonable.
In the same way, the conditions making each pattern reasonable are obtained asfollows.
(A123; B3): aA1dB1+aA2dB2 < aA3dA3(A12; B3): (aA2/aB2)aB3dA3 < aA1dB1+aA2dB2 < aA3dA3(A12; B23): (aA2/aB2)aB3dA3 < aA1dB1+aA2dB2and (aB2/aA2)aA1dB1 < aB2dA2+aB3dA3(A1; B23): (aB2/aA2)aA1dB1 < aB2dA2+aB3dA3 < aB1dB1(A1; B123): aB2dA2+aB3dA3 < aB1dB1

Although the size relations among the former three and among the latter three of six
inequalities are clear, the relations among all the six are not clear from the above
inequalities themselves. Let us set dA3 as a criterion for comparison and rewrite theabove inequalities. Then, the following is obtained.
(A123; B3)：(aA1dB1+aA2dB2)/aA3 < dA3(A12; B3)：(aA1dB1+aA2dB2)/aA3 < dA3 < aA1aB2dB1/(aA2aB3)+(aB2/aB3)dB2(A12; B23)：aA1aB2dB1/(aA2aB3) – (aB2/aB3)dA2 < dA3 < aA1aB2dB1/(aA2aB3)+(aB2/aB3)dB2(A1; B23)：aA1aB2dB1/(aA2aB3) – (aB2/aB3)dA2 < dA3 < (aB1/aB3)dB1 – (aB2/aB3)dA2(A1; B123)：dA3 < (aB1/aB3)dB1 – (aB2/aB3)dA2

Here, if we express (aA1dB1+aA2dB2)/aA3 as (1), aA1aB2dB1/(aA2aB3)+(aB2/aB3)dB2 as(2), aA1aB2dB1/(aA2aB3) – (aB2/aB3)dA2 as (3), and (aB1/aB3)dB1 – (aB2/aB3)dA2 as (4), therelation of 0 < (1) < (2), (3) < (2), and (3) < (4) are certain, but the relations between(1) and (4), (1) and (3), and (2) and (4) are uncertain, and moreover, (3) and (4) are
possible to be negative15. According to labor input coefficients and demand quantities,
there can be five arrangement patterns: (1) < (3) < (4) < (2), (3) < (1) < (2) < (4), (1) <(3) < (2) < (4), (3) < (1) < (4) < (2), and (3) < (4) < (1) < (2). We will illustrate therelations between parameters and the IDL patterns by taking “(3) < (4) < (1) < (2)” as
an example.

15

These can be confirmed by taking “aB3/aA3 < aB2/aA2 < aB1/aA1” into account.
19

Figure 3: Labor input coefficients, demand volumes, and the IDL patterns(A1; B123)(A123; B3)(A12; B23)

(A1; B23)(A12; B3)

dA3
(3)

(4)

(1)

(2)

In Figure 3, on the number line of dA3, four points are arranged according to theassumed order. This figure shows the relationship between values of dA3 and the IDL
patterns. Only (A123; B3) is possible on the right side of (2) , (A123; B3), (A12; B3),and (A12; B23) between (1) and (2), only (A12; B23) between (4) and (1), (A12; B23),(A1; B23), and (A1; B123) between (3) and (4), and only (A1; B123) on the left sideof (3). Although the values on the left side of (1) are possible to be negative, in such acase, some IDL patterns are excluded because the values of dA3 are always positive.
Under other arrangement patterns, other results follow. For example, all the five IDL
patterns are possible between (3) and (4) in the arrangement pattern (1) < (3) < (4) <(2), and between (1) and (2) in (3) < (1) < (2) < (4).
An important thing to be confirmed is that, as is evident from the inequalities andthe Figure 3, it is impossible for the limbo type to be the candidate of equilibrium
solution independently and the limbo type is always accompanied by one or more
linkage type adjoining the limbo type16. However, when the LQCs and URCs are taken
into consideration, it is possible that only one limbo type IDL pattern is the uniqueequilibrium solution. About this, we describe later (in Section 8).

6 Reason for the probability: An explanation using 2-country
2-commodity numerical examples
What is the reason why multiple equilibria are possible? We explain this by using a
2-country 2-commodity numerical example. Also here, we disregard the LQCs and

16

In Section 8, we conduct a simulation using a 3-county 4-commodity numericalexample. These are confirmed also by the simulation.
20

URCs for the same reason as Section 5. Suppose that labor input coefficients anddemand volumes are given as Table 2.
Table 2: Labor input coefficients and demand volumes
Labor input coefficients Demand volumes
Comm.1
Comm.2 Comm.1 Comm.2
Country A
1
2
10
10Country B
2
1
10
10

From the labor input coefficient, three reasonable IDL patterns, or (A12; B2), (A1;
B12), and (A1; B2), are derived. By setting commodity 1 as the numéraire and by
calculating production volumes, employment quantities, commodity prices, and wage
rates, we obtain three sets of solutions, which are shown in Table 3. All these solution
sets fulfill the conditions, and therefore, there exist multiple equilibria.
Table 3: Equilibrium solutions (case 1)
Production volumes Employment quantities Wage rates
Comm.1 Comm.2
Country A
20
530
1Country B
15
15
2
Commodity prices
1
2(A12; B2)

Production volumes Employment quantities Wage rates
Comm.1 Comm.2
Country A
15
15
1Country B
5
20
30
0.5
Commodity prices
1
0.5(A1; B12)

Production volumes Employment quantities Wage rates
Comm.1 Comm.2
Country A
20
20
1Country B
20
20
1
Commodity prices
11(A1; B2)

If we regard the obtained employment quantities as the labor endowments, thepseudo world production frontiers (PWPFs) are drawn as Figure 4: the pattern (A12;21B2) is expressed by the solid line, (A1; B12) by the dotted line, and (A1; B2) by thebroken line, which is shifted down a little for identification.
Figure 4: Pseudo world production frontiers (case 1)
Comm. 2
37.5

(A12; B2)
(A1; B2)30

20(A1; B12)

20

30

37.5

Comm. 1

Vertices on the three lines are the pattern (A1; B2), the left side lines of each vertex
are (A12; B2), and the right side lines are (A1; B12). The black point is the world
demand point. The same demand point lies on the three lines, and moreover, on the
different IDL patterns. Because, each country’s employment quantities are variable in
the underemployment version, a world production volume equal to world demand may
be attained by various combinations of each country’s production volume. Figure 4
expresses this situation.
However, the equilibrium is not always multiple. Suppose that country A’s demand
volumes for both commodities are 14 each and country B’s those are 6 each. Then, the
IDL pattern fulfilling conditions is only one, namely, (A12; B2). In the pattern (A1;
B12), commodity 1’s production volumes in country B are negative (–1) and, in thepattern (A1; B2), commodity 1 fails to pass the competitiveness test: country B’s wage
rate is 3/7 and commodity 1’s production costs in country B are 6/7, which are lower
than the price of commodity 1. Obtained mathematical solutions are summarized inTable 4. Country B’s employment quantities in the pattern (A1; B12) are 18, which is22

obtained by adding commodity 2 sector’s employment quantities (= 20) andcommodity 1 sector’s those (= –2). The PWPFs of Table 4 are drawn as Figure 5.
Table 4: Equilibrium solution and mathematical solutions (case 2)
Production volumes Employment quantities Wage rates
Comm.1 Comm.2
Country A
20
11
42
1
Country B
992
Commodity prices
1
2(A12; B2)

Production volumes Employment quantities Wage rates
Comm.1 Comm.2
Country A
21
21
1
Country B
20
(18)
0.5
–1
Commodity prices
1
0.5(A1; B12)

Production volumes Employment quantities Wage rates
Comm.1 Comm.2
Country A
20
20
1Country B
20
20
3/7
Commodity prices
1
3/7(A1; B2)

Figure 5: Pseudo world production frontiers (case 2)
Comm. 2

30
28.5

20
(21 , 18)
(42 , 9)

20

30

23

46.5

Comm. 1

The pattern (A12; B2) is expressed by the solid line, (A1; B12) by the dotted line,
and the frontier of (A1; B2) is omitted. The values in Figure 5 are coordinates ofvertices. The frontier of the dotted line shows the reason why the pattern (A1; B12) isdisqualified under the given demand volumes: in the pattern (A1; B12), country A
requires employment quantities of 21 to cover demand volumes 14 each; (commodity
1’s price) 1 x 14 + (commodity 2’s price) 0.5 x 14 = (wage rate) 1 x 21, while country
B requires employment quantities of 18 to cover demand volumes 6 each; 1 x 6 + 0.5 x
6 = 0.5 x 18. The PWPF with these employment quantities does not reach to thedemand point. That is to say, in the pattern (A1; B12), there is no combination of eachcountry’s production volumes that is able to meet the given demand while keeping
trade equilibrium.
Thus, even though the world total demand is the same, if the composition bycountries is different, the equilibrium solution is different, which can be multiple(Figure 4) or unique (Figure 5).

7 Effect of changes in demand on employment and wage rates
Changes in demand volumes cause adjustment processes. If the changes are large
enough to bring about changes in the IDL patterns, the equilibrium solutions are
re-determined newly. Although this re-determination is also important, in this section,
we assume that the IDL patterns do not change. Because the adjustment processes
differ between the linkage type and the limbo type, we examine it separately.
In the linkage type, when changes in demand occur, the wage rates/prices neverchange. This is shown by the fact that the wage rates/prices are expressed only by labor
input coefficients (see Section 4). In short, the quantity adjustments without pricechanges are conducted17.
17

We call the aspect of the quantity adjustments without price changes the Grahamcase, and the aspect of the adjustments with price changes the Mill case. To say in thisterms, the linkage type is the IDL patterns that the Graham case is exclusive, a largepart of the limbo type is the patterns that the Graham case and the Mill case coexist,
and a very small part of the limbo type or the perfect specialization patterns, which has
M–1 disconnections of the linkage and therefore have no link commodities, is thepatterns that the Mill case is exclusive.
24

How are the production volumes and the employment quantities adjusted? We
explain it by using the linkage pattern in Section 4. Define Eij as country i’s
employment quantities in commodity j sector. Because production volumes are
employment quantities divided by labor input coefficients and commodity prices arewage rates multiplied by labor input coefficients, we can deform pjxij into wiEij. In
addition, by replace commodity prices with labor input coefficients, we obtain thefollowing expressions from the trade equilibrium expressions in Section 4.
wA(EA1+EA2) = dA1+(aA2/aA1)dA2+(aB3aA2)/(aB2aA1)dA3+(aC4aB3aA2)/(aC3aB2aA1)dA4
wB(EB2+EB3) = dB1+(aA2/aA1)dB2+(aB3aA2)/(aB2aA1)dB3+(aC4aB3aA2)/(aC3aB2aA1)dB4
wC(EC3+EC4) = dC1+(aA2/aA1)dC2+(aB3aA2)/(aB2aA1)dC3+(aC4aB3aA2)/(aC3aB2aA1)dC4
By considering wA = 1/aA1, wB = aA2/(aB2aA1), and wC = (aB3aA2)/(aC3aB2aA1),
EA1+EA2 = aA1dA1+aA2dA2+(aB3aA2)/(aB2)dA3+(aC4aB3aA2)/(aC3aB2)dA4
EB2+EB3 = (aB2aA1/aA2)dB1+ aB2dB2+aB3dB3+(aC4aB3/aC3)dB4
EC3+EC4 = {(aC3aB2aA1)/(aB3aA2)}dC1+(aC3aB2/aB3)dC2+aC3dC3+aC4dC4These employment expressions show that employment quantities in each country are
determined only by domestic demand, and that even increases in domestic demand forcommodities produced in foreign countries increases domestic employment and only
this. Increases in foreign country’s demand never contribute to increases in domestic
employment. We explain this mechanism in the case of the increases of country A’s
demand for commodity 4 (dA4). By partially differentiating the above employment
expression of country A by dA4, we obtain∂ (EA1+EA2)/∂dA4 = (aC4/aC3)(aB3/aB2)aA2
This shows how much employment is increased by increases of one unit of dA4. The(aC4/aC3) is the opportunity cost of commodity 4 measured with commodity 3 incountry C, in other words, it is the reduced production volumes of commodity 3 on theoccasion of increasing one unit of commodity 4 by production switching in country C.
The (aB3/aB2) is the reduced production volumes of commodity 2 on the occasion ofincreasing one unit of commodity 3 by production switching in country B. The aA2 is alabor quantity required to produce one unit of commodity 2 in country A.
The mechanism that increases of one unit of dA4 increase country A’s employmentby (aC4/aC3)(aB3/aB2)aA2 units is as follows. The demand increases increase country C’s

25

production volumes of commodity 4 by one unit. The other hand, country C has todecrease production volumes of commodity 3 by (aC4/aC3) units in order to maintaintrade equilibrium18. Because the world demand for commodity 3 doesn’t change,
country B has to increase production volumes of commodity 3 by (aC4/aC3) units tocover the reduced production volumes. In turn, country B decreases production
volumes of commodity 2 by (aC4/aC3)(aB3/aB2) units. Finally, country A has to cover thevolumes, leading to the increase in employment quantities of country A by
(aC4/aC3)(aB3/aB2)aA2 units. This is the mechanism that only the Graham-type trademodel based on the linkage structure is able to have.
Note that, even though increasing demand volumes are the same, increasing
employment quantities may be different, because the combination of opportunity costs,
which influence employment quantities, is different according to the IDL patterns. We
give a concrete example in Section 10.
In the limbo type, changes in demand cause changes in the wage rates/prices insome cases, but not in other cases. We explain this in the case of the limbo type IDL
patter in Section 4. In this pattern, although country B and country C are linked, theboth countries are disconnected from country A. By solving the above-mentioned
equations for wB, we obtainwB ={dB1+dC1+(aA2/aA1)(dB2+dC2)}/(aB3dA3+aB4dA4)This expression shows that country B’s or country C’s demand increases forcommodities that are not produced in these country (increases of dB1, dC1, dB2, and dC2)
raises these country’s wage rate, and that country A’s demand increase for commodities
produced countries B and C (increase of dA3 and dA4) reduces the wage rates of bothcountries B and C. This moving of the wage rate may seem to be a little complicated.
To make it more understandable, we will apply this moving to the case of a perfectspecialization pattern with 2-country 2-commodity. The above moving of the wage ratemeans that domestic demand increases for the commodity produced in foreign country
raise the domestic relative wage rate19. This is the exact opposite of conventional trade18

Because the right side of the country C’s trade equilibrium expression (p3xC3+p4xC4= p1dC1+p2dC2+p3dC3+p4dC4) does not change, the increase of xC4 has to beaccompanied by the decrease of xC3.
19
The pattern 12) and 22) in Table 8 are concrete examples of this case.
26

models (e.g. Mill’s theory of reciprocal demand) that presuppose full employment20.
The mechanism is as follows. The domestic demand increases increase the import
volumes from foreign country. The other hand, domestic export volumes don’t change.
To hold trade equilibrium, the domestic commodity terms of trade have to worsen, andtherefore, the domestic relative wage rate has to decline.
Conversely, domestic demand changes for commodities produced in home country
or countries linked with home country (changes of dA1, dA2, dB3, dB4, dC3, and dC4) does
not change the wage rates/prices, just like the linkage type. The mechanism is also thesame with the linkage type.
Thus, the relation between demand and unemployment in the present model isclarified. Unemployment in the model is involuntary unemployment or Keynesian
unemployment that occurs due to a shortage of effective demand.

8 Excessive demand and unusual shortage of demand
According to demand conditions, there might be the case that the candidate of theequilibrium solution is only one and the candidate does not fulfill the LQCs or URCs,
and therefore, there is no equilibrium solution unless the conditions change. In thiscase, some kind of demand adjustment is inevitable because an excessive demand or an
unusual shortage of demand occurs.
We take country A’s LQC and URC in Section 4 as an example (reproduced below).
(1–αA)LA ≤ aA1xA1+aA2xA2 ≤ LA
At first, we consider the case not satisfying the URC. Suppose the unemployment rate
β. Then, the following equation expressing supply-demand gap holds.
wALA – wA(EA1+EA2) = βwALA
The first term of the left side is the potential income, the second is the actual income
determined by aggregate demand21, and the right side is the supply-demand gap, or the20

This is also contrary to the full employment version of the Graham-type trade model.
See Sato (2017b, pp. 15-16).
21
We can obtain “wA(EA1+EA2) = p1dA1+p2dA2+p3dA3+p4dA4” from the condition ofcountry A’s trade equilibrium.
27

income lost by unemployment. On the other hand, by multiplying the both side of the
URC by wA and deforming that, we obtain
wALA – wA(EA1+EA2) ≤ αAwALA
The right side is the upper limit of acceptable supply-demand gap. Therefore, if the
URC is not fulfilled, it means that the actual gap exceeds this upper limit (βwALA >αAwALA), that is, there is an unusual shortage of demand.
Next, by multiplying the both side of the LQC by wA and deforming that, we obtain
wA(EA1+EA2) ≤ wALA
If the LQC is not fulfilled, it means that country A’s demand exceeds the potential
income: wA(EA1+EA2) = the aggregate demand > wALA, that is, there is an excessive
demand which is over the potential income.
In the both cases, demand volumes have to be adjusted sooner or later.

9 Mitigation of trade equilibrium conditions: Trade imbalance case
So far, we have assumed trade equilibrium. This was done to close the model in a most
simple way, but we can mitigate the condition to introduce trade imbalance. As an
initial state, we use the linkage type IDL pattern in Section 4. Suppose that, in next
period, country B’s demand for commodities 1 and 4 increases by ΔdB1 and ΔdB4, andthis country covers the demand increment by trade deficit γ (= p1ΔdB1 +p4ΔdB4).
Country A’s demand volumes are unchanged but this country has trade surplus γ. Thereis no change in country C. Then, the equations in Section 4 are rewritten as follows.
Here, we change the naming of “Conditions of trade equilibrium” to “Conditions ofincome-expenditure balance” because there is now the trade imbalance.
Conditions of supply-demand balance (only three of the four are independent)
xA1 = dA1+dB1+dC1 +ΔdB1
xA2+xB2 = dA2+dB2+dC2xB3+xC3 = dA3+dB3+dC328

xC4 = dA4+dB4+dC4 +ΔdB4
Conditions of income-expenditure balance
p1*xA1+p2*xA2 = p1*dA1+p2*dA2+p3*dA3+p4*dA4 + γ= p1*dA1+p2*dA2 +p3*dA3 +p4*dA4 + p1*ΔdB1 + p4*ΔdB4
p2*xB2+p3*xB3 = p1*dB1 +p2*dB2 +p3*dB3 +p4*dB4 +p1*ΔdB1 + p4*ΔdB4 – γ= p1*dB1 +p2*dB2 +p3*dB3 +p4*dB4
p3*xC3+p4*xC4 = p1*dC1 +p2*dC2 +p3*dC3 +p4*dC4From the conditions of supply-demand balance, we can know that the production
volumes of commodities 1 and 4 increase, and from the conditions ofincome-expenditure balance, we can also know that employment of countries B and C
doesn’t change 22 and only country A increases employment. Country A, despite
unchanged domestic demand, increases employment and income by absorbing country
B’s demand increment through the trade surplus. However, this is effective only once.
If next period’s trade surplus remains γ, employment no longer increases. In order tocontinue increasing employment through trade surplus, any country has to continue
increasing trade surplus: so to speak, an economic growth depending on foreign
demand. In the real world, this is possible only in a short time. In a long time, domestic
employment increases are realized only by domestic demand increases.

10 Description using 3-country 4-commodity numerical examples
10.1 Identification of the IDL patterns
In this section, we set a 3-country 4-commodity numerical example and derive
equilibrium solutions practically. Labor input coefficients are given as Table 5. Here,units of the commodities are chosen in such a manner that all the labor input
coefficients of country A are one, and the commodities are numbered in order ofdiminishing country A’s comparative advantage between countries A and B23. Labor

22

See the employment expressions in Section 7 (p. 25).
Although, without this treatment, the number of the IDL patterns to be investigatedreaches 432, the number diminishes to 112 owing to this.
23

29

input coefficients of country C are given arbitrarily. The numéraire is commodity 1, sothat country A’s wage rate is always one.
Table 5: Labor input coefficients
Country A
Country B
Country C

Comm.1 Comm.2 Comm.3 Comm.4
1
1
1
1
5
4
3
2
60
25
30
7

From this table, 10 linkage type IDL patterns, 12 limbo type IDL patterns with one
disconnection, and 3 limbo type IDL patterns with two disconnections (perfectspecialization patterns) are identified as bellow. In the linkage type, 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. In the limbo type, commodity prices areomitted, and wB means country B’s wage rate. In the case of two disconnections,country C needs to fulfill two wage rates constraints.
Linkage type IDL patterns
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)
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)
10) (A12; B23; C24) (1; 1; 3/4; 7/25) (1; 1/4; 1/25)
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)

30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 disconnections23) (A12; B3; C4) (1; 1/4-1/3; 1/25-1/7 and wB/10-2wB/7)
24) (A1; B23; C4) (1; 1/5-1/4; 1/60-1/7 and 4wB/25-2wB/7)
25) (A1; B3; C24) (1; 1/5-1/3; 1/60-1/25 and wB/10-4wB/25)
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 will also give concrete examples of the judging method 2. For example, we canderive the pattern 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. As
it were, the limbo type is the derivative from the linkage type and the latter is the originof the former. Such a relation of derivation suggests that e.g. 1) and 4) adjoin eachother and 11) forms the boundary between 1) and 4) on the world production frontier(hereafter, WPF). According to Shiozawa (2017), the WPF of multi-country
multi-commodity has a shape of convex polytope which is covered by facets, thenumber of which is 10 in the case of 3-country 4-commodity, and each facet represents
each IDL pattern of the linkage type, and joints of the facets represent the IDL patternsof the limbo type. In two-dimensional graphs of 2-country (or multi-country)
2-commodity, line segments correspond to the linkage type patterns and vertexes thelimbo type patterns, and in three-dimensional graphs of 2-country 3-commodity,
surfaces correspond to the linkage type patterns and ridgelines the limbo typepatterns24.
We can also guess such an adjoining relation from wage rates. For example, if we
pay attention to the wage rate of 11) which is derived from 1) and 4), we can see that itlies between those of 1) and 4). The IDL patterns adjoining each other on the WPF

24

See e.g. Shiozawa (2017, p. 6) about the three-dimensional graphs.
31resemble each other only in the IDL patterns but also in the wage rates/prices(moreover, as stated in the next subsubsection, in employment quantities).
In the above list, another thing should be noted. There are very large wage rates
differentials according to the IDL patterns. The range of the wage rates differentials
reaches out from the minimum to maximum of the productivity differentials of
individual sectors, 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 calculated from the above list. From a viewpoint of wage rates differentials, notonly the production technique but also the IDL patterns are important.
10.2 Derivation of the equilibrium solutions
We give demand volumes as Table 6. The LQCs and URCs are introduced later.
Table 6: Demand volumes (case1)
Comm.1Country A
80
Country B
40
Country C
20World total
140

Demand volumes
Comm.2 Comm.3 Comm.4
90
100
110
60
70
110
30
40
50
180
210
270

By calculating in accordance with the steps 2 and 3 mentioned in Section 4, we haveseven candidates of the equilibrium solutions. The IDL patterns, employment
quantities, and wage rates of countries B and C are compiled into Table 7.
Employment quantities are rounded: the same applies hereafter.
Table 7: The IDL patterns, employment quantities, wage rates (case 1)
IDL patterns
A’s employment
B’s employment
C’s employment
B’s wage rates
C’s wage rates

4)
343
730
1295
1/3
2/21

5)
301
602
2600
1/3
1/25

7)
300
830
1470
1/4
1/14

32

10)
12)
16)
276
320
294
733
777
630
2350
1377
2534
1/4
15/52 109/350
1/25
15/182
1/25

22)
285
770
1890
1/4
4/77

The numbers of IDL patterns correspond to the numbers of IDL patterns in theprevious subsubsection: the four of 4), 5), 7), and 10) are the linkage type and the threeof 12), 16), and 22) the limbo type. As described in Section 5, it is impossible that thelimbo type is independently the candidate of equilibrium solution and the limbo type isnecessarily accompanied by one or more linkage type adjoining itself on the WPF: 12)
is accompanied by 4) and 7), 16) by 5) and 10), and 22) by 7) and 10). Let us pay
attention to each value of the limbo type in Table 7. Then, we realize that all the valuesof the limbo type, in employment quantities as well as wage rates, lie between thevalues of two adjoining linkage type patterns.
Although we have to proceed to the step 4, before that (while examples are many),
we examine the effect that changes in demand have on employment and wage rates.
Changes in demand volumes bring about changes in employment quantities. If thechanges are large, some of the candidates may disappear or new candidates may appear.
Here, let us suppose that, in Table 6, country A’s demand for commodity 4, which is
not produced in country A, increases from 110 to 130 and the others are unchanged.
Then, the candidates are unchanged and employment quantities and wage rates havevalues shown in Table 8. For convenience of comparing with Table 7, we put an up
arrow (↑) in cells with increasing values and a down arrow (↓) in cells with decreasing
values.
Table 8: The IDL patterns, employment quantities, wage rates (case 2)
IDL patterns
A’s employment
B’s employment
C’s employment
B’s wage rates
C’s wage rates

4)
↑ 357
730
1295
1/3
2/21

5)
↑ 306
602
2600
1/3
1/25

7)
↑ 310
830
1470
1/4
1/14

10)
12)
16)
22)
320
↑ 281
↑ 300
↑ 288
733
630
↑ 803
↓ 757
2350 ↑ 1423
2534 ↑ 2030
1/4
109/350
1/4↓ 15/56
1/25 ↓15/196 1/25
↓ 1/21In the four linkage type patterns and one limbo type pattern of 16) in whichcountries A and C are linked, the sole change is increases in employment of country
A25 and there is no change in the wage rates/prices. In the every five cases, the country
25

About the reason why, despite the same increases in demand, increases in
employment quantities differ, see Section 7.
33

increasing production of commodity 4 decreases production of another commodity,
and in turn, country A increases the commodity to cover the decreasing production,consequently leading to the increases in country A’s employment. To sum up, only the
Graham case (quantity adjustments without price changes) exists and the Mill case(adjustments with price changes) does not exist.
In the two limbo type patterns of 12) and 22), the wage rates of countries increasing
production of commodity 4 (countries B and C in 12) and country C in 22)), therefore
prices of commodities produced by these countries (commodities 3 and 4 in 12) andcommodity 4 in 22)) decline. These price changes are accompanied by changes inexport-import volumes, therefore production volumes and employment quantities, andconsequently trade equilibrium is kept. Anyway, the effect of demand changes on theemployment and wage rates is as described in Section 7.
Let us introduce the LQCs and URCs. Suppose that country A has 360 labor, country
B 800, and country C 2800 and the upper limit of unemployment rates is 25% in
common. Each country has to satisfy the following LQCs and URCs.
270 ≤ Employment quantities of country A ≤ 360
600 ≤ Employment quantities of country B ≤ 800
2100 ≤ Employment quantities of country C ≤ 2800
Under these conditions, the seven candidates are narrowed down to the elected threepatterns of 5), 10), and 16). However, if we assume the distribution of labor thatcountry A has 360 labor, country B 880, and country C 1600, the elected patternschange to the three of 4), 7), and 12). We can make intently a supposition that theequilibrium solution is unique. For example, under the assumption that country A has
340 labor, country B 810, and country C 1600, only the pattern 12) passes through the
LQCs and URCs. Which patterns survive depends on the assumption.
It should be kept in mind that, even though there are many candidates, equilibrium
solutions are narrowed down to a few of them by the LQCs and URCs. The reason is
as follows. Let us take two patterns arbitrarily from e.g. Tables 7 and compare eachcountry’s employment quantities between both patterns. Then, we can see the fact that,in the pattern which one country’s employment is more, one or two of the othercountries necessarily have less employment than the other pattern. Under the same34

world production (equal to demand) volumes, if one country’s production volumes are
more, other countries’ production volumes are naturally less. The above fact is no morethan a reflection of this natural thing. This means that, if an unemployment rate
declines in one country, it rises in other countries. Therefore, the number of
equilibrium solutions is not so many unless the upper limit of unemployment rates isvery high.
When there still remain multiple equilibria after screening by the LQCs and URCs,
the model itself is not able to determine which solution is realized finally. Various
reasons outside the reach of the model, e.g. path dependency, accident, and so on,
determine this. We have to note the following point, however. The equilibrium
solutions of the limbo type (limbo solutions) are almost always accompanied bychanges in the wage rates/prices in the face of changes in demand. Therefore,
especially when the limbo solutions are near the equilibrium solutions of the linkage
type (linkage solutions), switching from the limbo solutions to the linkage solutions isprobable. Contrary, switching from the linkage solutions to the limbo solutions isalmost improbable, because the linkage solutions don’t change the wage rates/prices in
the face of changes in demand. Considering these, probability that the linkage solutions
are realized finally would be high.
If the world demand differs, and if the composition by countries differs under thesame world demand, the equilibrium solutions may differ, which can be singular, notmultiple. For example, under the demand composition in Table 9, of which total is thesame as Table 6, the candidate is only one of the pattern 5).
Table 9: Demand volumes (case 2)
Comm.1Country A
50
Country B
30
Country C
60
World total
140

Demand volumes
Comm.2 Comm.3 Comm.4
60
100
110
40
50
110
80
60
50
180
210
270

Employment quantities are 241, 452, and 5350 in order from country A to C. In thecase that these employment quantities don’t fulfill the LQCs and URCs, as mentioned35

in Section 8, an excessive demand or an unusual shortage of demand occurs in the
countries not fulfilling the constraints.
10.3 Numerical simulation of multiple equilibria
Here, we conduct a numerical simulation using a 3-county 4-commodity example
about a probability of multiple equilibria. A starting point is as follows: each country’s
demand volumes for each commodity are as Table 5 except country A’s demand forcommodity 4 (dA4), which increases one by one from one. Then, the relations between
values of dA4 and the candidates of equilibrium solutions are as the list below. As to thelimbo type, we show the origins of the limbo type patterns in { } at the first
appearance: the pattern 23) is the limbo type with two disconnections and is derived
from the pattern with one disconnection.
Values of dA4:

the candidates of equilibrium solutions

1 – 50:5), 10), 16) {← 5) and 10)}51 – 75:5), 7), 10), 16), 22) {← 7) and 10)}

76–137:4), 5), 7), 10), 12) {← 4) and 7)}, 16), 22)

138–149:4), 5), 7), 10), 12), 16), 20) {←4) and 5)}, 22),23) {←12), 16), 20) and 22)}

150 – 175:4), 5), 10), 16), 20), 22), 23)

176 – 202:4), 5), 16), 20), 23)

203 – 211:4), 5), 20)

212 – 260:4)

261 – 389:1), 4), 11) {← 1) and 4)}

390 –

1)

:

The above transition of the candidates show concretely the content described in
Section 5. First, the candidates change with changes in dA4. Second, although thecandidates are usually multiple, sometimes singular. Third, there is no case that thelimbo type is a single candidate: the limbo type is always accompanied by one or more
its origins. Moreover, the origins (the linkage type) and the derivatives (the limbo type)36

simultaneously appear or disappear: for example, 7) and 22) appear together when thevalue of dA4 change from 50 to 51; 7) and 12) disappear together when the value of dA4change from 149 to 150. Thus, the analytical results obtained in Section 5 would bealso valid in the multi-country multi commodity case.
It should be noted that the transition of combinations of multiple equilibria and thechange of the realized IDL pattern are very different things. Suppose that, when dA4 is
130, for some reason or other, 4) out of seven candidates is realized and that only dA4changes. While dA4 changing from 130 to 140 and to 160, combinations of thecandidates or multiple equilibria change. The IDL pattern would remain 4), however,as long as the LQCs and URCs of country A are satisfied. Even though the
combinations of the multiple equilibria change, the IDL pattern once realized stabilize
owing to path dependency.

11 Concluding remarks
The present trade model determines not only relative commodity prices and relative
wage rates but also production volumes and employment quantities in each country.
Not being premised on full employment, the model tells us some important
information about the relation between demand and employment.
Changes in demand cause two kinds of adjustment process. Domestic demand
increases for commodities produced in home country or countries linked with homecountry increase domestic employment. The wage rates/prices and foreign employment
don’t change at all. On the other hand, domestic demand increases for commodities
produced in countries not linked with home country raise the domestic relative wage
rates to the countries exporting the commodities. This is contrary to the conventional
trade models presupposing full employment. As for employment, according to thesituation of linkage, various cases can occur. Domestic employment may increase or
may not change, and foreign employment may increase or decrease.
In the trade imbalance case, countries which continue to increase a trade surplus cancontinue to increase domestic employment without increases in domestic demand.
However, this is possible only in a short time. In a long time, any country must37

increase domestic demand to increase domestic employment. In the trade models
presupposing full employment, progress in trade liberalization never causes an
employment problem. In our model, this may cause a rise in unemployment rates if
domestic demand does not increase sufficiently. Domestic demand matters. We should
not rely on foreign demand. This is the most important message from our model.
Thus we have provided a new trade model. Of course, much remains to be done. In
order to closer to reality, we should introduce intermediate goods26 and non-tradable
goods which account for substantial share in the real world. Moreover, If we can set
any appropriate demand function, our model will be expand more and give us more
useful and suggestive information.

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