The production assignment problem assumes a central role in the multi-country, multi-good Ricardian trade model. However, resolution of the problem is not sufficient to illustrate the shape of the world production frontier, even if the examination is limited to efficient production of all goods. This study established the frontier pertinent to the three-country, three-good Ricardian model by connecting Ikema's (1993) illustration to McKenzie's (1954) efficient facets in order to classify the shape of the world production frontier. Our analysis indicates that the illustration of the Ricardian model has more explanatory power than the production assignment problem in identifying the shape of the world production frontier, at least in the three-country, three-good model.World production frontier, the Ricardian model, Classification, Efficient facet, Illustration

Takeshi Ogawa

English

Research Papers in Economics

Economics Bulletin, 2012, Vol. 32 No. 1 pp. 639-647
1.  Introduction 
Since Jones (1961), the production assignment problem—the determination of which good a 
country should specialize in for efficient production—has been central to the analysis of the 
Ricardian model characterized by many countries producing many goods. Jones (1961) 
demonstrated that within efficient specializations, there is usually only a single pattern of 
specialization determined by certain inequalities in the  n-country,  n-good model if every 
good is considered essential. Before Jones (1961), McKenzie (1954) had analyzed the model 
in the general form, and had proposed the efficient facet diagram using a three-country or 
four-country, three-good model. The McKenzie efficient facet diagram  is in the quantity 
space. 
After Jones (1961), Ikema (1993) (in Japanese) used another illustration, which is based 
on that of Amano’s (1966), representing the three-country or four-country, three-good model 
without intermediate goods.
1
Ikema (1993) focuses on information that the production assignment problem cannot 
solve, at least in the three-country, three-good model. The present study focuses on the entire 
world production frontier. In the frontier, there are many efficient production points satisfying 
the following parameters: (1) they can be achieved through diversifications, (2) they may 
occur due to various given prices, and (3) information regarding them cannot be obtained by 
the production assignment problem. To clarify these points, it is necessary to classify the 
entire frontier, including all essential information. 
  Ikema’s illustration concentrates on the price space and can 
analyze the model accurately with the technology parameter of each good for each country, 
whereas McKenzie’s efficient facet diagram is intuitive. Both McKenzie’s efficient facets and 
Ikema’s illustration provide useful information for this study. 
This study classifies the frontier by extracting the latent potential of Ikema’s illustration. 
While Jones focused on only the case where each country specializes in a single good, this 
study uniquely characterizes the entire frontier by connecting Ikema’s  illustration with 
McKenzie’s efficient facets. Although this study considers only the three-country, three-good 
model, its method of classification can be extended to a multi-country, three-good model 
similar to Ikema’s illustration, and further explains the entire production frontier.   
Section 2 describes  the production assignment problem by discussing Ikema’s 
illustration of the three-country, three-good model. Section 3 classifies the frontier in the 
model, and Section 4 concludes. 
 
2.  Model and Basic Illustration 
The analysis is based on the Ricardian model with three countries and three goods. The 
                                                 
1  Amano’s (1966) illustration is based on a two-country, three-good model with intermediate 
goods. 
640Economics Bulletin, 2012, Vol. 32 No. 1 pp. 639-647
production function is linear with one factor, labor. Let 
i
j l   be the amount of labor needed to 
produce one unit of the  j
th good in the  i
th country  ) 3 , 2 , 1 , ( = j i , 
i
j x   be the quantity of the 
j
th good in the i
th country, and 
i L   be the positive labor endowment in the  i
th country. 
Then the resource constraint is 
i i i i i i i L x l x l x l ≤ + + 3 3 2 2 1 1   for any  3 , 2 , 1 = i . If  ( ) 0 , , 3 2 1 >> p p p  
is assumed as the world price
2 ( ) 3 2 1 , , X X X X =   and    as the world production point, then a 
point  ( )
3
3
2
3
1
3
3
2
2
2
1
2
3
1
2
1
1
1 , , x x x x x x x x x X + + + + + + =   is on the frontier if and only if there is a 
vector  ( ) 0 , , 3 2 1 >> p p p   such that X maximizes 
                    ) 3 , 2 , 1 ( s.t. max
, 0
= ≤ ∑ ∑
≥
i L x l x p
i
j
i
j
i
j
j i
i
j j
x
i
j
.                (1) 
To understand Ikema’s (1993) illustration,  necessary for later analysis, this  study 
analyzes  the  relationship  between the world price and specialization. Specialization is 
characterized as follows: the  i
th country specializes in the  j
th good if and only if the  i
th 
country devotes all its labor to the production of the  j
th good. Thus, the set of prices at 
which the  i
th country specializes in the  j
th good can be expressed as 
( )
 



 



≠ > >> ) ( 0 , , 3 2 1 j k
l
p
l
p
p p p
i
k
k
i
j
j . 
Figure 1, which is originally derived from Amano (1966) and plays a key role in Ikema’s 
(1993) illustration, shows the good in which the  i
th country will specialize given the world 
price. The first good is defined as the numeraire. 
          
Figure 1                            Figure 2 
                                                 
2  Inequality signs are  >> > ≥ , , . 
641Economics Bulletin, 2012, Vol. 32 No. 1 pp. 639-647
The shape of Figure 1 crucially  depends on the information  regarding  the  technology 
parameter in the normalized world price  ( ) 3 2, , 1 p p P = . Thus, Figure 1 is in a one-to-one 
correspondence to the normalized world price P, where the value of every good’s marginal 
productivity is identical. Figure 2 presents the case for all three countries. Within area A of 
Figure 2, each country specializes in each good in the following manner: 
(first country, second country, third country) = (third good, second good, first good). 
Figure 2 demonstrates Ikema’s illustration.  Similar to the  manner  of one-to-one 
correspondence above, Jones (1961) considered the resolution of the production assignment 
problem to determine the assignment satisfying the condition that each country specializes in 
a different good at a world production point located on the frontier. To illustrate the frontier 
using Figure 2, the efficient facets developed by McKenzie (1954) are beneficial.   
In Figure 2, each country specializes in one good within each domain labeled A through 
J. These domains correspond to extreme points on the world production frontier, where an 
extreme point is defined as one  that  cannot  serve  as an internal dividing point  between 
different production points. When two domains in Figure 2 are adjacent to each other in a 
definite straight line, such as H and I, there is a one-dimensional degree of freedom in the 
price where both corresponding world production points represent maximization of the world 
output, leading the frontier to bend along this straight line. In contrast, a case in which two 
domains are tangential to each other at a point, such as C and F, there is only one price vector 
at which both world production points corresponding to the domains yield maximum world 
output. Thus, two world production points corresponding to C and F are in the same plane on 
the frontier. However, in a case in which two domains are neither adjacent nor tangential to 
each other, such as A and J, there is no price at which both world production points maximize 
world output. Thus, two corresponding world production points cannot occupy the same 
plane on the frontier. 
              
Figure 3                              Figure 4 
The  abovementioned  points  are  simplified in Figure 3 using McKenzie’s (1954) 
efficient facets. First, a triangle is drawn to identify each extreme point. Second, where two 
domains are identified as adjacent, such as H and I in Figure 2, they are connected with a 
642Economics Bulletin, 2012, Vol. 32 No. 1 pp. 639-647
straight line through the corresponding points shown in Figure 3. In this manner, one can 
illustrate McKenzie’s (1954) efficient facets in Figure 3, thus simplifying the original frontier 
as shown in Figure 4. 
With reference to the production assignment problem, point A can be characterized in 
Figure 4. Since Point A is located on various planes composing the frontier, information 
about the frontier can be obtained when point A  is characterized.  However, the case of 
triangle HIJ and the line segment HI must be considered. Although triangle HIJ composes 
one part of the frontier, point A is not located on HIJ, implying that no information can be 
obtained regarding HIJ by focusing only on point A. Moreover, the interior points of HI 
compose a part of the frontier, and these interior points can be achieved only by including 
diversification, because world output is maximized at these points with various prices. Figure 
5, viewed from the  direction of line HI,  illustrates  this case. The  points  also satisfy  the 
condition that every good is produced. However, because point A is not on HI , it is important 
to focus on classifying the aspect of the frontier for which the resolution of the production 
assignment problem seldom provides information. 
           
Figure  5                            Figure  6 
The production assignment problem derives the efficient specialization pattern from the 
three-country, three-good model. As Jones (1961) argued, without the case of ties between 
different specializations, the pattern exists and is unique. Area A shows the assignment in 
Figure 2. Moreover, in Figure 2, there are areas like E (all countries specializing in the first 
good), J (all countries specializing in the second good), and B (all countries specializing in 
the third good), where a specific good’s price is very high. Because three countries exist in 
the model, three borders and two areas typically exist between areas B and E, a situation 
similar to that between EJ and BJ. Thus, there are 10 areas in Ikema’s figure describing the 
three-country, three-good model. Area A is surrounded by nine other areas. Each area can be 
connected to the neighboring areas by a line segment. If now the efficient facets shown in 
Figure 6 are considered, it can be observed that CF, DG, and HI are the only possible pairs 
that can be linked with each other without passing through point A. The frontier can be 
classified regardless of whether or not CF, DG, and HI are linked. Consider Figure 3 as an 
643Economics Bulletin, 2012, Vol. 32 No. 1 pp. 639-647
example. While C and F, and D and G are not linked to each other, H and I are. 
Focusing solely on Ikema’s (1993) illustration, the difference in the shape of the frontier 
cannot be analyzed. However, if the focus is solely on McKenzie’s (1954) efficient facets, 
even if the difference can be discerned intuitively, an effective tool cannot be determined for 
analyzing it. The classification of the frontier can be analyzed only by connecting Ikema’s 
illustration and McKenzie efficient facets. In the following section, the shape of the frontier is 
classified using Ikema’s illustration in a manner beyond its original scope. 
 
3.  Classification 
First, it should be noted that several characteristics of the model are shown in Figure 1. Each 
country has three borders whose location has a one-to-one correspondence to the price 
(denoted by P) at which the value of the marginal product for any good is the same. In Figure 
7, therefore, the location of another country’s point P can be found within six areas labeled 
a  through  f . As the analysis of  d   is similar to that of  a  and the analysis of e c, , and  f  
is similar to that of  b , the results of the analysis of  a  and  b   can be generalized to the 
other areas.   
          
Figure 7                            Figure 8 
In case  a , the shapes of the frontiers can be classified as shown in  Figure 8. 
Specifically, given the borders that define the specialized production of goods for two 
countries in Figure 7, the frontiers can be classified with the position of point P of the third 
country, using notations such as 1/3, 2, and 1/2 to illustrate the shape of the frontier.   
For example, if one considers that point P of the third country falls within the area 
denoted by 1/3, then E (all countries specializing in the first good) and A (derived by the 
resolution of the production assignment problem) would occupy the same plane on the 
frontier. Similarly, B (all countries specializing in the third good) and A would occupy the 
same plane on the frontier. However, J (all countries specializing in the second good) and A 
would not occupy the same plane on the frontier. When neither “A and B” nor “A and E” nor 
644Economics Bulletin, 2012, Vol. 32 No. 1 pp. 639-647
“A  and  J”  occupy the same plane, the symbol “-”  is used in  Figure 8. The detail 
corresponding to the notations is presented in Table 1. 
Notations  A  and  E   A  and  J   A  and  B  
1  Same Plane  Not  Not 
2  Not  Same Plane  Not 
3  Not  Not  Same Plane 
-  Not  Not  Not 
1/2  Same Plane  Same Plane  Not 
1/3  Same Plane  Not  Same Plane 
2/3  Not  Same Plane  Same Plane 
1/2/3  Same Plane  Same Plane  Same Plane 
Table 1: Notations (1) 
These conditions correspond to the conditions shown in Figure 6. For example, in the 
case of notation 1/3, C and F are linked and D and G are linked, but H and I are not linked. 
There are 8 such cases. While the detailed correspondence to notations is presented in Table 2, 
the rules regarding Figure 6 are given below. The three rules are as follows:   
(1) When A and E occupy (do not occupy) the same plane on the frontier, D and G are 
not linked (are linked) to each other.   
(2) When A and J occupy (do not occupy) the same plane on the frontier, H and I are not 
linked (are linked) to each other.   
(3) When A and B occupy (do not occupy) the same plane on the frontier, C and F are 
not linked (are linked) to each other. 
Notations  D  and  G   H   and  I   C   and  F  
1  Not linked  Linked  Linked 
2  Linked  Not linked  Linked 
3  Linked  Linked  Not linked 
-  Linked  Linked  Linked 
1/2  Not linked  Not linked  Linked 
1/3  Not linked  Linked  Not linked 
2/3  Linked  Not linked  Not linked 
1/2/3  Not linked  Not linked  Not linked 
Table 2: Notations (2) 
 
Figure 8 (area  a)  1st Country  2nd Country  3rd Country 
White Area  3rd Good  2nd Good  1st Good 
Light-Gray Area  3rd Good  1st Good  2nd Good 
Dark-Gray Area  1st Good  2nd Good  3rd Good 
Table 3: Explanations of Figure 8 
645Economics Bulletin, 2012, Vol. 32 No. 1 pp. 639-647
In accordance with Ikema (1993), Figure 8 shows which good each country specializes 
in per the resolution of the production assignment problem. The result is shown in Table 3. 
Q is the point of intersection of these three borders. Therefore, with reference to Figure 8, 
this study can classify the shape of the frontier in area  a. 
The shape of the frontiers can be classified for area b by reference to Figure 9, which 
uses the same symbols used in Figure 8.   
 
Figure 9 
Similar to Figure 8, Figure 9 shows which good each country specializes in per the 
resolution of the production assignment problem. The result is shown in Table 4. 
Figure 9 (area  b )  1st Country  2nd Country  3rd Country 
White Area  3rd Good  2nd Good  1st Good 
Light-Gray Area  1st Good  3rd Good  2nd Good 
Dark-Gray Area  1st Good  2nd Good  3rd Good 
Table 4: Explanations of Figure 9 
With reference to Figure 9, this study can classify the shape of the frontier in area  b , and 
with reference to both Figures 8 and 9, one can classify the shape of the entire frontier. 
By using this means of classification, essential information can be derived regarding the 
frontier in the three-country, three-good Ricardian model. Ikema (1993) shows the difference 
of efficient specialization patterns derived by the production assignment problem, which is 
illustrated in different shades (white, light gray, and dark gray) in Figures 8 and 9. However, 
he does not mention these classifications, which are important for ascertaining the shape of 
the frontier. By following the approach described in this study, Ikema’s (1993) illustration can 
be extended to the multi-country, three-good model.   
 
646Economics Bulletin, 2012, Vol. 32 No. 1 pp. 639-647
4.  Conclusion 
This study focused on addressing a problem that cannot be resolved solely by the production 
assignment problem in the three-country, three-good Ricardian model. Specifically, this 
problem was addressed by introducing Ikema’s (1993) illustration and relating it to 
McKenzie’s (1954) efficient facets to classify the shape of the world production frontier. 
Following Jones (1961), the present study discussed the production assignment problem as a 
useful means of analyzing the Ricardian model featuring many countries and many goods. 
However, if the production assignment problem was solely addressed, as did Jones (1961), all 
the information could not have been clarified regarding the frontier. Ikema (1993) focused on 
this point but did not provide an explanation. In response to Ikema’s argument, the present 
study focused on classifying the shape of the frontier using Ikema’s illustration.   
Although the present study is based on Ikema’s illustration and on the benchmark of the 
multi-country, multi-good Ricardian model, it uniquely applies Ikema’s diagram to the entire 
frontier in the three-country, three-good case. Thus, this study’s approach is more powerful 
than not only Jones’ (1961) production assignment problem but also Ikema’s (1993) original 
explanation of the illustration.   
 
References 
Amano, A. (1966) “Intermediate Goods and the Theory of Comparative Advantage: A 
Two-Country, Three-Commodity Case” Weltwirtschaftliches Archive 96, 340-345 
Ikema, M. (1993) “Determination of the Patterns of Specialization in International 
Production: A Multi-Country, Multi-Commodity Case” (in Japanese) Hitotsubashi 
Ronso 110, 873-894   
Jones, R.W. (1961) “Comparative Advantage and the Theory of Tariffs: A Multi-Country, 
Multi-Commodity Model” The Review of Economic Studies 28, 161-175 
McKenzie, L.W. (1954) “Specialization and Efficiency in World Production” The Review of 
Economic Studies 21, 165-180 
647

Intermediate Goods and the Theory of Comparative Advantage: A Two-Country, Three-Commodity Case” Weltwirtschaftliches Archive 96,

Classification of the frontier in the three-country, three-good Ricardian model

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Classification of the frontier in the three-country, three-good Ricardian model

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