Some new perspectives on the intercountry analysis of the world production system
Abstract
When production theory is discussed in the traditional economic custom, somehow, either intermediate goods or price determination is often not examined precisely. The prices are usually fixed, while scarcity of resources is exceptionally attached importance. The efficiency of production is consequently forced to be adapted to a given price system of goods. However, this kind of restrictive treatments of production came up against several difficulties when the theory of international trade is argued in terms of Ricardo and Heckscher–Ohlin, i.e., comparative cost theory. The object of international transactions may not be limited to such resources as regulations are often imposed. It is rather important to trade not only among the different intermediate goods but also even among the same kinds of intermediate goods empirically shown in Ikeda et al. (RIETI Discuss Pap Ser 16(E26):1–35, 2016). Thus, the introduction of intermediate goods for production is indispensable to develop the theory of international trade. Even in the 21st century, however, economists were still frustrated to escape from a special two country–two commodity case. Fortunately, by remarking geometry, Shiozawa (Evolut Inst Econ Rev 3(2):141–187, 2007; Evolut Inst Econ Rev 12(1):177–212, 2016; A new construction of Ricardian theory of international values: analytical and historical approach, pp 3–73. Springer, Berlin, 2017) has been successful to establish a price determination in a more general case of three country–three commodity. In Shiozawa (Evolut Inst Econ Rev 12(1):177–212, 2016), he smartly employed a subtropical geometry to examine a general case of international trade, i.e., three country–three commodity case. Behind his idea, there is the idea of Minkowski space of production, in particular, zonotope. This approach will give a different view of production set, and possibly suggest a further generalization more than three of the number of country and commodity. First of all, this article gives a brief look of the new essence of Shiozawa’s theory, and then gives by some numerical simulation a new characterization of international trade in line with Shiozawa’s theory. Furthermore, this article examines the effect of the introduction of free international trade. The introduction of the optimization rule to international trade results in drastic changes in network structures. Finally, the link with the network analysis and econophysics will be argued.
Keywords
Subtropical geometry International trade Network structure Deregulation Intermediate commodityJEL Classification
F1 C6 D5 B12 L111 An introductory review on the international trade
1.1 An integration of two production systems through specialization
It is well known that an integration of any two production systems \(\{A, B\}\) will bring some more efficient production as a whole than a simple sum of the production amounts over the two independent systems. For simplicity, we suppose that each system can produce two goods \(\{x, y\}\) common to the two systems. We then give a simple example. We have two states of each production system where any system can specialize her production to some particular good of any two goods. Thus, we may have a state with specialization and a state without specialization to some particular good.
It is shown that specialization can increase the total production of A and B for both goods x and y. An efficient total production can only be achieved if either A or B specializes in producing the good in which they have a comparative advantage. We assume that a production possibility frontier is linear. In comparison between given two systems, a comparative advantage of any system will be then revealed as for the production of one of two goods. For instance, the system A has a comparative advantage in the production of y. The system B has a comparative advantage in the production of y. If it is the case, the production possibilities frontier of the systems A and B will be then depicted in Fig. 1. Thus, the frontiers of both countries have their maximum production on each good, respectively. We denote the system A’s maximum capacity on good x by \(X_{\max A}\), on good y by \(y_{\max A}\), for instance. The same notations are applied to \(X_{\max B}\), on good y by \(y_{\max B}\) for the system B.
Numerical instance
System  Independent  Integration  

Commodity  A  B  A  B 
x  15  7.5  20  4 
y  2.5  30  0  34.7 
In our instance shown by Table 1, there are two modes of production: independent mode and integrative mode, i.e., the mode “without specialization” and the mode “with specialization.”
In an integrative mode, A can produce 50 units of X, while B can produce 100 units of Y. Here, \(x \in X\) and \(y \in Y\). On the other hand, either A or B can only produce less than her maximum amount which they can specialize their production. We set \(\alpha = \frac{Y_{\max B}}{X_{\max A}}\). Hence, it depends on the ratio \(\alpha\) whether A or B should specialize. In our example, if \(\frac{Y}{X} \ge \alpha\), A must specialize in the production of X, and vice versa. We may then allow an intermediate case that A can produce both goods but B can specialize to produce Y. This illustrates how a comparative advantage in integration for specialization can occur given some independent systems. The result is represented by the activity analysis of Fig. 1. In Fig. 1, the specialization to a particular commodity due to its comparative advantage in the system A is demonstrated. However, this kind of specialization does not refer to and/or specify at all how the outputs will be exchanged among the systems. The transactions among the systems should be complex if the outputs were also used as the intermediate goods for production. This is the reason why the classical international trade theory was not updated until Shiozawa (2007, 2016) has mathematically innovated by the use of subtropical geometry.
1.2 Production set in the Minkowski space
1.3 Shiozawa’s innovative use of zonohedron
2 Ricardian economy with international trade
A Ricardian trade economy, or Ricardian economy, is an economy with M countries and N kinds of commodities, where M and N are positive integers. The usual convention \([M] = {1, \ldots , M}\) and \([N] = {1, \ldots , N}\) is used. The economy is given a set of production processes or production techniques, expressed by a rectangular matrix A with M rows and N columns. An entry \(a_{ij}\) of A denotes the labour input coefficient for the production technique in country i producing commodity j. Thus, \(a_{ij}\)’s of labour is required to produce s units of commodity j in country i. Each country has a fixed labour force \(q_i\). This is the maximum quantity of labour available for production in a country. E comprises the set \(\{A, q\}\), where A is a set of \(M \times N\) positive numbers \(a_{ij}\) (i.e., \(A = [a_{ij}]\) and the vector q is a set of M positive numbers \(q_i\). Thus, a Ricardian economy is simply the couple \(\{A, q \}\).
No body has successfully proved the existence proof in its general case of M countries and N kinds of commodities with intermediate inputs. Jones (1961) only gave a sufficiency proof in the special case of \(M = N\). However, the number of commodities is actually far bigger than the number of countries. We were keen to generalize Ricardian trade theory. A general case must contain a production of commodities by means of commodities. In this point of view, Yoshinori Shiozawa was the first scholar who successfully proved a general case of Ricardian trade theory in Shiozawa (2007). Here, he employed barycentric coordinates for his proof.
2.1 Shiozawa’s Ricardian international trade theory of subtropical form
Shiozawa (2016), second, confirmed in the Ricardian economy that the subtropical hyperplane arrangement with full commodity apexes (in case of mintimes semiring) and with full country apexes (in case of maxtimes semiring) could determine the range of wage rate vectors and price vectors in a selfreplacing steady state. Here, Shiozawa regarded the competitive type at a point as spanning, namely, sharing and covering. Therefore, we get the diagrams of spanning core in commodity simplex and spanning core of country simplex. In this elegant arrangement, Shiozawa formulated the world production set in terms of Minkowski sum. This formulation is also innovative for mathematical economies. It is noted that Fig. 4 is driven by Fig. 5 by incorporating the above major conjugate relations.
2.2 The application of hyperplanes of tropical geometry to the international trade
Leaving a rigorous proof given by Shiozawa (2017) in Shiozawa et al. (2017), we can easily summarize how the goods and countries generate a complex network structure. In the following, by the use of simplex geometry, we first depict that domains in which each country has cost advantage. We employ this to embed relative prices into this simplex to establish the price simplex of the world production system diagrammatically. We then incorporate the wage simplex into the price simplex to prove “commodity’s cost advantage” in countries. In this procedure, we finally find competitive types of country domains in a superposition of the price simplex and the wage simplex.
Commoditywise view
Commodity 1  Commodity 2  Commodity 3 
\(\downarrow\)  \(\downarrow\)  \(\downarrow\) 
Country 2, 3  Country 2  Country 2 
Countrywise view
Country 1  Country 2  Country 3 
\(\downarrow\)  \(\downarrow\)  \(\downarrow\) 
\(\phi\)  Commodity 1, 2, 3  Commodity 2 
Selection of cost advantage allocation
Country 1  Country 2  Country 3 
\(\downarrow\)  \(\downarrow\)  \(\downarrow\) 
Commodity 2  Commodity 3  Commodity 1 
The apex of a tetrahedron is degenerated/flattened on a plane. Thus, V is called the apex of commodity space represented by a twodimensional triangle. We argue the cost advantage when the wage vector \(w = (w_1, w_2, w_3)\) is given in the triangle \(V^1 e_1 e_2\). The cost of production for commodity 1 is then the lowest in country 3. As the wage vector changes to move to \(V^1 e_2 e_3\), country 1 has the lowest cost of commodity 1. If the wage vector is found in \(V^1 e_3 e_1\), country 2 has the lowest cost of commodity 1. Similarly, as in commodity 1, we can also argue the cost advantages for commodity 2 as well as commodity 3. Therefore, as shown in Fig. 7, we have three apexes such as \(V^1\), \(V^2\), \(V^3\), and 10 domains in the triangle, which correspond to the different types of cost advantages, i.e., the different competitive types.
3 A verification of Shiozawa’s international trade theory by simulation
In the first section, when we argued a comparative advantage of an integration of any two independent systems, we noticed that the effects of international trade would give rise to much more complex interactions. Shiozawa (2016) successfully found that this kind of interactions was revealed by some new mathematical transformation like subtropical geometry. In short, the essence is no other than wageprice variations among countries through international trades. Needless to say, there is a big difference between a simple comparative advantage of productivity of commodities and the cost advantages between the relationship of commodities and countries. Moreover, we also observed that an idea of comparative advantage would no longer bring any pure specialization of production each country. As an empirical test showed that there may occur an internationally bilateral transaction on the same commodities between any two countries. This must invalidate the old justification of international trade by specialization of production each country, as Shiozawa (2007, 2016) mathematically showed. Thus, we finally verify an occurrence of transaction of the same commodity between countries and some associated meanings of the network order of production. We will achieve this verification by the countrybased simulation.
Nettleton, moreover, has given a new insight to randomize the net make less use tables. We show the case of three commodities and three countries randomly generated, for instance.the model is quite sophisticated in implementing tableau networked production functions across multiple industries in multiple countries, all interconnected by trade, settled simultaneously in volume and price through linear programming.
State of before deregulation of Country 1
Country 1  Country 1 infrastructure  Net export from Country 1 to 2  Net export from Country 1 to 3  Country 1  

Limited trade  Indst.1  Indst.2  Indst.3  Indst.1  Indst.2  Indst.3  Indst.1  Indst.2  Indst.3  Consumption 
Industry 1  51  −12  −15  −10  0  0  −4  0  0  −10 
Industry 2  −18  33  −10  0  2  0  0  3  0  −10 
Industry 3  −16  −13  31  0  0  10  0  0  −2  −10 
Labour  −59  −30  −62  0  0  0  0  0  0  0 
State of before deregulation of Country 2
Country 2  Net export from Country 2 to 1  Country 2 infrastructure  Net export from Country 2 to 3  Country 2  

Limited trade  Indst.1  Indst.2  Indst.3  Indst.1  Indst.2  Indst.3  Indst.1  Indst.2  Indst.3  Consumption 
Industry 1  10  0  0  25  −15  −12  2  0  0  −10 
Industry 2  0  −2  0  −16  34  −15  0  9  0  −10 
Industry 3  0  0  −10  −15  −12  49  0  0  −2  −10 
Labour  0  0  0  −17  −100  −27  0  0  0  0 
State of before deregulation of Country 3
Country 3  Net export from Country 3 to 1  Net export from Country 3 to 2  Country 3 infrastructure  Country 3  

Limited trade  Indst.1  Indst.2  Indst.3  Indst.1  Indst.2  Indst.3  Indst.1  Indst.2  Indst.3  Consumption 
Industry 1  4  0  0  −2  0  0  42  −14  −20  −10 
Industry 2  0  −3  0  0  −9  0  −17  51  −12  −10 
Industry 3  0  0  2  0  0  2  −17  −16  39  −10 
Labour  0  0  0  0  0  0  −12  −43  −89  0 
3.1 The make minus use table and the network diagram
We focus on the number of oriented spanning trees rooted at a vertex i in our tables. According to Kirchhoff theorem on the adjacency matrix, usually, the offdiagonals of the matrix are \(1\). However, we note that our table are not an adjacency matrix but rather a weighted adjacency matrix. The make minus use table is fitted to the definition of weighted Kirchhoff matrix. Thus, we apply the weighted Kirchhoff matrix to our table.^{2} Incidentally, it may be useful to depict a directed network structure of the make minus use tables by the directed graph if the table is asymmetrical.^{3}Kirchhoff’s theorem for directed multigraphs
Kirchhoff’s theorem can be modified to count the number of oriented spanning trees in directed multigraphs. The matrix Q is constructed as follows:
 1.
The entry \(q_{ij}\) for distinct i and j equals \(m\), where m is the number of edges from i to j;
 2.
The entry \(q_{ii}\) equals the indegree of i minus the number of loops at i. The number of oriented spanning trees rooted at a vertex i is the determinant of the matrix gotten by removing the \(i_{th}\) row and column of Q.
3.2 The traditional view of the landscape of the production system
Now, first of all, we deal with the quantity system of supply and demand. The production system of country k is then represented each country as follows:
Here, the wage rate of country k is denoted by \(w^{[k]}\).

Type \([IA]^{1}\)

Type \([I[IM]A]^{1}\)

Type \([IA_D]^{1}\)
3.3 Dual linear programmings of production and consumption in case of the international trade of the intermediate goods
The above formulation never contains any selection process of international transaction of intermediate goods (industries) between countries. It could no longer stand the world system of international trade. We must move to a new augmented profile to embrace all the interaction of import and export between countries. We then construct a new array, i.e., commodities matrix B. The matrix is “arrayed” by a specific rule taking into account the mutual interaction of productive relationship between countries by the international trade.
The mutual interaction of multiple countries and many industries will generate a more complicated network difficult to grasp a whole relation. To avoid difficulties of new notations, it seems easily understandable to employ a numerical example to present how to construct the competitive process of international transaction of intermediate goods between countries. This operation will suggest the introduction of the free trade system to all the interregional system. We, only for convenience of illustration, take the smallest case of 3 industries and the 2 countries, although our simulation refers to a more general case of the 3 countries–3 industries case later.
3.3.1 The commodity matrix of 2 countries by 2 industries case
Table of Country 1
Country 1  Country 1 infrastructure  Net export from Country 1 to 2  Country 1  

Limited trade  Industry 1  Industry 2  Industry 1  Industry 2  Consumption 
Industry 1  25  −12  −3  0  −10 
Industry 2  −10  25  0  −5  −10 
Labour  −30  −18  0  0  0 
Table of Country 2
Country 2  Net export from Country 2 to 1  Country 2 infrastructure  Country 2  

Limited trade  Industry 1  Industry 2  Industry 1  Industry 2  Consumption 
Industry 1  3  0  17  −10  −10 
Industry 2  0  5  −15  20  −10 
Labour  0  0  0  −12  −1 
Corresponding activity levels of Country 1 in the smallest international economy
Country 1  Country 1 infrastructure  Net export from Country 1 to 2  Consumption  Slack variable  

Industry 1  Industry 2  Industry 1  Industry 2  Consumption  S  
Industry 1  \(b_{11}\)  \(b_{12}\)  \(b_{13}\)  \(b_{14}\)  \(b_{15}\)  \(b_{16}\) 
Industry 1  \(y_{11}\)  \(y_{12}\)  \(y_{13}\)  \(y_{14}\)  \(y_{15}\)  \(y_{16}\) 
We thus specified the labels of the activities by the next rules to Country 1 in the smallest economy of the world production system containing the international trade. It then follows our optimization problem of the world production system.
3.3.2 The establishment of shadow prices including the wage rate
Shadow prices of the 2 countries by 2 industries case
Countries  Commodities  Shadow prices 

Country 1  Commodity 1  0.047 
Country 1  Commodity 2  0.053 
Country 1  Labour  0.041 
Country 2  Commodity 1  0.047 
Country 2  Commodity 2  0.053 
Country 2  Labour  0.008 
After deregulation table of Country 1
Country 1  Net export from Country 1 to 2  Country 1 infrastructure  Country 1  

Limited trade  Industry 1  Industry 2  Industry 1  Industry 2  Consumption 
Industry 1  0  −32  43  0  −11 
Industry 2  0  67  0  −56  −11 
Labour  0  −48  0  0  0 
After deregulation table of Country 2
Country 2  Country 2 infrastructure  Net export from Country 2 to 1  Country 2  

Limited trade  Industry 1  Industry 2  Industry 1  Industry 2  Consumption 
Industry 1  −43  0  67  −9  −15 
Industry 2  0  56  −59  18  −15 
Labour  0  0  −8  −65  0 
Even in the case of the smallest world production system of the 2 countries by 2 industries, the simple idea of mutual specialization of industry composition between two countries does not hold. Thus, we will focus on a network analysis on the structural change after deregulation int the final subsection.
3.3.3 The 3 countries by 3 industries case
Corresponding activity levels of Country 1
Country 1  Country 1 infrastructure  Net export from Country 1 to 2  Net export from Country 1 to 3  Consumption  Slack variables  

Indst.1  Indst.2  Indst.3  Indst.1  Indst.2  Indst.3  Indst.1  Indst.2  Indst.3  \(S_2\)  \(S_1\)  
Industry 1  \(b_{11}\)  \(b_{12}\)  \(b_{13}\)  \(b_{14}\)  \(b_{15}\)  \(b_{16}\)  \(b_{17}\)  \(b_{18}\)  \(b_{19}\)  \(b_{10}\)  \(b_{11}\)  \(b_{12}\) 
Industry 1  \(y_{11}\)  \(y_{12}\)  \(y_{13}\)  \(y_{14}\)  \(y_{15}\)  \(y_{16}\)  \(y_{17}\)  \(y_{18}\)  \(y_{19}\)  \(y_{10}\)  \(y_{11}\)  \(y_{12}\) 
State of after deregulation of Country 1
Country 1  Country 1 infrastructure  Net export from Country 1 to 2  Net export from Country 1 to 3  Country 1  

Limited trade  Indst.1  Indst.2  Indst.3  Indst.1  Indst.2  Indst.3  Indst.1  Indst.2  Indst.3  Consumption 
Industry 1  10  −56  0  36  0  0  36  0  0  −26 
Industry 2  −4  153  0  0  −62  0  0  −62  0  −26 
Industry 3  −3  −60  0  0  0  45  0  0  45  −26 
Labour  −12  −139  0  0  0  0  0  0  0  0 
State of after deregulation of Country 2
Country 2  Net export from Country 2 to 1  Country 2 infrastructure  Net export from Country 2 to 3  Country 2  

Limited trade  Indst.1  Indst.2  Indst.3  Indst.1  Indst.2  Indst.3  Indst.1  Indst.2  Indst.3  Consumption 
Industry 1  50  0  0  0  0  −64  50  0  0  36 
Industry 2  0  58  0  0  0  −80  0  58  0  −36 
Industry 3  0  0  −112  0  0  261  0  0  −112  −36 
Labour  0  0  0  −144  0  0  0  0  0  0 
State of after deregulation of Country 3
Country 3  Net export from Country 3 to 1  Net export from Country 3 to 2  Country 3 infrastructure  Country 3  

Limited trade  Indst.1  Indst.2  Indst.3  Indst.1  Indst.2  Indst.3  Indst.1  Indst.2  Indst.3  Consumption 
Industry 1  −86  0  0  −86  0  0  216  −27  0  −17 
Industry 2  0  4  0  0  4  0  −87  98  0  −17 
Industry 3  0  0  68  0  0  68  −87  −31  0  −17 
Labour  0  0  0  0  0  0  −62  −82  0  0 
3.4 Validity of the comparative advantage among countries mutually transacting

The deregulation may generally expand the net outputs of the world.^{5}

However, the expansion of the international trades will not necessarily generate either the specialization of a particular industry or the balanced growth of interindustries.

It may be rather observed that the productive network in some country is entirely destroyed to prevent from selfreproducing ever before.
3.4.1 The transition of the landscape before and after deregulation
3.4.2 The transition of the Kirchhoff diagram before and after deregulation
4 Concluding remarks
The economy essentially is sustained periodically through mediating industries (intermediate commodities), which is preparing for the next year’s growth. In this process, on the other hand, the international trade must play a decisive role to reconnect the interindustrial activities. The networked production and the international trade are then the cooperative factors for the growing/decaying process of the national/regional economy. In this point of view, Ikeda (2016) seems smartly navigating an empirically based analysis of the worldwide economy by employing the world input–output data and the international trade data. In Ikeda (2016), the fundamental set of the analysis for this purpose is neatly well defined in view of econophysics. The community structure of the system is interestingly driven according to the idea of the modularity benefit function, for instance. An undirected multiplex network may then be employed to detect the community structure in a timedependent network.
To deal with the dynamics of the world economy and to explore the structural controllability of the world system, it is also pointed out by Ikeda (2016) that the idea of “driver nodes” (partially and globally) is “identified by the maximum matching in the bipartite representation of the network” due to Newman (2006). Here an integer linear programming is applied.
 1.
“[T]he increase of the number of driver nodes during economic crisis is explained qualitatively by the heterogeneity in terms of degree distribution.”
 2.
These community structure means that international trade is actively transacted among same or similar industry sectors.
The effect of the international trade is decisively different from a merger of any independent units. However, an analysis of the world production system requires a more complicated idea of production system. A possibility set of production composed of two countries may be generated as a zonotope of two facet of country’s production. The sphere the activity analysis targets is merely a part of the zonotope set. Thus, according to Shiozawa (2016), we employ a new tropical geometry and were successful to introduce a hyperplane of wageprice coordinates, and argued a price competition in a general case of the international trade of multiple intermediate commodities between multiple countries. Hence, Shiozawa was successful to demonstrate properly that any idea of industry specialization in country is irrelevant to confirm any cost advantage of the international trade. The cost advantage holds without any idea of optimization. We, however, examined the effects to introduce the framework of free trade, i.e., deregulation each country. According to a network analysis, the introduction of free trade suggests a drastic change of industrial networks each country. The result may be disastrous for employment of some country.
Footnotes
 1.
Under the projection \((t_1,\ldots ,t_r) \mapsto \sum _{i=1}^{r} t_i a_i\)), the shadow of the rdimensional cube \([0,1]^r\) is Z(Y).
 2.
The function titled “KirchhoffGraph” of Mathematica is quite helpful to depict the strength of the oriented spanning trees. See, for example, “Weighted Kirchhoff Matrix?” (https://mathematica.stackexchange.com/questions/23122/weightedkirchhoffmatrix).
 3.
 4.
As for the empirical econophysics verification, see Ikeda (2016):
 5.
It is apparent, because deregulation implies maximizing the total net output common to all the participating intercountry systems. Also see an example of the simulation by Nettleton (2011).
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