Jun 25, 2015 - term âtropicalâ was coined in honor of Imre Simon (1943â2009), a Brazilian mathematician and computer scientist, because the Tropic of ...
International Trade Theory and Exotic Algebras Yoshinori Shiozawa Emeritus, Osaka City University
☆ This is the second draft of my paper and was submitted to EIER. The first version of this paper was titled Trade Theory and Exotic Algebra. ☆ This paper was published as follows in Evolutionary and Institutional
Economics Review with some revisions. For the citation, please ask me for the
published copy. 2015.6.25 Shiozawa, Y. 2015 International Trade Theory and Exotic Algebra. Evolutionary
and Institutional Economics Review 15(1): 177-212.
DOI 10.1007/s40844-015-0012-3. F1 C6 D5 B12 §1. Introduction §2. Exotic algebras and their geometry §3. Ricardian trade theory and subtropical algebra §4. Barycentric representation and competitive types §5. Conjugacy between wage rates and prices §6. Hyperplane arrangements and spanning core §7. Minkowski sum and the world production set §8. Pattern of specialization and graphs of competitive types §9. McKenzie-Minabe diagram and the Cayley trick §10. Jones theorem and permanents of tropical square matrices
§1. Introduction International trade has fascinated economists because of its pure theory and its policy implications. Approximately 200 years ago, David Ricardo first presented his famous numerical example (Ricardo, 1817, Chapter 7). Since then, economic theories have developed tremendously, and yet the logical structure of international trade theory has not been well explored. Thus, it was surprising to find that an exotic algebra lies behind the Ricardian trade theory. Many traditional problems have been translated into the language
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of an exotic algebra, including the determination of specialization patterns, price determination, combinatorial questions, and graph theory. Exotic algebras, which include tropical algebra, is a relatively new topic, even in mathematics. Major topics in tropical algebra, and its associated geometry, have resulted from research in this area during the past two decades. Tropical algebra has become one of the most productive fields of applied mathematics (e.g., timed event graphs), and provides indispensable tools for pure mathematical studies, such as algebraic geometry (Sturmfels, 1994). Because they have a rich body of literature, trade economists have many ready-made theories that provide us with concrete information on combinatorics and the geometry of production possibility sets. Here, topics cover theories such as tropical-oriented matroids (Ardila and Develin, 2009), the number of vertices and facets, the Minkowski sum, and zonotopes (Huber, Rambau, and Santos, 2000), and cephoids (Pallaschke and Rosenmüller, 2004). These studies provide knowledge on the geometry and combinatorics of the production possibility set of a Ricardian trade economy. Such knowledge was unattainable prior to these theories. Thus, they have resulted in significant gains in terms of Ricardian trade theory. On the other hand, Ricardian trade theory provides mathematicians with interesting mathematical entities that connect various tricks in tropical geometry, such as the tropical matroid, Minkowski sum, permanents, and transportation polytopes (De Loera et al., 2009). For example, the Cayley trick, hitherto studied from a topological point of view, now has a new, quantitative interpretation. In this sense, tropical algebra and classic polytope theory are new mathematical objects with a rich structure that need to be studied. Note this paper serves more as an introductory text on trade theory and exotic algebra. A book is currently being written that provides a thorough discussion, including details of all formulations and proofs (Shiozawa, to appear). To this end, I focus on three-country, three-commodity cases, or smaller models, although the theory is applicable to any number of countries and commodities. Sections 2 and 3 introduce exotic algebras and Ricardian trade theory, respectively, two fields that have thus far been treated separately. Section 4 offers a graphical representation of a Ricardian economy. Section 5 describes the conjugate relation between wage rate and prices as a first application of subtropical matrix operations. Section 6 prepares the mathematical part of our examination. Note that Sections 4 to 6 are only concerned with
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wage rates and prices. Then, Section 7 introduces production. The world production possibility set is defined as a Minkowski sum of each country's production possibility sets. Section 8 gives an almost complete description of possible types of specialization, and relates these to the faces of the maximal frontier of the world production set. Section 9 describes a little-known result on a modified McKenzie diagram. The new diagram and its construction illustrate the intrinsic connections between various entities in tropical geometry. Lastly, Section 10 discusses Jones’ classic results and compares them with the new formulations. Richter-Gebert, J., B. Sturmfels, and T. Theobald (2003) and Joswig (2014) give introductory knowledge on tropical geometry, whereas Ziegler (1995) gives basic knowledge on convex polytopes.
§2. Exotic algebras and their geometry Exotic algebra refers to any sets with operations that differ from the ordinary plus and product operations. A well-known exotic algebra is tropical algebra. This is a set of real numbers R, with exotic operations a
⊕ b = min {a , b}
and a
⊕ and ⊙. The two operations are defined as follows:
⊙ b = a + b.
The operations are commutative and associative, and in particular, are distributive: a
⊙ (b ⊕ c ) = a ⊙ b ⊕ a ⊙ c.
Thus, algebraic operations using
⊕ and ⊙ can be calculated as ordinary real numbers. The
difference is that there is no inverse element for the operation
⊕. In other words, there is no
number that satisfies the equation a
⊕ x = b,
when a > b. On the other hand, the equation a
⊙x=b
always has a solution, because x = b－a by the ordinary minus operation. The number 0 is a unit for the
⊙ operation. In fact, we have
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a
⊙0=a
∀a ∈ R.
This algebraic entity is called a semi-ring, and is known as tropical algebra. The entity is also known as a max-plus semi-ring when we explicitly cite the two operations. Furthermore, this algebra has the following special property: a
⊕ a = a.
By virtue of this property, the algebra is also referred to as an idempotent semi-ring. It is not an (algebraic) field like the standard real field R or the complex field C, which include the inverse a - b for the plus operation, and a/b for the multiply operation (except when b = 0). However, the tropical algebra can generate objects similar to vector spaces for a field and, thus, we have a geometry over these objects. However, the exotic algebra that plays a crucial role in Ricardian trade theory is not tropical algebra, but a similar algebra defined on the set of all positive real numbers R+, with two operations defined as follows:
a ⊕b = min {a , b}
and
a ⊙b = a・b.
Note that this is not completely an exotic algebra as the
⊙
operation is the ordinary
multiplication of real numbers. Therefore, I call it a subtropical algebra, in the same sense that the subtropics lie between the tropics and temperate climate zones. The mathematical term “tropical” was coined in honor of Imre Simon (1943–2009), a Brazilian mathematician and computer scientist, because the Tropic of Capricorn passes over São Paulo, where he lived. The meaning is somewhat modified, but we add the “sub” prefix to indicate that the multiply operation refers to ordinary multiplication. Another possible name is the min-times semi-ring. When we need to distinguish between a min-times semi-ring and a max-times semi-ring (e.g., when we discuss conjugate properties in Section 5), this explicit notation is more convenient. However, we use the terms tropical and subtropical when max and min are interchangeable. Tropical and subtropical algebras are isomorphic because positive numbers can be transformed, one-to-one, by the log function onto the set of all real numbers. Furthermore, the operations are conserved by this transformation. Indeed, we have
log(min{a, b})=min{loga, logb} and log(a + b)=(loga)・(log b). In this regard, the subtropical algebra is a special variety of tropical algebras.
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As coordinate-wise operations, ⊕ and ⊙ are generalized over the set of positive numbers Rd. Explicitly,
(x 1, x 2, ... , x d) ⊕ (y 1, y 2, ... , y d) = (x 1⊕y 1, x 2⊕y 2, ... , x d⊕y d,) and
λ⊙ (x1, x2, ... , xd) = (λ⊙x 1, λ⊙x 2, ... , λ⊙x d). Because ⊙ is the same as the classic multiply operation, there is no fear of confusion if we omit the symbol ⊙. We keep symbol ⊗ for the case of matrix multiplication (as defined later). Moreover, vectors in R+
d
can be mapped by applying the logarithmic function
coordinate-wise into vectors in Rd, and the subtropical operations are mapped to the corresponding tropical operations. The map is invertible, and the properties of tropical mathematics are transmitted to subtropical mathematics. In this way, almost all properties of tropical mathematics (e.g., oriented matroids and tropical hyperplanes and arrangements) are transferred to subtropical mathematics. Thus, we can construct a new and separate series of theories for subtropical mathematics. The special relation between subtropical algebra and Ricardian trade theory is evident if we consider that only two operations are necessary to describe price relations. First, we need to take a minimum and, second, we need to calculate the product of a country’s wage and labor input coefficient. We discuss this point in more detail in the next section.
§3. Ricardian trade theory and subtropical algebra 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, …, M} and [N] = {1, … , N} is used throughout this paper. 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 labor input coefficient for the production technique in country i producing commodity j. Thus, a ij s of labor is required to produce s units of commodity j in country i. Each country has a fixed labor force q i. This is the maximum quantity of labor available for production in a country. Formally, the (M, N) Ricardian economy E comprises the set {A, q}, where A is a set of M・N positive numbers a ij (i.e., A = { a ij }) and the vector q is a set of M positive numbers q i. Thus,
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a Ricardian economy is simply the couple { A, q } 1. In a Ricardian economy, there is no choice of techniques in a real sense. If there are several different techniques that produce the same product, the most efficient technique is the one with the minimal coefficients. Note that we treat the production technique in a different country as a different technique. Thus, an (M, N) Ricardian economy (i.e., an M-country,
N-commodity Ricardian economy) has M・N production techniques. Here, we simply refer to technique (i,j), which indicates the production technique of country i that produces commodity j. We first need to determine the wage–price vector for a Ricardian economy that satisfies certain conditions. A wage–price system, or international value (w, p) is a couple comprising an M-row vector w = (w i ) and an N-row vector p = (p j ). Component w i gives the wage rate for country i, and component
p j gives the price of commodity j. Note that wage rates and
prices are expressed by a single international currency (e.g., US dollars). There is only one price for a commodity. This means that any goods of the same commodity have the same price in any country. This means we assume that transport costs are zero2. Wages and prices are positive. The unit cost for production technique (i, j) is w i a ij, which we compare with the price p j of commodity j. When p j = w i a ij, we say technique (i, j) is competitive. An international value v = (w, p) is admissible when the conditions
w i a ij ≧ p j, ∀ i ∈ [M] and j ∈ [N]
(3-1)
are satisfied. Condition (3-1) means there is no technique with excess profit, or that no technique satisfies p j >
w i a ij. If an international value is admissible, some techniques are
competitive, while others are in deficit. Investigations in international trade theory sometimes require more refined conditions than (3-1). Indeed, what we need is an international value that has at least one competitive technique for each commodity. How can this be formulated? Subtropical expressions are useful here. First, we introduce subtropical matrix multiplication. The classic matrix multiplication of Ricardian trade economy thus defined excludes production processes that require material inputs and, as a consequence, input trade or trade in intermediate goods. The economy that incorporates input trade is referred to as Ricardo-Sraffa trade economy. It has many similar properties as Ricardian trade economy, but the present paper's results do not directly apply to Ricardo-Sraffa trade economy. 2 This assumption has serious consequences, but we do not discuss these in detail here. 1
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vector w and matrix A is defined by the formula w A = (∑ i=1 M w i a ij ) = (w 1
a 1j + w 2 a 2j + …+ w N a Nj).
Two binary operations + and ・ are used here. If we replace these two operations with subtropical operations, we obtain the definition of matrix multiplication ⊗, as follows:
⊕
w⊗A=( where
i=1
N w i ⊙ a ij ) = ( Min{w 1 a 1j, w 2 a 2j, … , w N a Nj ),
⊕ is the operation of taking the minimum. Using subtropical expressions, the
condition that an international value is admissible can be written as
w ⊗A ≧ p.
(3-2)
Now, the condition that an admissible international value has at least one competitive technique can be expressed as
w ⊗A = p .
(3-3)
The change from (3-1) to (3-2) is a formal transcription. However, with expressions (3-2) and (3-3), we now obtain new concepts. In (3-3), we construct a matrix product in subtropical algebra. This provides a change of view and opens a new direction for reasoning. The parallelism between classic and tropical algebras works well, and many confusing exotic algebra questions can be interpreted via analogies to classic geometry.
§4. Barycentric representation and competitive types In the case of three countries, a graphic representation is useful when analyzing price problems in trade theory. There are several ways to represent such problems, but here we use barycentric coordinates. This implies that we are only concerned with relative wage rates between countries. The absolute value of a wage rate or a price is important when we want to analyze monetary problems, but here we are concerned only with exchange ratios, or relative wages and prices. Cases involving more than three commodities cannot be represented graphically on a plane, but a graphical conceptualization is useful, and theorems can be generalized in those cases.
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e3
e3
Barycentric representation Country 2
Domains in which each country has cost advantage
B1 B2
Country 3 V(v1, v2, v3)
V(v1, v2, v3)
C2
C1
Country 3
W(w1, w2, w3)
e1
e1
e2
B3
Figure 1
e2
Figure 2
A barycentric representation is a triangle (or simplex of any dimensions), with weights as coordinates. Normally, we use an equilateral triangle as a frame. Let the vertices of the frame triangle be at e1, e2, and e3, and a positive vector v = (v 1, barycenter of the triangle with relative weights v 1,
t (v 1
e1
+ v2
e2
+ v3
v 2, v 3) be given. Then, the
v 2, and v 3 at each vertex is given by
e3),
where t = 1/(v 1+v 2+v 3).
Figure 1 illustrates point V(v 1,
v 2, v 3). Note that each vertex of the frame simplex is a
point where the weight is concentrated in one country. Each vertex should then be labeled by countries. Conversely, if a point is given in the frame triangle, the barycentric coordinates are uniquely determined for the point. Let B1, B2, and B3 be the points in base lines (sides opposite to a vertex) at which each extended straight line connecting a vertex and point V crosses its baseline. Then we have three equalities of six proportions: e1 B3 : e2 B3 = v2 : v1, e2 B1 : e3 B1 = v3 : v2 and e3 B2 : e1 B2 = v1 : v3. These equations are not independent, but, together with
v 1+v 2+v
3
= 1, they fully
determine the barycentric coordinates. A barycentric representation has a simple intuitive interpretation. If we embed the equilateral triangle in RM and the vertices are situated at segment 1 for each Cartesian coordinate, we may interpret a point of the barycentric representation as a half-line that starts from the origin. The point defined earlier is the crossing point of this half-line with the frame triangle. In this interpretation, the vector (v 1,
v 2, v 3) determines a half-line that
starts from the origin and passes through the point (v 1, v 2, v 3). Any points on the half-line
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(except the origin) are proportional to each other. Relative wage rates determine this half-line. In the following, when we speak of barycentric coordinates, we assume they sum to 1, unless otherwise stated. Then, the frame triangle is replaced by a simplex (i.e., a generalized triangle and a tetrahedron), which is conveniently called a subtropical simplex because it represents a subtropical projective space. A barycentric representation is convenient when we want to know which country is most competitive given a certain wage vector w = (w 1,
w 2, w 3). Consider the case of commodity 1. Here, we have the set of production techniques a 11, a 21, and a 31 for each country. Let v 1 = c /a 11, v 2 = c / a 21, and v 3 = c / a 31
where c is a scalar value to make the sum v 1 +
v 2 + v 3 equal to 1. That is,
c = 1 / a 11 + 1 / a 21 + 1 / a 31. Let V be a point thus determined. This is called the apex for commodity 1. If the point w =
(w 1,
w 2, w 3) lies in the triangle e1 V e2, then country 3 (or, more precisely, production technique (3, 1)) has the greatest advantage when the wage rates are w = (w 1, w 2, w 3). The above assertion is confirmed as follows. Draw a straight line from e1 passing through point W, and take the point where this line crosses the base line e1 e3. If we name this point C1, we get
w 2 :w 3 > v 2 : v 3 by comparing the two proportions of C1 and B1 as two internally dividing points of the line e2 e3. Because
v 2 : v 3 = 1/a 21 : 1/a 31, the previous inequality means
w 2:w 3 > a 31 : a 21 or w 2 a 21 > w 3 a 31. Similarly, if we compare the two dividing points of line e3 e1, we get
w 1 a 11 > w 3 a 31.
In conclusion, when the wage vector w = (w 1,
w 2, w 3) is in the triangle V e1 e2, the cost of
production for commodity 1 is lowest in country 3. Thus, the wage vectors are divided into three domains V e2 e3, V e3 e1, and V e1 e2. In each of them, countries 1, 2, and 3 have the lowest costs of production for commodity 1 (see Figure 2). Note that point V is the apex of a degenerated tetrahedron flattened on a plane. This is why it is called the apex for a commodity. We have examined the case of commodity 1. The same divisions of domains are possible for commodities 2 and 3. Figure 3 shows the divisions of domains for all commodities. Note that
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the names of the apexes have been changed from V to v1, v2, and v3. There are now 10 domains, and each domain has different types of cost advantages. The labels such as (1, 2, 1) denote the competitive countries for each commodity, and are called competitive types. Thus, (1, 2, 1) means commodity 1 has the lowest cost in country 1, commodity 2 has the lowest cost in country 2, and so on. Note that there is only one domain that includes all three countries.
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Competitive types of domains
(1,2,1) (2,2,2) (1,2,2)
v2
v1
(1, 1,1)
C (3,2,2)
(1,3,1) (1,3,2) (3,3,1)
(3,3,2) (3,3,3)
v3
e1
e2
Figure 3 Thus far, we have only examined points within divisions, but not those that lie on one of the lines. For example, consider a point P that lies on e1v1. For commodity 1, countries 2 and 3 have the same production cost, both of which are less than that of country 1. Therefore, countries 2 and 3 are both competitive for commodity 1. Thus, point P has competitive type ({2, 3}, 2, 2). Similarly, point v1 has competitive type ({1, 2, 3 }, 2, 2). In summary, each point has its own competitive type. A competitive type can take many different forms. The aforementioned expression uses set notation, or more precisely a notation that uses a list of commodity-wise subsets of country labels. The second method is to use a list of country-wise subsets of commodity labels. Another elegant representation uses a bipartite graph (see for an example Figure 10 in Section 7). This is a graph in which the vertices are divided into two mutually exclusive sets.
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In our case, one of two is the set of country labels and the other the set of commodity labels. A bipartite graph is a set of edges or links that connect two sets. In a graphical representation, we put the country-label set in the left side and commodity label set in the right side. A link that connects country 1 and commodity 2 is denoted (1, 2). The first entry of a couple refers to a country number, and the second entry refers to a commodity number. Thus, the commodity-wise competitive type ({2, 3}, 2, 2) takes the form (φ, {1, 2, 3}, 1) in country-wise notation, whereφ is an empty set. The linking edges comprise (2, 1), (2, 2), (2,3), and (3,1). The point C on the the cross point of two segments e1 v2 and e2 v1 has commodity-wise list ({1,3}, {2,3}, 2). In country-wise representation, the list take the form (1, {2,3}, {1,2}). Figure 10 gives the bipartite graph of this competitive type.. A bipartite graph representation is often preferable, because it shows more explicitly the conjugate relationship between countries and commodities and between wage vectors and price vectors. We discuss this further in section 5. Another merit of bipartite graph representation is we can get a more direct definition of competitive type. Indeed, if T is the competitive type of a point w in a barycentric representation and p = w ⊗ A, we have
w i a ij = p j
if and only if (i, j) ∈ T.
Thus we can take this as the definition of competitive type for an international value v = (w, p). Formerly the competitive type of an admissible value v is a subset of all indices (i, j) that satisfies the equation
w i a ij = p j . This definition plays an important role in Section 7. An important point in determining wage rates is whether all countries have at least one competitive commodity. If there is no such commodity for a country, the prices of all commodities are lower than the production cost for the country. In this case, the country has no products to export, and all industries will lose out to competition. Furthermore, in the long term, all jobs will be lost. However, this state of an economy does not continue for a long time, and either the exchange rate or the wage rate decrease. If we are interested in a stable wage–price system, such wage rates should be excluded. This is a very weak condition and, thus, it seems better to find a stronger condition. In reality, this condition is necessary and sufficient for a steady self-replacing state to exist within a wage–price system. With regard to terminology, I refer to the aforementioned condition as “sharing,” and apply it to different types of entities, such as competitive types, admissible wage–price systems,
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world production activities, and world net products. However, the concept is defined primarily for competitive types. We say a competitive type is sharing when the (set represented) type includes all country numbers or, in the language of bipartite graphs, all countries have linking edges. If a competitive type defined in this way is sharing, then we say that the respective entities are sharing. In the same spirit, we say various entities are “covering” if the competitive type in the set representation (i.e. the list of commodity-wise subsets of countries) has no empty subset, or when all commodities have at least one linking edge. Finally, if a competitive type is covering and sharing, we say the type is “spanning.” The question now is how a Ricardian economy is related to subtropical geometry. In reality, the three lines connecting apex V and the vertices of the unit simplex in Figure 2 form a subtropical hyperplane. Note that the “Y”-lines in Figure 2 can be interpreted both as subtropical lines and as a subtropical hyperplane. The hyperplane interpretation better suits our purpose, because the entity separates the total space into exclusive domains. The subtropical hyperplane can be mapped to a tropical hyperplane in a tropical projective space using a logarithmic function. Figure 3 is an example of subtropical hyperplane arrangements. Here, the competitive types are fine types of tropical-oriented matroids. Many problems on specialization patterns can be translated into the language of tropical mathematics. However, a Ricardian economy cannot be reduced to a specific type of tropical geometry, because it has an intrinsic structure that is not usually associated with tropical geometry constructions. We discuss this in Section 7 and later sections.
§5. Conjugacy between wage rates and prices We have so far been focusing on wage rates. Let us now change our eye onto prices (of commodities). This has been much more conventional approach and it deserves examining this possibility. Focusing on wage rates, we are lead to use min-times semi-ring. When we focus on prices, we are lead to use max-times semi-ring. If we observe and compare these two subtropical formulations, there emerges an interesting symmetry between wage rates and prices, which I now call conjugacy. The max-times semi-ring is also a set of all positive numbers, with two operations, and
⊙, defined as follows: 12
⊕
max
a ⊕max b = max {a, b} and a ⊙ b = a ・b. We can construct a subtropical geometry based on this semi-ring, and matrix multiplication can be defined in a similar way to that of the min-times semi-ring. Consider the (M, N) Ricardian economy E = { A = (a ij), q= ( q i) }. The wage–price system (w, p) = (w 1, w 2, ... , w M; p 1, p 2, ... , p N) is admissible when the following conditions hold:
w i a ij ≧ p j, ∀ i ∈ [M] and j ∈ [N]. This condition can be transcribed into subtropical matrix multiplication form, as follows:
⊗
w ≧ p max B, where B is a matrix in which the (j, i) element is 1/a ij. The symbol
⊗ is used to indicate this
is a matrix multiplication in subtropical algebra. Matrix B is the transposed matrix of reciprocals. Here, I use the expression
⊗max in order to distinguish it from the ⊗ operation
in the min-times semi-ring. Note that the latter must be written as
⊗min if we use both
operations. If the equation w ＝ p
⊗
max
B
(5-1)
holds, every country i has a commodity j = j (i ), such that
w i = p j / a ij
and
p j / a ij ≧ p k / a ik ∀k ∈ [N].
This means that every country has an industry in which employers can pay the maximal wage rate. During the discussion when we were preparing Fujimoto and Shiozawa (2011-12), Fujimoto hinted that the condition (5-1) must be equivalent to the condition (3-3). It seemed doubtful for me but, as he insisted it, I worked a week or so if his contention was true and finally found that Fujimoto was right. This fact led me to a discovery of conjugate relations between wage rates and prices. Starting with wage rate vector w, the condition that it is sharing is essential. Then, starting with price vector p, the condition that it is covering is essential. If these conditions are fulfilled, then the following equations hold: w = w ⊗min A ⊗max B, p = p ⊗max B ⊗min A.
(5-2) (5-3)
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Conversely, these conditions can be used to search for sharing wage vectors and covering price vectors. Moreover, a couple comprising a sharing wage vector w and a covering price vector p, connected by these equations, makes a kind of saddle point (see Shiozawa, to appear, Lemma 4.8). Changing from the min operator to the max operator changes the meanings of the frame simplex and the appearance of subtropical hyperplanes. A point of the frame simplex (i.e. frame triangle in the case of N = 3 ) now expresses relative prices. If the fame simplex of Section 4 is to be called wage simplex, the present frame simplex is to be named price simplex. Vertices of price simplex are of course labeled by commodities (instead of countries in the case of wage simplex). The hyperplane for max-times semi-ring also separates the simplex into three domains of different types, just as in the case of subtropical hyperplanes with the min-times semi-ring. The only difference is that separating segments are now extensions of the lines that connect apex V and the vertices. As a result, each domain is a quadrilateral rather than a triangle (see Figure 4). In Figure 5, another subtropical hyperplane is added. This corresponds to a (2,3) Ricardian economy, and the label in each open domain corresponds with a competitive type. However, in this case, the first and second entries indicate the commodities for which country A and country B can offer the highest wage, respectively. d3
d3
Max-times hyperplane
Max-times hyperplane arrangement and competitive types
u(A): (123,1) u(B): (2, 123) C: (23, 13) (3,3) (3,1)
C (2,3)
V
u(A) u(B) (1,1)
d1
(2,2)
d1
d2
Figure 4
(2,1)
d2
Figure 5
The conjugacy that appears between wage rate vectors and price vectors is an example of wider conjugate relations. The following is a list of major conjugate relations: main indices
Countries
commodities
value vectors
wage rate vector
price vector
algebra
min-times semi-ring
max-times semi-ring
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competitive types matrices
Sharing
covering
A
B
If we are only considering economic meanings, it will be difficult to imagine that countries and commodities or wage rates and prices can have interchangeable roles between them. This fact is revealed because we have reformulated trade problems in a highly abstract way.
§6. Hyperplane arrangements and spanning core In this section, we return to Figure 3 and Figure 5. Because there are several subtropical hyperplanes, we call this a subtropical hyperplane arrangement or subtropical arrangement in short. Formally, a subtropical hyperplane arrangement is a set of subtropical hyperplanes. Although their definition is simple, hyperplane arrangements (either in classical or tropical sense) are interesting entities from the point of view of both pure theory and applications. In a Ricardian economy, a subtropical hyperplane arrangement with full commodity apexes (in the case of a min-times semi-ring) and with full country apexes (in the case of a max-times semi-ring) determines the range of wage rate vectors and price vectors that can be associated with a self-replacing steady state. Let us call the collection of these sets (in both the min-times and the max-times cases) the spanning core. This terminology reflects that the competitive type at a point is spanning (i.e., sharing and covering). From Figure 3 and Figure 5, we can easily determine the spanning cores. In the case of Figure 3, the domain with inscription (1, 3, 2) is conspicuous. In the case of Figure 5, there is no such open domain, but the spanning core is not empty. In fact, all points on the segment u(A)C and u(B)C have spanning competitive types. For example, the point C, which is the unique crossing point of two hyperplanes, has competitive types (A,2), (A,3), (B,1), and (B,3). Here I use the indices A and B to explicitly denote these as country labels. Determining a competitive type of a point not inside an open domain is straightforward if we know the competitive types of the domains surrounding the point, because the type is simply the union of all competitive types of the surrounding domains. Points other than those on two segments are not spanning. For example, a point in the extension segment of u(A)C toward the lateral line d2d3 has the competitive type {(A, 2), (A,3), (B, 2)} and is not spanning. Thus, the spanning core of the hyperplane arrangement of Figure 5 is the point set indicated in Figure 6.
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In the same way, we can determine the competitive core of the hyperplane arrangement for a wage simplex. Figures 7 shows the spanning cores for the hyperplane arrangement shown in Figure 3. The core comprises not only points in the domain that has (1,3,2) type but also all points on the segments between v4, v2, v3 and the domain.3 d3
e3
A spanning core in a price simplex: (2, 3) Ricardian economy case
A spanning core in a wage simplex: (3,3) Ricardian economy case
C
v1 u12
u(B)
v2
u23
u(A)
u13
v3 d1
e1
d2
Figure 6
e2
Figure 7
How can we characterize the spanning core? Although defining the spanning core as a set of points that have spanning types may be clear, calculating the competitive types for all points is not straightforward. One method is to use equations (5-2) and (5-3). It has been proved that any point satisfying equation (5-2) has a sharing type and, therefore, is spanning. This is because the competitive types of any points in a wage simplex are covering. In the same way, any point satisfying equation (5-3) has a spanning type. Another characterization of the spanning core is to use the subtropical convex hull concept. Let V be a set of points in a wage simplex. The convex hull of the set of generators V is defined as sconvV :＝{λ u⊕ μ v | λ, μ∈R+, u, v∈V }.
(5-4)
At first, this definition may seem strange. Classic geometry defines convexity as including all convex combinations of generators. The expression on the right-hand side of equation (5-4) seems to include internally and externally dividing points. Recall that a point in a simplex represents a half-line that starts from the origin. Positive number multiplication only means that we choose any point on the half-line.
3
The two figures show different economies and are not conjugate. 16
When V is composed of two points, (5-4) defines a “segment” that connects the two points. When V is composed of three points, (5-4) defines a subtropical triangle. Figure 8 shows the subtropical segment that connects points u and v. Interestingly, the two hyperplanes cross only at point r. It is kinked at a point r, because ⊕ is a coordinate-wise operation. In a higher dimensional space, a subtropical segment may have several kinks. Subtropical hyperplanes in a two-dimensional projective space are actually subtropical straight lines. Thus, the classic geometry axiom that two straight lines cross only at a point is conserved. In the case of subtropical (and tropical) geometry, any two straight lines always cross. In this sense, subtropical geometry is more similar to non-Euclidean elliptic geometry. e3
e3
Segment which connects two points u and v
A subtropical triangle
v
v
r
u e1
Figure 8
t
r
u e2
e1
w s e2
Figure 9
Subtropical triangles take various forms. Figure 7 shows one of them. Figure 9 shows another example of a subtropical “triangle,” which is the classic hexagon. There is a third way to characterize a spanning core. Consider Figure 3, which is a hyperplane arrangement that we can view as a set of various parts. The parts contain zero-dimensional points, one-dimensional segments, and two-dimensional polygons. In higher dimensions, we would have higher dimensional parts, but we do not consider those here. We call these parts “faces” following the naming custom of polyhedral subdivisions. The formal definition of a face is the set of points that have the same competitive type. A face is internal if it is contained in the interior of a simplex. The interior of a simplex is the set of points whose coordinates are all positive. By this definition, no different faces have a point in common. We can say that a hyperplane arrangement is composed of faces, each of which has a different fixed type. In another definition, a face is a closed set which comprises all boundary points of the above defined face. A proposition (Develin and Sturmfels, 2004, Lemma 10; Joswig, 2009, Proposition 4) confirms that the points of a fixed type of (sub)tropical arrangement are not only tropically convex, but also form the relative interior
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of a polyhedron in the classic sense. Note that polyhedrons in tropical geometry correspond to polytopes in a subtropical simplex. Here, the boundedness of a polyhedron can be interpreted as the property of lying in the interior of the simplex. If a face of a subtropical hyperplane arrangement lies in the interior of the simplex, or if it is bounded in the above sense, it has been proved that the competitive type of the face is spanning (Develin and Sturmfels, 2004). As a consequence, the spanning core is the collection of all bounded faces of the hyperplane arrangement. The hyperplane arrangement in Figure 3 is composed of (0) 6 points, (1) 15 segments, and (2) 10 polygons. It has 31 faces in total. The spanning core is the set of all faces (or, more precisely, the set of all points that belong to faces) that lie completely within the interior of the simplex. Thus, segments extending to vertices, domains adjacent to a lateral of the simplex, and domains approaching one of the vertices are excluded. Then, the interior faces comprise (0) 6 points, (1) 6 segments, and (2) 1 domain. The above usage of the term “boundedness” is not so strange if we consider the points on the boundary of the simplex as points at infinity. We can safely say interior faces or bounded faces interchangeably. It is amazing that a combinatorial property, such as a spanning type, is fully characterized by being in the interior or by being bounded. In a (3,3) Ricardian economy, an interior domain is unique if it exists. More generally, in an (N,N) Ricardian economy, there is only one interior open domain, if it exists (Jones, 1961). This topic has not been examined in any depth in tropical arrangement analysis. We return to this issue in Section 10. Ricardian trade theory economists have been particularly interested in interior open domains for historical reasons (Shiozawa, 2014, Chapter 4). However, it is easily shown that when N > M, there are no such domains. Because it is evident that the number of commodities is far bigger than the number of countries, this aspect of interior open domains is not well researched.
§7. Minkowski sum and the world production set Earlier, we defined a Ricardian economy as a set of labor input coefficients and labor forces. However, labor forces have not appeared in our analysis until now. This is because we have been working only on determining wage rates and prices. Another important topic of the trade economy is the world production possibility set of an economy. Labor forces play an
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important role in the determination of the world production possibility set. The production possibility set of a Ricardian economy E = (A, q) is not given in a formula using matrix A, either in classic or subtropical matrix multiplication. In a Ricardian economy, the world production possibility set, or world production set, is the Minkowski sum of the production possibility sets of all countries. Note that in a Ricardo–Sraffa economy, where inputs are traded, this proposition does not hold. We begin by defining the production possibility set of a country. The production possibility set of country i is defined as follows:
P ( i ) = { y= (y h ) | ∑j=1N y h・a ij ≦q i, y h≧0 ∀j∈[N] }.
(7-1)
A Minkowski sum is defined on two or more sets of vectors. Each vector in one set is added to each of the vectors in the other sets. The individual results are the elements of the Minkowski sum. If P (1), P (2), ... , P (k) are subsets of a vector space, the Minkowski sum P is defined by
P = { y | y = ∑h=1k y(h) , ∀ y(h) ∈ P (h) }, and is denoted as
P (1) + P (2) + ... + P (k).
s ij. This means that
A production scale vector s is a collection of non-negative numbers production technique ( i,
consuming s ij
j ) is in activity level s ij, producing s
ij
of commodity j and
a ij of labor in country i. Using production scale vectors, the world production
set can be written as
P = { (∑i=1M s ij ) | ∑i=1N s ij a ij ≦ q i, s ij ≧ 0, ∀i ∈[M ] and ∀j ∈[N ] }.
The world production set is the Minkowski sum of the production sets of all countries. By this definition, it may seem that world production set is rather simple. In reality, the Minkowski sum can be highly complex. However, the production set of each country has a simple structure. As definition (7-1) shows, the maximal production of commodity j of country i is ηj = q
i
/ a ij. In this case, all other production is zero. Hence, country i 's
production set is spanned by ηj e(j) and the origin 0, where e(j) is the unit vector in the j-th coordinate. This is sometimes called a prism4. The Minkowski sum of a set of prisms is called a cephoid (Pallaschke and Rosenmüller, 2004; 2007). Thus, world production sets are cephoids. The world production set as a whole has some simplifying features. First, the world production set P is included in a non-negative orthant. In other words, 4
This word may also have other meanings. 19
ｙ ∈ P
⇒ y ≧ 0.
Another prominent feature is that P includes the origin 0. The intersection of a non-negative orthant and a neighborhood of the origin in R+M is also included in P. Let S be a non-empty subset and E be a subspace of RN, which is composed of points satisfying conditions x j = 0 for all j ∈ S. Then P ∩E is the world production set of a Ricardian economy. Indeed, if we delete commodities included in S, we obtain such an economy. Thus, the question of determining the shape of a world production set is reduced to studying the maximal frontier Fr, namely the set of points of P in a positive orthant and that are maximal in the order ≦. Formally,
Fr = { y | y ∈ P , ∄ x ∈ P such that y ≦ x }. A point on the maximal frontier is called a maximal point of the world production set. The following theorem characterizes these maximal points. Theorem 1 (Characterization of a maximal point of the production set) Let E = { A, q } be an (M,N) Ricardian economy, F the maximal frontier of the world production set, and d a point in Fr. There exists an international value v = (w, p), i.e. a couple of a positive wage rate vector w and a positive price vector p, that satisfies two conditions: (i) w ⊗ A = p. (ii) 〈 w, q 〉= 〈 p, d 〉. Conversely, if there exists a value (w, p) that satisfies conditions (i) and (ii), and output d with production scale vector s = (s ij) satisfies the condition that
s ij > 0 only if w i a ij = p j , then d lies in the maximal frontier Fr.
□
An equivalent theorem, although expressed differently, is given in Shiozawa (2007, Theorem 5.2), and applies to a wider Ricardo–Sraffa economy. Two conditions (i) and (ii) implies that 〈 p, y 〉 ≦ 〈 p, d 〉
(7-2)
∀ y ∈ P,
where P is the production possibility set. This is proved as follows. A production scale vector s that gives output y satisfies the condition ∑j=1 N s ij a ij ≦ q i for all i. Form condition (i), we have inequality p j ≦ w i a ij for all ( i, j ). Using these relations and the fact that s ij is non-negative, we have 〈 p, y 〉 = ∑j=1 N p j ( ∑i=1 M s ij ) = ∑i=1 M
≦ ∑i=1 M ∑j=1 N w i a ij s ij ≦ ∑j=1 N w i q i
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∑j=1 N p j s ij
= 〈 q, w 〉= 〈 d, p 〉. The last equality is the condition (ii). If the price vector p satisfies the inequality (7-2) for all production point of a polytope P, p is said to be normal to P at point d. Condition (i) means that the wage-–price system is admissible. Then, we can define the competitive type of the value (w, p). On the other hand, we can define another set of couples of indices (a bipartite graph) that comprises all (i, j) that satisfies
s ij. > 0. This is called
activity type of an output d. It is natural to ask what kind of relationship these two types have. As we will see soon, these two types are not necessarily identical, although they have a close relationship. In fact, we have the next proposition. Proposition 2. If an output d with production scale vector s lies in the maximal frontier of a Ricardian economy and the couple (w, p) satisfies the condition (i) and (ii) of Theorem 1, then
Ts ⊆ Tv, where Tv is the competitive type of (w, p) and Ts is the activity type of s.
□
This proposition is proved by contradiction. Indeed, suppose that there is a couple (i,j) that is included in Ts but not in Tv. This means there exists a couple (i,j) that satisfies
s ij > 0 and w i a ij > p j. Note that to produce a maximal point requires full employment. This means that we have qi = ∑j=1N
s ij.
Then, 〈w, q〉 =∑j =1 N ∑i =1M w i a ij s ij < ∑i =1M ∑j =1 N p i s ij = 〈p, d〉. This contradicts to condition (ii). Therefore, (i, j) ∈ Ts implies (i, j) ∈ Tv. The proposition is proved. The inclusion in the opposite direction does not hold in general. However, if we focus on a face instead of a point of the maximal frontier, we get an equivalence theorem, For the formulation of the theorem, we need some definitions. A face of a polytope P is the common set of P and a half space that includes P (this is the classical concept of faces for polytopes). A half space in RN is expressed as the set H = { x ∈ RN | 〈 p, x 〉≦ c },
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where p is a vector in (the dual of) RN and c is a real number. With c = 〈 q, w 〉, a price vector p that satisfies conditions (i) defines a half space. As this half space contains a maximal point of the production possibility set, the face thus defined is not empty. We define the competitive type of this face by that of value (w, p). The activity type of a face F is the maximal activity type of production scale vectors that output a point of F. This maximal activity type is given by any point of the relative interior of face F5. With this preparation, we can state the following theorem. Theorem 3 (Identity of the competitive type and the activity type of a face) Let E = { A, q } be an (M,N) Ricardian economy and F a face of the maximal frontier of the world production possibility set, then the competitive type of F is identical to the activity type of F.
□
We now return to Theorem 1. If the point d of the maximal frontier is positive, the activity type Ts of the production scale vector s that produces d is spanning. In fact, as d is positive,
Ts must contain a link that connects all commodities. If Ts contains no link that starts for a country i, it means that s ij = 0 for all commodity j. Then it is impossible that country i satisfies full employment condition. So, the type Ts is spanning. In view of proposition 2, the competitive type Tv is spanning, because Ts ⊂ Tv. When we know all spanning faces of the wage simplex or price simplex, we also know all the faces of the maximal frontier. Suppose a face of the wage simplex or price simplex has a spanning type T. Then, with w or p contained in the face, together with its conjugate, we have an admissible wage–price system (w, p) and, by virtue of Theorem 1, a face in the maximal frontier. As the face has the spanning activity type, it has a positive production point. This is possible only if the face is not totally included in the border of the maximal frontier. Let us call this type of face interior face of the maximal frontier. Then, for each spanning face of the wage simplex or price simplex, there exists in the maximal frontier an interior face that has the same competitive type. Conversely, let F be an interior face in the maximal frontier and d be a point in the relative interior of F. Vector d is necessarily positive, because it is in the relative interior of an interior face. Then there is a value (w, p) that satisfies conditions (i) and (ii) of Theorem 1. Theorem 3 implies that (w, p) has the competitive type that is identical to the activity type of The relative interior of a face F is the interior of F in a space that includes F as full dimensional subset. Note that the interior of a subset included in a space whose dimension is less than that of the whole space is always empty.
5
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d. As d is positive, the competitive type of (w, p) is spanning. Therefore, we get the following theorem, which gives us the basis of our further analysis. Theorem 4. (Type preserving correspondence between spanning and interior faces) Let E = { A, q } be an (M,N) Ricardian economy. There is a competitive-type preserving bijection (or one-to-one correspondence onto) between the set of spanning faces of subtropical arrangement, either in wage simplex or price simplex, and the set of interior faces of the maximal frontier.
□
Theorem 4 can be illustrated by the following examples. First, refer to Figure 3 once again. The interior open domain has the competitive type (1, 3, 2) in commodity-wise list. The commodity-wise list takes the same form (1, 3, 2.). Linking edges are (1,1), (2,3), and (3,2). This means that country 1 produces commodity 1, country 2 produces commodity 3, and country 3 produces commodity 2, all at full capacity. In this case, world production reduces to a point P. The wage rates and corresponding prices can move freely within certain limits. This means that P is an extreme point of the production set. As another example, consider in Figure 3 the crossing point C of e1 v2 and e2 v1. The competitive type of point C is {(1,1), (2,2), (2,3), (3, 1), (3,2) }. The bipartite graph of this type is shown in Figure 10. In the language of graph theory, terms nodes, edges, and paths are more often used, but we prefer using vertices, links, and chains when we speak of graphs of a competitive type. All vertices have a link and the bipartite graph is spanning. There is no cycle in the sense that we cannot trace the links and arrive at the same vertex without passing an identical link twice (formal definition of a cycle or cyclic chain will be given in Section 8). Another remarkable fact is that the graph is connected. As we will see in Section 8, these two facts (i.e., no cycles and connectedness) insure existence of a unique wage–price system (w, p).
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Country number
Commodity number
1
1
2
2
3 3 Figure 10 We can determine the shape of the face that corresponds to point C. Country 1 produces commodities 1 at full capacity, country 2 produces commodities 2 and 3, and country 3 produces commodity 1 and 2. The production of country 1 is fixed, whereas countries 2 and country 1 produce segments that are not parallel. Then, the Minkowski sum of these sets forms a parallelogram. Similarly, vertex v1 forms a triangle, because the competitive type in country-wise list is (1, {1, 2, 3}, 1). Country 1 amd 3 produce commodity 1 and countries 2 produces commodities 1, 2 and 3. As a whole, the maximal frontier is covered by three triangles and three parallelograms. A point between vertex v1 and point C has the competitive type (1,1), (2,2), (2, 3), (3, 1). This gives a segment that forms a common border of a triangle and a parallelogram.
x3 Spanning core of a (2,3) Ricardian economy
eA
v1
v2 v3
B A
x2
x1
Figure 11
Figure 12
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eB
Describing the world production that corresponds to Figure 3 is a bit complicated. Let us take a simpler case of (2,3) Ricardian economy shown in Figure 5 and 7. Figure 11 gives a perspective in the positive orthant that roughly corresponds to the subtropical arrangement of Figure 5. Figures 5 and 6 are given in price simplices. The same economy can be expressed by the spanning core in wage simplex. Conceptual relations are expressed in Figure 126. There are two triangles. The first represents the triangle where country A produces commodities 1, 2, and 3, and country B produces commodity 1. The second represents the triangle where country B produces commodities 1, 2, and 3, and country A produces commodity 2. The parallelogram on the top shows that country A produces commodities 2 and 3 and country B produces commodities 1 and 3. Link relations of zero-dimensional faces in a subtropical arrangement provide information on the adjacent relations. Point C is connected by segments to points u(A) and u(B). This signifies that the parallelogram is adjacent to triangles A and B, separately. A facet of a polytope is a face with codimension 1, or with dimensions N－1 where N is the dimension of the whole space. The maximal frontier is covered by a finite number of facets and completely determined by these facets. The above situation in the previous paragraph illustrates the more general following proposition. The set of 0-dimensional faces of the subtropical arrangement of an economy (i.e. vertices and crosspoints in the case of 2-dimensional simplex) gives all corresponding facets of the maximal frontier. Details will be given in Section 8. Competitive types of various domains of a subtropical arrangement give us information on the shape of maximal frontier. By the bijection of Theorem 4, each domain corresponds to a vertex of the maximal frontier. For example, the domains at the bottom with types (1,1), (1,2), and (2,2), in bipartite graph representation, give the vertices of the following productions: (i) both countries produce commodity 1; (ii) country A produces commodity 1 and country B produces commodity 2; and (iii) both countries produce commodity 2. Note that in all cases, production is at full capacity. Case (i) gives a vertex on the x 1-axis, case (ii)
give a vertex on the x 1
x 2 plane, and case (iii) gives vertex on the x 2-axis.
As is well known, a convex polytope can be defined in two ways. The first method gives the 6
Proportions are not exact. 25
vertices and defines the polytope as a convex hull of these vertices. The second method gives half-spaces and defines the polytope as a common set of those half-spaces. From the domain types, we have the first representation, and from the types on the zero-dimensional faces, we have the second representation7. Converting from the first to the second representation, and vice versa, requires rapidly increasing computation time that grows with the number of vertices and facets (Fukuda, 2004). Thus, we can see how it takes time to describe the types for all domains and vertices, because it is equivalent to describing both representations.
§8. Pattern of specialization and graphs of competitive types A pattern of specialization associated to an admissible international value v = (w, p) in a (M,
N) Ricardian economy E = {A, q} is given by the competitive type T of value v. As v is admissible, w i a ij = p j if ( i, j ) ∈ T and w i a ij > p j otherwise. The latter case means that country i cannot produce commodity j without loss. Examination of pattern of specialization is now decomposed into two parts. (1) How is the international value of a economy determined? (2) What kind of properties does the corresponding competitive type have? The first question is closely related to the world final demand. If we suppose that the world final demand d is on the maximal frontier Fr, the international value given by Theorem 1 is a good candidate for stable values. If the value (w, p) does not satisfy the two conditions, one of two situation results: condition (i) is violated or condition (ii) is violated. First, suppose that condition (ii) is violated, i.e. 〈 p, y 〉 ≠ 〈 w, q 〉, but condition (i) holds. Taking the first and the last but one member in the derivation of (7-2), we get an estimation 〈 p, y 〉
≦ 〈 w, q 〉. Combining the two, we have 〈 p, y 〉
