Free Trade
Our next example for Cephoids deals with a more than 200 years old concept in Macroeconomics. This theory is considered to be the basis of Free Trade Theory. It is the model of David Ricardo. Ricardo’s theory establishes a first version of efficiency gains when Free Trade is admitted. as compared to production in autarky. The author is greatly indebted to Wolfram F.
Richter who pointed out to him the relevance of the subject in context with the theory of Cephoids. As a result, the subsequent presentation is a version of [24].
183
1 Ricardian Free Trade
As to the basis for our presentation, we refer to David Ricardos Volume
“On the Principles of Political Economy and Taxation”, see [23]. There are quite a few modern reprints, we also cite McKenzie [14], Jones [11]
or – for a textbook reference – Caves et al. [2]. Ricardo considers a model of two countries (Britain and Portugal), each of them producing two commodities (cloth and wine) but at different production costs (essentially in labor). We exhibit the general structure if one admits an arbitrary number of commodities produced by an arbitrary number of countries – which turns out to be a Cephoid.
We start with a single country producing in autarky. Let I := {1, ...n} denote the commodities and let L denote the country’s supply of labor. Then we set up a “Ricardian” model of production and trade as follows.
Let ˆbi >0 denote the input coefficient i.e., the amount of labor required to produce one unit of commodity i ∈ I. Then ˆai := ˆb1
i > 0 is the productivity of labor with respect to commodity i ∈ I, i.e., the number of units of commodity i that can be produced with the input of one hour of labor.
We writexi for a quantity of commodityi∈I. Thenx= (x1, . . . xn)∈ n+is aplan according to which the amountxi of commodityiis being produced.
This plan results in an aggregate amount Xn
i=1
xiˆbi ,
of labor reqired to produce the vector x. The plan x is feasible if the aggregate demand does not exceed the total supply of laborL. Consequently, the feasible plans are represented by the (deGua) Simplex
(
x∈ n+
Xn i=1
xiˆbi ≤L )
.
We introduce the notation ai := Lˆai, hence bi := a1
i = Lˆ1a
i = ˆbLi is the relative amount of labor neccessary in order to produce a unit of commodity i. Accordingly a= (a1, . . . , an) is the capacity vector.
Then the above feasible set can as well be rewritten as
(1) (
x∈ n+
Xn i=1
xiˆbi ≤L )
= (
x∈ n+
Xn i=1
xiˆbi
L ≤1
)
= (
x∈ n+
Xn i=1
xi
Lˆai ≤1 )
= (
x∈ n+
Xn i=1
xi
ai ≤1 )
=: Πa . That is, we obtain the DeGua Simplex generated by the capacity vector a.
The efficient plans are given by the Simplex
∆a = (
x∈ n+
Xn i=1
xi
ai
= 1 )
which is the Pareto surface of Πa.
An efficient planxis said to be supported by aprice vector p= (p1, . . . , pn) if pxmaximizes th linear funtional x→pxover Πa.
The efficient production plan ai := aiei ∈ ∆a allots all labor available to the production ofone commodityi; these plans representcomplete special-ization of the economy. The total amount of labor available is employed to produce just one commodity i.
Figure 1.1 represents the familiar DeGua Simplex – now reinterpreted as a feasible set of production plans for 3 commodities. The completely special-izing production plans appear as the vertices of the Simplex. Consequently, the lengthai (i∈I) of the line segments from the origin to a vertex represent the capacities.
Recall that input coefficients are given bybi = a1
i. The input coefficients can be seen as the coordinates of a price vector
n := ( 1 a1
, . . . , 1 an
) = (b1, . . . , bn)
which is supporting for all efficent production plans of the Simplex ∆a. For short, n is supporting ∆a. Geometrically, this vector is the normal to ∆a. All price vectors supporting ∆a are multiples ofn, thus up to a multiple, the supporting prices to ∆a are uniquely determined.
We now enhance our model by introducing several countries into the pro-duction scene, we assume that there are K countries each one producing
a1 a2
a2
Πa
∆a
Figure 1.1: Efficient Plans – the DeGua Simplex Πa
the same commodities i ∈ I. Then K := {1, . . . , K} denotes the list of these countries. Each countryk ∈K is characterized by acapacity vector a(k) = (a(k)1 , . . . , a(k)n ) ∈ n. Accordingly, for each country the feasible production plans are provided by the DeGua Simplex Π(k) := Πa(k).
A set of feasible production plans, one for each country, is aproduction plan schedule, that is, a list (x(1), . . . ,x(K)). The aggregate production of the world economy resulting thereby is given by thesum x(1)+. . .+x(K). Thus,
(2) Π = Πa• :=
XK k=1
Πa(k)
=
x(1)+. . .+x(K) x(1) ∈Π(1), . . . ,x(K) ∈Π(K)
is the set of aggregates of production schedules, for short theglobal plans. Obviously, we obtain a sum of deGua Simplices, that is, the aggregate pro-duction schedules constitute a Cephoid.
We recall the sketch presented inSection1 ofChapter1. Now we interpret the first two sketches in Figure 1.2 as representing the situation for two countries and two commodities. Then the feasible production sets for each country are represented by the triangles ∆a(1) and ∆a(2). The global plans appear as the (algebraic) sum of these triangles in 2+, represented by Π.
The Cephoid Π0 is a sum of 4 triangles.
a(1)1+a(2)2
a(2)2
a(1)1 Π
Π0
∆a(2) ∆a(1)
Figure 1.2: Construction of global plans
According to Theorem 2.3 inChapter1, all polyhedral production plans (i.e., compact, convex comprehensice polyhedra with a piecewise linear Pareto boundary) are being obtained as aggregate production plans of a certain set of countries with suitable capacity vectors.
From an Economical viewpoint, the idea of “comparative advantages” is relevant here: not every aggregation of complete specializations results in a vertex of the cephoid, that is in a Pareto efficient global plan. The sum of two vertices is a vertex if and only if they admit of a common normal . That is to say, a production schedule of complete specializations in all countries results in an efficient global plan if and only if there is a common price vector supporting the plan of each country involved.
E.g. in Figure 1.2 we observe that a(1)1 +a(2)2 is Pareto efficient but, say, a(1)2+a(2)1 is not. Even more significant, consider the case that, in autarky, both countries are producing at the center of their capacities, i.e, country k chooses
a(k)1+a(k)2
2 = (ak1, ak2)
2 = a(k) 2
Then the global plan resulting if each country sticks to its production sched-ule would be
a(1)
2 +a(2)
2 = (a11+a21
2 ,a12 +a22
2 ) < (a11, a22) = a(1)1 +a(2)2 , as a21 < a11 and a12 < a22. There are no prices at which countries 1 and 2 can jointly and efficiently produce when each of them chooses the central production plan.
1
2 3
∆a+b1
∆b+a2
∆a23+ ∆b13
Figure 1.3: Incomplete Specialisation
Similarly Figure 1.3 (again taken from Chapter 1) refers to the production of 3 commodities in 2 countries. The resulting cephoid of global plans Π = Πa+ Πbhas a Pareto surface∂Π consising of translates of Simplices (∆a+b1 and ∆b+a2) and the parallelogram or rhombus ∆a23+ ∆b13.
In the first case, (in order to produce efficiently) one country is completely specialized while for the other one any efficient plan is admitted.
In the second the rhombus is a sum of two Subsimplices. Such a Subsimplex, say ∆a23, consist of vectors that are convex combinations of extremals a(2)2 and a(2)3 hence represent production plans involving commodities 2 and 3 only but not commodity 1. This we interprete naturally as a partial or incomplete specialization of the country reflected by a on commodities 2,3 while the country reflected by b partially specializes in commodities 1 and 3.
Generally, if x¯ = PK
k=1x(k) is efficient in Π = PK
k=1Πa(k), then we use an appropriate notation: we say that country k is completely specialized if x(k) =a(k)i ei = a(k)i holds true for some i∈I.
Also, given x, country¯ k is said to be partially specialized if there is a nonempty subset of commodities J(k)⊆I such thatx(k) ∈∆(k)J(k).
Thus, plans at which economyk is partially specialized w.r.t the same subset
J(k)of commodies used in production constitute the subsimplex ∆(k)J(k) ⊆∆(k). For more than two countries and 3 commodities we repeat Figure 2.8 from Chapter 1 which illustrates the situation.
Figure 1.4: Four Countries producing 3 commodities
Forn = 3 the shape of maximal faces does not change. any Pareto face is the sum of either 3 vertices and a full Simplex (the translates of a Simplex) or else the sum of 2 vertices and two straight lines (the rhombi). The vertices involved do not appear explicitly in the geometrical description. Generally, with n commodities and K economies we obtain various patterns of partial specialization.
Within this context we recall the reference vector which describes the great abundance of possibilities: we know from Corollary 1.4 in Chapter 6 that any such vector indicates uniquely a Pareto face, hence a possible market equilibrium and hence the corresponding versions of specialization.
Thus an integer vector r = (r1, . . . , rK) satisfying (3) 1≤rk ≤n (k∈K)
XK κ=1
rk = K+n−1
demonstrates a possible distribution of production among the countries, where economy 1 specializes in r1 commodities, . . ., economy K specializes in rK commodities. All reference vectors appear as possible partial special-izationswhen production takes place “in” a Pareto face or equivalently in the corresponding equilibrium.