Economic modelling by lump algebras

Math1Comput. Modelling, Vol. 10, No. 5, pp. Printed in Great Britain. All rights reserved

ECONOMIC

0895-7177/88 S3.00+0.00

389-393,1988

Copyright Q 1988Pergamon Press plc

MODELLING

BY LUMP

ALGEBRAS

S. TAUBER Department of Mathematics, Portland State University, Portland, OR 97207, U.S.A. (Received

December

1985; accepted for publication

Communicated

February

1988)

by X. J. R. Avula

Abstract-The elements of an economic entity are written in vector form, the exchanges are expressed by matrices. By generalizing the Leontieff matrices the exchanges are represented by transform matrices in a way similar to chemical reactions, The model is then used to represent the merger of entities, thus introducing a scator structure.

1. INTRODUCTION

The use of linear algebra in the modelling of economic problems is certainly not new. Through the introduction of new algebraic structures one may expect to give a further impetus to this kind of modelling. In several papers [l-5] the author has used such structures to model some flow and exchange problems in chemistry, hematology and air pollution analysis. In the present paper an attempt is made to apply these methods to economic systems. The idea is justified by the flowing character of goods and money in economic systems.

2.

BASIC

NOTATIONS

Let (E) be an economic entity, i.e. a structure that exchanges with its environment goods in the form of merchandise, information or money. The goods may be transferred by (E), transfer with gain of some kind, they may be transformed, and then resold, or they may be used up. Let A = [A (l), A (2), . . . , A(n)] be the ordered set, written as a vector, of all the goods which (E) deals with in one form or another, and let B = [B(l), B(2), . . . , B(n)] be the amounts expressed in convenient units of A (k), 1 < k < n, involved in the operations. If several operations are to be considered separately then SZ= [Q(l), R(2), . . . , Cl(p)] is the set of operations, written again as a vector and B,,, 1 < h
= Bh. C,, = k B(h, m)C(h, m=l

In case the individual extends for each A(k)

WW)l=B,~G=P(k

m).

(1)

are needed we can write the extend vector as

l)C(h, l>,B@,2)C(h,‘&. ..,B@,n)C(h,n)l,

where use has been made of the Hadamard product of two vectors [cf. 61. 389

(2)

Brief Notes

390

3. L- AND T-MATRICES Consider (E), A and B, as defined in Section 2, and assume that (E) involves basically the transformation of each of the A(k) into other A (k)s. The matrix corresponding to the transformation of the unit quantity of each A(k), written in the form Lu = /I

(3)

whereuT=[l,l,..., 11 and B’= [8(l), B(2), . . . , /3(n)], IL being the transpose of II, is called the Leontieff matrix of (E), for short the L-matrix of (E). In a more general vein we shall define the T-matrix for (E). Assume (E) to be a plant using certain of the A(k), and producing others. Let Bi be the input vector, i.e. the quantities of A absorbed by (E) during unit time, or for a unit extend at an instant of given C. Let B,, be the output vector of (E), i.e. the quantities of A produced by (E) for given Bi. The T-matrix (transform matrix) is defined by the relation B, = %Bi.

(4)

We shall show how the matrix U can be constructed.

U is actually the sum of three matrices

%=O+P-Q,

(5)

where 0 is the identity matrix, i.e. 0 = [i(m, k)], i(m, k) = d,,, = 0, if m #k, 6,,,, = 1, if m = k; P is a matrix with positive terms representing the produced amounts, - Q is a matrix with negative terms representing the absorbed amounts. To clarify, consider the simplest case of four components, symbolically writing A(I)+A(2)+A(3)+A(4),

(6)

meaning that A (1) and A (2) are used to produce A (3) and A (4). Clearly, A=[A(l),A(2),A(3),A(4)1

and

B=PJ(l),B(2),B(3),B(4)1.

We write symbolically P(l)IA

(1) + P(2)lA (2) + P(3)lA (3) + P(411A (4).

(7)

It is convenient to use one of the As as the reference substance, and take the correspondent B(k) = 1. This can be considered as normalizing the symbolic equation (7). In order to differentiate between the coefficients on the 1.h.s. and the r.h.s. of the equation we shall use a prime for the ones on the r.h.s. Let us then assume that there are f equations of the form (7) which we can write as

wherej=1,2

,...,

m$, [B(j, m)lA (m) + i tB’(j, m)lA (m), m=I

(8)

B( j, m)B’( j, m) = 0;

(9)

f,and

since A (m) cannot appear on both sides of equations (8), if it appears on the r.h.s., B’( j, m) # 0, B(j, m) = 0; if it appears on the l.h.s., E( j, m) # 0, B’( j, m) = 0; if A (m) does not appear in the jth equation, B( j, m) = B’(j, m) = 0. Let A (ri) be the reference substance of the jth equation of equations (8). We then normalize the relations by writing i

P(j,m)lB(j,r,)lA(m)

+ i

P’(j,m)lB(j,

rj)lA(m),

(10)

m-l

m=l

wherej=1,2 ,..., f. It can be easily verified that with the notation so introduced the elements of the matrices P and Q are given by the following equations: j=

(11) I

391

Brief Notes

and q(m,k) = k [~(_M(_L m)lB(.L

j=l

(12)

~jaq.k,

where 6 represents the Kronecker delta, and o(j) represents the proportion of A (rj) absorbed in the jth equation of equations (8). Following our general tendency we define the vector

o=[0(1),O(2),...,o(f)l.

(13)

In toto the elements of the matrix U will be given by the following expression: t(m,k)=&.,+P(m,k)-q(m,k). Since all the elements of the vector (which could be called the proportion of the system. It is clear that amongst the different depreciation, maintenance, overheads

(14)

B, must be positive, the input vector B and the vector o vector) must be chosen properly to satisfy the conditions components etc. 4.

of the set A may appear such items as labor,

PROFIT

From the preceding section it follows that if C is the vector of unit prices (in the case of a production) of A then the total cost of the production will be C * Bi, and the total revenue of the production will be C . B,, so that the profit (all calculated for a unit time period) will be Pr = C. B, -C.

Bi = C. [(O+ P - Q;P)Bi-Bi] =C. [~ - aS]Bi.

(15)

For given C, and Bi the profit will depend on the matrix P - Cl, in particular its maximum can be obtained by playing on the vector o. The determination of o usually leads to a problem in linear programming. 5.

MERGER

AND

ADDITIVITY

Consider first two economic entities (E,) and (E2), and the problem of their merger. We define for each entity its A vector, say A(1) = [A(l, l), A(l, 2), . . . , A (1, nr)l

and

A(2) = [A (2,1), A (2,2), . . . , A (2, ndl.

There are several possibilities: (i) n, =n2 =n, and A(l,k) A( 1) = A(2) = A.

=A(2,k)

Vk, such that

1 Sk
then

simply

(ii) n, en,, or n2
,...,

A(l,n,),A(l,n,+l),...

. . . A (1, d, A (2, n3+ 11,. . . , A(2, dl; A has in this case n3 + (n, - nj) + (n2 - n3) = n, + n, - n3 = n elements. (iii) n, # n2, and A(l,k)#A(2,h) A=[A(l,l),A(1,2)

Vk, h such that 1
and 1
, A (1, nd, A C&l), A (2,2), . . . , A (2,

dl;

A has in this case n = n, + n2 elements. In all three cases we end up with a vector A of n elements. We associate to each A (1, k) and A(2, k) a B(1, k) and B(2, h). In case (i): B(1, k) # 0 and B(2, k) # 0, for 1 B k d n; but B(l, k) and B(2, k) are not necessarily equal. In case (ii): B(1, k) # 0 and B(2, k) # 0, for 1 < k G n,; B(1, k) 2: 0 and B(2, k) = 0, for n3 < k < n,; B(1, k) = 0 and B(2, k) # 0, for n, < k d n,. In case (iii): B(l,k)#O and B(2,k)ZO Vl dk
Brief Notes

392

We have thus for A the set of all items involved in both (E,) and (E2), with two vectors B(1) and B(2), giving the quantities involved in (E,) and (E2) during a unit time. For practical reasons both Bs will be output vectors, although they could be input vectors in certain special cases. To represent (E,) by B(1) and (E,) by B(2) only would be rather superficial. To make the representation more flexible we introduce an additional scalar s(l) for (E,) and s(2) for (E,). This scalar can depend on a number of conditions such as the geographic, historic aspects of the entities, their relative ways of functioning etc. We thus represent both entities by a scalar and a vector which we call a scalar-vector or scator. We have already mentioned the use of stators in the Introduction. We thus write symbolically S, = [s(l),

BUN,

s, = b G9,W)l

(16)

and assuming a merger of (E,) and (E2) we write S, + S, = S, = [s(3), B(3)].

(17)

The laws of addition of stators in simplified form without specification of analytic conditions are

~(2K

s(3) =f[s(l),

B(3) = g]s(l), s(2), B(l), KU,

f[S(l),S(2)1=f[s(2),s(l)l

(18) (19)

and g[s(l),s(2),

B(l), B(2)1=g[s(2),s(l),B(2),B(l)l.

(20)

Stators are said to be linear if f[s(l),

s(U= ~(1)+ ~(2) and gb(l),s(2),.W), WY= ab(l>, a(dlB(1) + bbU), s(WV).

According to equation (2) it follows that in the case of linear stators b [s (l), s (2)] = a [s (2), s(l)], so that for linear stators

(21) and &(l),s(2),B(l),

B(2)1=a[~(l),s(2)1B(1)+a[~(l),s(2)1B(2), (22)

where a(p, q) is a given scalar function. If several stators are added, which would correspond to the merger of several entities, the addition is commutative, but not in general associative, i.e. S, + [S, + S,] is not in general equal to [S, + S,] + S,. In the case of linear stators it can be proved that the sum is associative iff the function a@, q) satisfies the condition a&, 0) = g(u)c”/g(a

+ u),

(23)

where g(u) is an arbitrary function such that g(0) # 0, and c is a positive constant. As far as the technical applications of stators are concerned the only stators so far applied in flow problems have been the so-called linear elementary stators or line1 stators: in this case a[s(l), SW1 = s(l)/[s(l) + s(2)]. Line1 stators in which the parameter s and the vector B are functions of a scalar variable (usually time) have derivatives. It can be shown that if s = s(t), B = B(t), then [cf. 61 -$S[s,B]=S

$B+$; [

,1 dt

(24)

i.e. the derivative of a scator is a scator. We summarize by saying that stators can be used to model economic entities and their mergers. We shall see in the next section that the merger model can be refined by using other algebraic structures somewhat more complex than stators.

393

Brief Notes 6.

LUMPS

When several parameters in a mathematical discussion are considered as a whole we say that they are lumped together and are following an algebra of their own. The simplest example of a lump is the n-tuple, an ordered set of n elements considered a vector, and following the laws of vector algebra. The next lump to be considered is an n x m matrix of which a special case, i.e. an n x 1, or a 1 x n matrix is in fact a vector. Stators defined in Section 5 are lumping together scalars and vectors, and therefore lumps for which we have so far defined addition only, although much more can be done to create a complete scator algebra. Considering again two economic entities, as in Section 5, we characterize them now by an output vector and several scalars instead of only one. We lump these scalars into another vector and thus characterize the entity by two vectors; one of them is the output vector B, the other one is a set of ordered scalars v, and we call the total lump a double vector. (We cannot use the term bivector since such a structure has already been used in another context.) We thus establish an additive law for double vectors as follows: B [v, &,I +

Cb, Cl = D [u,DoI,

where u = u(v, w) and D, = D,,(v, w, B,, C,) can be chosen to satisfy the modelling requirements of the problem. The merging model can be improved by replacing in the lump the output vector by the input vector and the T-matrix. We obtain a new structure containing a scalar, a matrix and a vector. We call this new structure a vector-matrix, or vectrix. Thus an entity (E) is to be represented by a vectrix I/ such that I/ = V[s, T, Bi]. Considering again the problem of the merger of two economic entities (E,) and (E,) we have I’, = I’l[s(l), a(l),

f’*= J’2is (2),T(2), 8 (2)1,

&(l)l,

(25)

where in general the elements of the two vectrices are different. For each entity B, = TB,, and s has the same meaning as in the case of the scator representation. We clearly have the relationship V[s, 8, Bi] = S[s, B,] = V[s, 0, B,], between vectrices and stators, but we can do better in the merging process by writing I’, + I’, = I’s = I’3[~(3), T(3),

&(3)1= Ss[s(3), B,(3)],

(26)

where s(3) =f[s(l),

sm

T(3) = G[s(l), s(2), T(l),

TG91,

Bit31=H[s(l), ~(213Bi(l),Bi(2)1,

(27)

where the T(3)-relation will establish the essential part of the merger since it gives in matrix form the resulting work of the new entity. It is clear that more complex relationships can be expressed by introducing new lumps and their algebras as needed for the modelling problem at hand. REFERENCES 1. S. Tauber, Application of algebra to concentration calculations. Atmosph. Enuir. 11, 571-573 (1977). Br. Hydromech. Res. Ass. Fluid Engng Absrr. No. 61926. S. Tauber, An algebraic model for blood concentrations. Res. Commun. Chem. Path. Pharmac. 30(3), 409-417 S. Tauber, Lumped parameters and stators. Math. Modelling 2, 227-232 (1981). S. Tauber, An algebraic model for blood chemistry. Math/ Modelling. 4, 11l-l 16 (1983). 6. S. Tauber, Algebraic models for merging systems. Inc. J. gen. Syst. 11, 79-88 (1985).

2. 3. 4. 5.

(1980).