The third-world contingency problem

Matrices are used to represent curves on the Cartesian plane. Consider this matrix:

1 0
0 1

It's a 2x2 identity matrix that can be represented on the Cartesian plane in terms of the x and y coordinates.

   x y
x 1 0
y 0 1

On the Cartesian plane (or a graph paper), this can be represented by a line that goes from [0,0] to [1,1] with a slope of 45 degrees. Now, this is a two dimensional matrix where x and y are the dimensions. Let me add a new dimension, z, of depth. Now, if I were to represent the same line in space, it would be

    x y z
x  1 0 0
y  0 1 0
z  0 0 0

To imagine this, think of a graph sheet where a line is drawn from [0,0] to [1,1]. By adding the z dimension, this line is drawn from [0,0,0] to [1,1,0]. To paraphrase, it is a line in a 3-dimensional space with a direction. In the first case, the line was a line: because of the starting point, the ending point and the slope being defined, it was scalar. Now, by adding one more dimension, the line has become a vector. This is a popular notion in mathematics: when an entity exists as a scalar in n dimensions, it will exist as a vector in n + 1 dimensions.

In order to prove this theorem, the origin of the n + 1 vector space must be found. Considering the above 3x3 matrix as a determinant and using Kramer's rule to solve it,

    x y z
x  1 0 0
y  0 1 0
z  0 0 0,

It becomes evident that-

D = 1 0 0
       0 1 0
       0 0 0
Dx = 1 0 0
         0 0 0
         0 0 0
Dy = 0 0 0
         0 1 0
         0 0 0
Dz = 0 0 0
         0 0 0
         0 0 0

Thus, the solution as given by Kramer's rule:

x = Dx / D
y = Dy / D
z = Dz / D

In doing so and substituting the [z,z] value with 0, 1 and -1, the value of the D determinant goes from 1, 1 and -1. In other words, the value of D on the number line goes from one side of zero to the other side. Since the number line is one dimensional, it has 2 directions within the dimension: positive and negative. Therefore, the same line which was a scalar in n = 2 dimensions now becomes a vector in an n+1 vector space because it now has directions as well as magnitude.

This problem, along with its many requirements and conclusions, is used widely used in economics. Before I state the problem, however, there is one more important concept: the Laplace transformation. Every vector space, in order to exist as a vector space, must conform to the basic mathematical operations like addition, subtraction, multiplication and division. If it conforms, then it joins the vector space club.

Say there is a vector space denoted by V5 (i.e., having 5 dimensions). It conforms to the basic operations as well as 5 parameters of its own, three of which are length, breadth and height, one of which could be time, and the fifth one could be, for example, weight. All these dimensions are possessed by, say, a chair.

In order to transform this V5 space into a V6 space, the following rules must be met:

1. The V6 space must conform to the basic mathematical operations
2. 5 of the 6 dimensions of the V6 space must be length, breadth, height, time and weight

The Laplace transformation is pure genius because it allows the transformation from V5 to V6 with the maintenance of 5 dimensions, while the 6th dimension may or may not yet be defined, and allows us to work with a vector space. The transformation, in other words, creates a space out of nowhere given only that the V5 space exists and is defined.

Take the chair. It has dimensions in a V3 space, a time component in a V4 space, and a weight component in a V5 space. In the V6 space, it may now assume a monetary value component.

Now, here is the problem:

Developing nations all around the world have a contingency plan for crises that is defined by 3 conditions: intrinsic non-monetary (Inm) parameter, extrinsic monetary (Em) parameter, and globalization(Glo).

Most economics models today, if not all, deal with the variations in Inm and Em exclusively. However, we know from reality that the variation in Inm and Em are not independent. To whit, if there is no rain during a particular season, the agricultural produce volume falls, agricultural exports fall, the imports increase, inflation within the country rises, the purchasing power parity falls, the income of the earner goes from enough to not-enough and foreign direct investments fall.

So, why can't economic models deal with them together? In mathematical terms, the solution for the [Inm, Em, Glo] vector space is found by excluding the set [Inm, Em], while in realistic terms, the [Inm, Em] space exists very much.

This is done so because of two reasons:

1. The matrix denoting the relation between Inm and Em has too many parameters: there is no one factor that yields a proportionality relationship between Inm and Em.
2. The relation between Inm and Em is a relation and not a function, a function being defined by a set variation pattern. While, one day, Inm and Em might be defined by a function f1, the very next day they become instances of another function, f2.

So, how is an economic model that deals with [Inm, Em] formulated?