Foundations of Mathematical and Computational Economics

Many believe that mathematics is one of the most beautiful creations of humankind, second only to music. The word creation, however, may be disputed. Have humans created something called mathematics? Then how is it that we increasingly discover that the world, and indeed the universe around us, obey one or another mathematical law? Perhaps to the faithful, the answer is clear: A higher power is the greatest mathematician of all. A worldly answer may be that humans discovered, rather than created, mathematics. Thus, wherever we look, we see mathematical laws at work. Whether humans created mathematics and it just so happened that the world seems to be mathematical, or the universe is a giant math problem and human beings are discovering it, we cannot deny that mathematics is extremely useful in every branch of science and technology, and even in everyday life. Without math, most likely we would be still living in caves. But what is mathematics? And why are so many afraid of it?

We have already encountered two-dimensional numbers and variables in the case of complex variables. Two, three,..., and n dimensional numbers are simply extensions of the concept of a number. For example, we can speak of the length of a string, which would be a one-dimensional number. Or we could talk of the length and width of a page, which would be two-dimensional. A three-dimensional number could represent length, width, and height of a room. Multidimensional numbers are of great importance in all sciences including economics. Such numbers provide economy in exposition and facilitate the manipulation and analysis of complex questions.

In Chap. 8 we saw that by taking the derivative of the total cost function with respect to the amount of output, we can obtain the marginal cost function. Because an integral is the inverse of the derivative, we may ask if the reverse is true. The answer is, yes, almost. By taking the integral of the marginal cost function we can get the total cost function up to an additive constant (indefinite integral). Whereas the mathematical logic for this will become clear later, the economic logic should be evident to the reader. A knowledge of fixed cost is not contained in the marginal cost function, and, therefore, we will know the total cost function, except for the amount of the fixed cost that we will show by the unknown constant C.

In a scene in The Godfather, the late Marlon Brando, playing Don Vito Corleone, tells his son, “Well, this wasn’t enough time, Michael. It wasn’t enough time.” There is never enough time, nor is there ever enough money. In personal life, in the affairs of a company or university, and in the government budget, there are never enough resources, be it time, money, or energy. Economics has always been concerned with optimal allocation of scarce resources to competing and unbounded wants. We can imagine that if resources were infinite or wants were limited, there would be no economic problems. Thus, in deciding the family consumption, the hiring for a university, the number and types of courses to offer in a discipline, the amount of R&D expenditures in the company, the allocation of the federal budget among national defense, education, and welfare programs, we face the same problem of allocating scarce resources to achieve the best result possible. But if the behavior of economic decision makers is determined by choosing the best allocation subject to budget constraints, then the starting point of economic theory of households and firms ought to be constrained optimization.

The rationale behind dynamic models in general and in economic analysis in particular was discussed in Chaps. 14, 15, and 16. The reason for a system of difference or differential equations follows the same logic, except that here more than one dynamic process is at work and, therefore, we need more than one equation to describe these processes. Analysis of dynamic systems is a vast field of inquiry and could be the subject of a multivolume book. Here we confine ourselves to two topics that are deemed most useful for economic analysis: the solution of linear systems of homogeneous differential equations with constant coefficients and qualitative analysis of a system of differential equations using phase portrait.