Levels of Generality

In discussing the various types of function, wc have without explicit notice introduced examples of functions that pertain to varying levels of generality In certain instances, we have written functions in the form y - 7 y = 6x + 4 y = x2 - 3x + 1 (etc.)

Not only are these expressed in terms of numerical coefficients, but they also indicate specifically whether each function is constant, linear, or quadratic. In terms of graphs, each such function will give rise to a well-defined unique curve. In view of the numerical nature of these functions, the solutions of the model based on them will emerge as numerical values also. The drawback is that* if we wish to know how our analytical conclusion will change when a different set of numerical coefficients comes into effect, we must go through the reasoning process afresh each time- Thus, the results obtained from specific functions have very little generality.

On a more general level of discussion and analysis, there are functions in the form y — a y = a + hx y = a + hx -f ex1 (etc.)

Since parameters are used, each function represents not a single curve but a whole family of curves. The function y = a, for instance, encompasses not only the specific cases y = 0, y = 1, and y = 2 but also y = y — -5,..., ad infinitum. With parametric functions, the outcomc of mathematical operations will also be in terms of parameters. These results are more general in the sense that, by assigning various values to the parameters appearing in the solution of the model, a whole family of specific answers may be obtained without having to repeat the reasoning proccss anew.

In order to attain an even higher level of generality, wc may resort to the general function statement y = /(x), or z = #(x, y). When expressed in this form, the function is not restricted to being either linear, quadratic, exponential, or trigonometric—all of which are subsumed under the notation. The analytical result based on such a general formulation will therefore have the most general applicability. As will be found below, however, in order to obtain economically meaningful results, it is often necessary to impose ccrtain qualitative restrictions on the general functions built into a model, such as the restriction that a demand function have a negatively sloped graph or that a consumption function have a graph with a positive slope of less than 1.

To sum up the present chapter, the structure of a mathematical economic model is now clcar. In general, it will consist of a system of equations, which may be definitional, behavioral, or in the nature of equilibrium conditions/ The behavioral equations are usually in (he form of functions, which may be linear or nonlinear, numerical or parametric, and with one independent variable or many. It is through these that the analytical assumptions adopted in the model are given mathematical expression.

In attacking an analytical problem, therefore, the first step is to select the appropriate variables—exogenous as well as endogenous—for inclusion in the model. Next, we must translate into equations the set of chosen analytical assumptions regarding ihe human, institutional. technological, legal, and other behavioral aspects of the environment affecting the working of the variables. Only then can wc attempt to derive a set of conclusions through relevant mathematical operations and manipulations and to give them appropriate economic interpretations.

T Inequaliti-es may also enter as an important ingredient of a model, but we shall not worry about them for the time being.

Chapter

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