Functional Equations in Behavioral and Social Sciences

Theories in the behavioral, social, and natural sciences are often formalized by equations involving unknown functions, i.e., by functional equations. For instance, a theorist may be reluctant tomake specific assumptions regarding the form of functions involved in a mathematical model, but the qualitative formulation of the model itself may impose constraints on the initially unknown functions. Such constraints often reduce the possibilities and occasionally are so severe that they restrict the possible forms to a few. A recent general reference to functional equations is Acze l and Dhombres (1989) where references to earlier and still useful surveys are given. In the simplest cases there is one unknown, realvalued, function . An example, named after the famous French mathematician A. L. Cauchy, is the Cauchy equation (x y) (x) (y), where x, y are in the set of real numbers . More restrictive domains are sometimes studied. With no further restrictions, the solutions to the Cauchy equation can be wild. But assuming that is monotonic, or continuous, or bounded over a finite interval, the solution reduces to (x) cx for some constant c. By taking ψ exp , we get ψ(x y) ψ(x)ψ(y), which is the so-called ‘lack of memory property’ and it is easy to see from the previous case that the strictly monotonic solutions are ψ(x) ecx, c 0. A celebrated psychophysical example is the connection between empirical just noticeable differences and G. T. Fechner’s hypothesis that sensation corresponds to subjective differences being equal. By one interpretation of what he meant, one is led to the family of functional equations, named Abel equations after the famous Norwegian mathematician N. H. Abel, η[x g(k,x)] η(x) h(k), where the variable x and the parameter k are in the set of nonnegative real numbers + . For the special case of Weber’s law where g(k, x) (k 1)x, this equation reduces to η(kx) η(x) h(k) η(k) η (1), i.e., with (z) η(exp z) η (1), k log k, x log x, we have (k x ) (k ) (x ), which if k is treated as a variable is Cauchy’s equation. The restriction of strict monotonicity yields η(x) a lnx b, a 0, b constants, as the solution. For references and a general discussion of closely related issue see Falmagne (1985, Chap. 4). Sometimes a functional equation has multiple, qualitatively different solutions. Yet the scientist arriving at the functional equation has the strong intuition that only one of these solutions is really appropriate for the scientific problem in question. Whenever this happens, the challenge is to discover additional behavioral properties that seem to be empirically correct and that serve to isolate the desired solution. A reason for fully determining the several solutions is that the same functional equation may arise in an entirely different empirical context and, for that context, one of the previously unacceptable solutions may be appropriate. So the complete characterization is clearly of interest. As will be seen, functional equations arise in the social sciences in at least three main ways. One occurs when one knows how tomeasure numerically the same attribute in two different ways, which is the case more often than not. Then the two measures are related by an unknown strictly increasing function. An empirical law linking the two underlying measurement structures manifests itself as an equation restricting that unknown function. Functional equations also arise when some invariance condition holds. The Fechner hypothesis above is an example. Others are given later. Economic aggregation problems are a third source of functional equations. Consistent aggregation both rules out some ad hoc aggregation functions and leads to families of functions that are indeed consistent. Despite the intuitive appeal of additive aggregation, that assumption is inconsistent with the most common production functions used in economics. Examples are given of production functions and of permissible aggregation rules.