On Comparative Dynamics

Abstract

Lately, there has been an increased interest in stability of growth paths, see e.g., Brock and Scheinkman [1974]. The problem has been stated in terms of properties of stationary paths. In order to appreciate the difficulty of the general stability problem, one must realize that there are two types of "time" involved in the analysis: stability "time” and path “time.” Thus, the appropriate mathematical field is that of differential equations defined on a space of functions rather than a finite dimensional space. Naturally, if one restricts one’s attention to stationary paths, then the usual stability analysis is appropriate. However, we would be then discussing the asymptotic behavior of the asymptotic state of the economy. This note strives to put the problem of path stability in the proper perspective by discussing the much simpler problem of comparative dynamics. Unfortunately this term has been used in the economic growth literature to discuss the basically comparative statics problem of comparing stationary growth paths. By comparative dynamics, we mean the determination of the “direction” of change in the optimal path of decision variables due to a change in the exogenous variables. The traditional method of deriving comparative statics results has been to use second order conditions for optimality. However, if one is willing to assume concavity, these results could be derived in a more direct way by utilizing the fact that a differentiable concave function lies below its tangent plane. We shall use this concept in deriving the main inequalities of this note. By way of motivation, we first derive two inequalities of comparative statics. Then we derive the comparative dynamics results and finally we discuss some economic theoretical examples.