The Stability of Dynamic Models - Eckaus, R. S. (The Review of Economics and Statistics, 1957)
COMPARATIVE dynamics is a relatively unworked but highly significant area of economic analysis. It is important to appreciate the implications of changes in the structure of a dynamic system and, thus, the extent to which the analytical results depend on the particular, restricted forms employed. These are the problems with which comparative dynamics deals in general 1 and with which this paper is centrally concerned, with particular respect to the stability of difference-equation models of income determination. Dynamic difference-equation models of economic fluctuations have been useful tools for the economist. Though not intended to provide, in themselves, a complete description of aggregate economic activity, such models have led to a better appreciation of cycle-producing forces and the character of cyclical movements. These models have been developed in a variety of forms based on different assumptions about time sequences and using different types of expenditure functions as components. Inevitably, however, each of the models represents a special case whose analysis provides conclusions of only limited applicability. There has not, I believe, been adequate recognition of the important qualifications which must as a result be applied to many of the existing dynamic models of economic fluctuations. In this paper greater generality will be sought in two ways: first, by the construction of a number of alternative systems based on different assumptions about the lag sequences involved; second, by detailing somewhat more than has been done the expenditure functions which are the components of the models. I believe that it will be possible in this way to provide further insights which are important in themselves and in qualifying previous dynamic analyses. The variety of results which can be obtained will be illustrated rather than treated exhaustively. The procedure of this paper will be to construct and compare a series of alternative models. In section I the general method of analysis will be illustrated in presenting models based on the Robertsonian lag of expenditures behind the receipt of income.2 Section II will begin to break new ground in the analysis of models which use the Lundberg lag of output behind sales.3 Finally, in section III models will be developed using both types of lag. The investigation is confined to models of effective demand in a closed economy. The expenditure functions used are, in general, adaptations of a simple consumption function and the acceleration principle -admittedly gross, but useful, simplifications of reality. Even though thus confined to simple lag relations and spending functions, the analysis becomes quite complicated. After constructing each model, the types of movements which it may generate and the conditions which it must satisfy for stability will be investigated.4 The systems constructed will be subject to the well-known limitations of linear models. It would not be difficult, however, to apply simple types of nonlinear restraints to the systems, and the effects of such restraints will be noted at several places. The mathematical techniques of stability analysis which are the basis of this paper have been introduced into economics by Professor P. A. Samuelson. Their usefulness has, perhaps, not been fully appreciated, and this paper provides an example of their application. A general comment on the use of stability conditions may be in order here. We want to investigate the economic implications of a number of alternative dynamic systems; we know that such systems are able to produce a variety of motions consisting of monotonic and