## Dynamics and The Correspondence Principle

Dynamics and The Correspondence Principle, Paul Samuelson

The previous discussion concerning the usefulness of the Le Chatelier-Samuelson principle in deriving comparative static theorems, was predicated upon a knowledge of the dynamic properties of the model of the imperfectly competitive firm and the assumption that the firm operated under conditions of stability. Samuelson has named the fundamental relationship between static and dynamic analysis, the 'Correspondence Principle'. In giving this name, Samuelson has drawn, yet again, upon his scientific intellectual capital. As enunciated by the Danish physicist, Niels Bohr, in the 1920s the correspondence principle requires that new theories should also explain all of the phenomena for which a preceding theory was valid. Samuelson's adaptation stipulated that the 'new' theory of dynamics, brought about by examining the behaviour of a model in disequilibrium situations, would adjust over time in such a manner as to restore the equilibrium properties of the 'old' static theory. For this hypothesis of stability to be valid, the slopes of the various functions within the model must conform to certain magnitudes, magnitudes which are subject to empirical corroboration.
By investigating the (sufficient) conditions for stability in theoretical models, it is therefore often possible to obtain the information necessary to determine the sign of the effects of a parameter shift in a comparative statics experiment (Samuelson, 1947).

The correspondence principle, therefore, emphasises the import­ance of the dynamic properties of, for example, a market model with Walrasian price adjustment. If such a market is in dis­equilibrium (see Figure 1b), the market clears through the assumption that excess demand/supply leads to a price increase/fall. It can be demonstrated (Figure 1a) that the sufficient condition for the market to have stable equilibrium is that the absolute value of the slope of the demand function should be less than the absolute value of the slope of the supply function. (Note that this allows for the supply function to be negatively sloped.) In such a case, the comparative static exercise of seeing what happens to price as demand increases, yields a 'normal' positive relationship.

Figure 1a Figure 1b

Figure 1a (Stability Of Walrasian Adjustment)

Figure 1b (Instability Of Walrasian Adjustment)

Does this example of the correspondence principle tie in with the Maximum principle? The answer is yes, in the sense that 'lying behind' the demand curve is the assumption that the consumer behaves as if intent upon maximising his satisfaction.

Samuelson's original contributions to the field of dynamics and maximising have gone far beyond this simple case. One of his most ambitious undertakings has been the attempt to apply the type of optimal control techniques employed in lunar rocket missions (the determination of an optimal trajectory over time, given initial payload, thrust and terminal descent) to the deriv­ation of intertemporal paths of production that satisfy the usual microeconomic efficiency conditions and the intemporal paths of consumption that are optimal in the sense of yielding maximum rates of consumption per head over time. In negotiating the mathematical hazards of intertemporal optimal growth, Samuelson made some interesting conjectures about turnpikes (1958). All Americans know that should Paul Samuelson plan a West Coast lecture tour, beginning in Los Angeles, the short drive from Cambridge to the Massachusetts Turnpike is the only part of the journey, other than the concluding short distance off the highway into Los Angeles that has to be made away from the optimal, uninterrupted path, that comprises a succession of turnpikes and interstate highways. Could the same kind of procedure be applicable to the problem of formulating optimal growth policies in an economy? Samuelson's answer was, in principle, yes;

to develop a country most efficiently, under certain circumst­ances it should proceed rather quickly toward the configur­ation of maximum balanced growth, catch a ride so to speak on this fast turnpike and then at the end of the twenty year plan move off to its final goal . . . as the horizon becomes large, you spend an indefinitely large fraction of your time within a small distance of the turnpike (Samuelson, 1971a).

It would be foolish to pretend that this avenue of Samuelson's work gives rise to operationally meaningful theorems with any degree of ease; the specification problems are daunting, to say the least. Nevertheless, some attempt has been made at examining Soviet economic development in the context of turinpikes. Finally, in this section, it must be pointed out that the determination of stability conditions need not be associated with maximising behaviour. Samuelson's famous demonstration of the interaction between the multiplier and the accelerator (1939) and the conditions under which the time-path of national income would exhibit convergent or divergent fluctuations, is a case in point.