Working Paper
of the
Graduate Institute of the Americas
College of International Studies
Tamkang University
US Studies Division
October 2008
The Precarious Nature of
Samuelson’s Correspondence Principle
Professor David Kleykamp
The Precarious Nature of
Samuelson’s Correspondence Principle
One of the hallmarks of a good education in economics is when a student shows a close familiarity with Paul Samuelson’s Correspondence Principle (CP). The core assumption of this principle is that economic phenomenon are stable over time and therefore any disturbance to a long run equilibrium will result in an eventual return to a (possibly) new long run equilibrium. The comparisons made between these two long run equilibria are what we call comparative statics. It follows that, without stability of long run equilibria, it would not be not valid to compare stationary points in any dynamic system, which has under gone some parametric change. Fortunately, in some cases, economic models are found to be inherently stable due to highly intuitive theoretical restrictions placed on the parameters in the model. The simple Solow-Swan model of economic growth is a good example, depending as it does on the diminishing marginal productivities of labor and capital.
At other times, the issue is moot whether or not a particular dynamic system is stable. In applying what he had learned from physicists, Samuelson noted that the necessary conditions for stability of long run equilibrium could in some cases be used to determine the qualitative behavior of the related comparative statics. That is, by a priori assuming the stability of the dynamical system under consideration, definite qualitative results could be obtained on the comparative statics. This represented an important methodological breakthrough in economics and it remains today an important tool for economists to use.
Despite the overwhelming acceptance and use of the CP by economists, there is a conceptual problem involved in freely applying it to all cases. Physicists used this principle because they knew that the models they constructed were not generally subject to missing variables. The problems they confronted were usually carefully constrained and highly controlled. Any stability which was posited must naturally be inherent in the system since it was a system complete in itself. By contrast, economic models unquestionably omit many variables, due to the tremendous complexity of the phenomena being studied. And, it is the omission of variables that generate doubts over the advisability of using the CP to sign our comparative statics. That is, the stability (or instability) of dynamic systems in economic models may be due to variables being left out of the model, rather than some inherent property of the model itself or some appeal to the fact that markets in the “real world” are typically stable. Imposing stability on such models that omit key variables can lead to entirely different comparative static results.
A small example may help to clarify the problem involved here.
Suppose that we have a simple linear differential equation representing the reduced form of a much larger model
where we may assume for now without loss of generality that b > 0. A standard application of Samuelson’s CP would first identify the stationary point, which in this case is equal to
Note that b is often referred to as an exogenous parameter and it follows that this long run value of x will now depend on the sign of θ. Suppose next that there is an infinitesimal change in b (say an increase). How will change in response to this change in b? The answer once again depends on the value of θ. If θ > 0 , then < 0. On the other hand, if θ < 0, then the derivative is positive. It is here that the CP comes to our aid; and much welcomed it is. By a priori imposing stability on the long run equilibrium, we are forced to accept θ < 0 and the signing of the comparative statics becomes determinant.
It follows that for this small one variable model, with stability imposed by the CP, we have θ < 0 and > 0.
But, suppose now that we embed this small model within a slightly larger model by adding (in a particularly contrived way) a formerly excluded endogenous variable, y. Suppose that this new 2 X 2 model can be written as
which is just a system of linear differential equations commonly used in economics. Note how that if we completely ignore the existence of y, (i.e., y ≡ 0), then the model reverts to the simple, one variable model above it. Naturally, economists are loath to exclude important variables from their models, but mathematical convenience and the so-called “need to focus” often trump other considerations that might produce a more complete model. What is interesting about the enlarged model is that global stability of long run equilibrium now requires that θ > 0. With respect to the coefficient matrix, the trace is always negative and the determinant is equal to θ. This is precisely opposite of the CP assumption made for the smaller one variable model above! What is worse, the signs on at least some of the comparative statics change. The long run equilibrium values are equal to
and
meaning that < 0, if the system is stable (i.e., θ > 0). Extending models, even by a single variable, can drastically change the stability conditions and comparative statics.
What is more, the mere inclusion of a previously excluded variable can sometimes stabilize the system making it invalid to impose the CP on the system. For example, suppose that we change ever so slightly the above 2 X 2 model in the following way
Note that in this case, the system is globally stable for ALL values of θ since the trace of the coefficient matrix is equal to -1 and the determinant is equal to 1. By adding an excluded variable in a particular way we have completely stabilized the system.
If we viewed y as an “error” term, then we could write the first equation as
and this resembles a simple first order autoregression in x (where the derivative is to be interpreted as a first difference). However, despite its appearance, the εterm is clearly not an exogenous variable. It depends likewise on the behavior of x. It is not simply a residual, anymore than x is a residual. Harvey and others have claimed that an appropriate modeling strategy for econometric time series is to treat ε as a leftover. By contrast, methods such as Box-Jenkins ARMAX and generalized Cochrane-Orcutt see the ε as a perfectly legitimate object to model dynamically. Indeed, it is the ε which will generate an error correction back to stable equilibrium, NOT x alone.
How should we now view the simple, single variable model before? Naturally, we should be suspicious of simply blindly applying the CP to solve our problems. It may be that there are many excluded variables that if only included in the model would render it stable, thus possibly nullifying this use of the CP and the comparative statics that follow from its imposition.
Economists have for too long simply set up models and run through the comparative statics mindlessly invoking the CP without once thinking that the simple nature of their model might very well produce false results. For many models, perhaps all models, it may be that the variables we exclude are precisely the ones that stabilize our systems. Professor Samuelson was quite obviously aware of this, although he apparently thought it insufficiently important to delineate clearly in his Foundations. Unfortunately, not only is the real world more complicated than our theory, but also our theoretical world can be quite unforgiving at times. There is always a cost to simplicity.