Sunday, January 10, 2010

Paul Samuelson on Nonlinear Dynamics

There have been a number of tributes to Paul Samuelson over the past couple of weeks applauding both his intellectual contributions and his character. In his appreciation, Paul Krugman identifies eight distinct seminal ideas, "each of which gave rise to a vast and continuing research literature." An even more comprehensive list of accomplishments spanning six decades may be found in Avinash Dixit's moving eulogy.
One of the articles mentioned in passing by Dixit is a 1939 paper that was published in the Review of Economics and Statistics when Samuelson was just 24 years old. Dixit describes it as the "first workhorse model of business cycles" but that is a bit too generous: earlier contributions by FrischSlutsky, and Kalecki each have a stronger claim. Furthermore, the model in this paper is linear and therefore generates oscillations that are either damped or explosive.
A far more interesting paper by Samuelson appeared a few months later in the Journal of Political Economy. By coincidence, Barkley Rosser mentioned this work in an intriguing comment on Mark Thoma's page just two weeks before Samuelson's death. I recently took another look at the paper and it does indeed contain one of the earliest models capable of generating persistent oscillations without exogenous shocks, thus anticipating the seminal work of Richard Goodwin by more than a decade.
Samuelson took the linear multiplier-accelerator model of his earlier paper and extended it in two ways. First, he allowed for a nonlinear consumption function with the property that the marginal propensity to consume decreased with income, "approaching zero in the limit." Second, he observed that "net investment can only be negative to the extent of deferred replacement or consumption," which necessarily implies a nonlinear investment function. If the steady state is locally unstable, this model generates fluctuations that are bounded and persistent even in the absence of exogenous shocks.
Samuelson recognized the possibility that in his two-dimensional difference equation system "successive cycles need not be similar in timing or amplitude." We now know that highly irregular trajectories are possible even in one-dimensional discrete time models (though at least three dimensions are required in continuous time.) Furthermore, in footnote 7 of the paper, Samuelson made the following cryptic comment:
There remains one interesting problem still to be explored. Mathematical analysis of the nonlinear case may reveal that for certain equilibrium values of α and β a periodic motion of definite amplitude will always be approached regardless of initial conditions. Such a relation can never result from systems of difference equations with constant coefficients, involving assumptions of linearity. This illustrates the inadequacy of such assumptions except for the analysis of small oscillations.
Here Samuelson is not only conjecturing the possibility of a stable limit cycle, but also arguing that the existence of such a cycle may be proved mathematically. In a continuous time model this would be possible using the Poincaré-Bendixson Theorem, but this result has no counterpart in discrete time systems. Hence the existence of a limit cycle in Samuelson's model would have to be demonstrated numerically rather than analytically.
Samuelson's model is outdated in many respects, and one could raise objections to a number of his core assumptions. But the paper does offer a perspective on economic dynamics that stands in sharp contrast to the currently dominant Frisch-Slutsky approach, and is worth reading for that reason alone.


Update (1/11): Barkley Rosser (via Mark Thoma) has more on the subject. My earlier discussion of Buiter, Goodwin, and nonlinear dynamics may also be of some interest; this is the post to which Barkley was originally responding.


  1. Too bad you didn't add the example of a simple non-linear difference equation with the properties you describe.

  2. The simplest is probably the logistic map.