optimal growth :Keynes, Ramsey and Bliss


ineteconomics.org   2020  Janeway on Ramsey and Keynes: A Comment     By Lance Taylor and William Janeway


Optimal growth

Nowadays, Ramsey’s 1928 paper on “A Mathematical Theory of Saving” is seen as the launch pad for the theory of optimal economic growth. In fact, the Cold War has the stronger claim. In the late 1950s air forces on both sides of the Iron Curtain developed a strong interest in optimizing flight paths of fighter aircraft.[2] Teams of mathematicians around the RAND Corporation in Santa Monica CA and the Steklov Institute in Moscow took on the task. The latter worked out a reformulation of the classical calculus of variations called optimal control theory based on a distinction between state and control variables (see below). Initial publication in Russian in 1961 (English in 1962) by Lev Pontryagin and others put the new mathematics into the public domain.

Given their fascination with shiny mathematical toys, economists rapidly took up control theory.[3] In 1965, more or less independent papers by David Cass and Tjalling Koopmans introduced optimal economic growth. A publication waterfall followed. Only later was Ramsey’s forty-year-old model brought fully into the picture.

A comment on Janeway is not the place to review optimal growth. But two points stand out, illustrating how even Ramsey and Keynes combined could be wrong.

The growth models involve ordinary differential equations for changes over time of two “state variables” – the capital stock and its asset price (analogs in the traditional calculus of variations are the position of a particle subject to a force, and its momentum). The Pontryagin formulation also includes “control variables” such as consumption (or saving = output minus consumption) which influence the dynamics of the states.

Optimal growth is interpreted as meaning that ”society” or an immortal “representative agent” maximizes the discounted utility of consumption over a perfectly foreseen infinite time horizon. The mathematics says that there is only one time path of capital and its asset price that will solve this problem. Its trajectory will converge to a state of steady growth in which new capital formation equals the sum of depreciation per unit of capital and the growth rates of labor productivity and population.

Along all other paths capital will diverge toward zero or infinity. For an initially given capital stock the asset price (or the level of consumption) has to be specified exactly to get the economy on the optimal path. The fact that selecting a single number from a continuum is beyond mere human agency is not widely discussed, even though the models are being used to analyze contemporary issues such as climate change.

An exogenous positive “pure” rate of discount enters standard calculations. One justification is that future generations may be richer than we are; another is that people consistently undervalue future in comparison to present consumption. More practically, a solution to the model will not exist unless the pure discount rate exceeds the labor force growth rate.

As Janeway observes, Ramsey rejected a pure discount rate. To get his model to converge, he came up with an alternative – at some level of consumption its utility saturates with marginal utility falling to zero. He called this situation Bliss. At Bliss with constant consumption, no depreciation and zero productivity and population growth, the profit rate would be zero.

Dynamics of Bliss

Shortly after Ramsey’s untimely death, Keynes published an article called “Economic Possibilities for Our Grandchildren” in the Yale Review. The gist is that capital accumulation and labor productivity growth will boost output per capita over generations, allowing less work and much more leisure for the masses. In the GT a few years later, he foresaw the euthanasia of the rentier due to a falling rate of profit. Textual evidence is scant, but both projections are consistent with an economy at Bliss.

Abstracting from climate change (if that makes any sense), what sort of path should the economy follow to get there?

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