S1E3: "They do it in completely arbitrary ways"

Discounting in optimal growth theory

This post is part of a series on the history of how economists model the future with the Ramsey formula, based on joint work with Pedro Garcia Duarte. See episode 1 and episode 2. Full Paper here

The evolution of growth theory in the 1950s and 1960s, particularly the reformulation of growth models using intertemporal maximization and dynamic optimization tools, often brings to mind the Ramsey-Koopmans-Cass one-sector growth model with endogenous savings (which historians have argued should also credit Malinvaud). The now-canonical model is built on the principle of maximizing discounted utilities of consumption over some indefinite time. However, a closer examination of the model's development reveals a complex history marked by trials, failures, and persistent doubts about the role of discounting.

In these works, as in any dealing with savings since Ramsey (with exogenous or endogenous savings, one or many sectors), the tension between theoretical consistency, ethical preferences and tractability constraints becomes visible upon closer inspection. The evidence in this post had been meticulously gathered by Pedro in his wonderful article on the history of time discounting in 1960s growth models with an evocative title, “A path through the wilderness” (gated pub version, draft version) Do read it for full analysis of the contributions mentioned here (and more on discounting in multi-sector models and activity analysis).  

An interesting intermediary case is the work of Dutch economist Jan Tinbergen, who essentially reframed public policy in the language of optimal control and built the first macroeconometric model of an economy. A League of Nations expert in the interwar period, he then took his models as he traveled to countless developing countries and advised international organizations. In a 1956 contribution on the problem of optimal savings in developing countries, he considered Ramsey's bliss, then reflected that "there need not be, in principle, any difference between the choice an individual makes and the choices to be made for the nation as a whole" and endorsed a discounted intertemporal utility function. But four years letter, while discussing planning in Asia, he refused to do so, citing both theoretical and ethical reasons. He doubled down in a 1962 coauthored book:

“No discount for future consumption was applied in the belief that for a country’s planning, future generations should count as much as present generations. According to this philosophy, a discount may be realistic for the individual’s plans but not necessarily for a nation’s…Instead of a discount, a finite horizon T may be introduced.”

Finite horizons and bliss were two alternatives to discounting that former Tinbergen student turned Cowles director Tjalling Koopmans and Stanford graduate student David Cass also considered when developing their own model in the early 1960s. Another was simultaneously proposed by Hiroshi Atsumi and Carl Christian von Weitzsäcker. Very roughly, the so-called “overtaking criterion” instructed economists to check whether after some specific date, a consumption path provided greater utility than another at every subsequent date. In this case, an unbounded problem could be turned into a bounded one. Koopmans showed that the criterion failed to produce a complete ordering, but was handy “if one wishes to consider the no-discounting case, for ethical or other reasons.”

Koopmans approached growth models through contributing to the axiomatization of time preferences. He demonstrated that reasonable postulates about the utility function for consumption programs extending over the indefinite future (some unrelated to time preferences), imply that agent are impatient. He also showed how a modifying one of those postulates turns the intertemporal utility function into the sum of future utilities over the indefinite future with a constant discount rate. He thus provided theoretical justification for discounting. While his seminal contribution discussed all the modeling strategies above, and in spite of admitting “an ethical preference for neutrality as between the welfare of different generations,” he eventually settled on the utilitarian objective function with discounting. He explained that such function could indifferently endow an individual or a central planner.

So did Cass in 1965,  briefly arguing that “planning obligation is stronger to present and near future generations than to far removed future generations” to support his choice of an objective function with discounting. In the final paragraph of his paper, he however recognizes that his "(somewhat artificial) positive effective social discount rate glosses over a difficult problem, the proper weighting of future generations in the concept of social welfare.”

During this transitional period, thus, theoretical arguments on intertemporal individual choice and social planners were raised. Ethical commitments were contemplated. After all, some these discussions were held in Vatican, during a 1963 conference on “the Econometric Approach to development planning” at the Pontifical Academy of Sciences (telegram exchanges with the Pope below). But ultimately, tractability prevailed in the collective decision to use positive discounting. The key debate was whether theorists would endorse intertemporal utility maximization over other types of objectives (like output or consumption maximization, productive efficiency), and discounting was, in a sense, a by-product of winning this fight. It was the easiest way to get such objective function to converge in an indefinite future setting.

Economists’ 1963 telegram to which the Pope “deigned to respond”

The tractability appeal of discounting had not escaped observers. One was Richard Bellman, the Carnegie mathematician who proposed the recursive methods to solve those dynamic models, and to whom you, reader, may owe your fame and fortune. While discussing cost minimization over the indefinite future in his book Dynamic Programming (1957), he explained that:

“If we wish to consider an unbounded period of time over which this [cost minimization] process operates, we must introduce some device to prevent infinite costs from entering. The most natural such device is that of discounting the future costs. This possesses a certain amount of economic justification and a great deal of mathematical virtue.”

But to other participants in the growth modeling soul-searching of the decade, such economic justification had been altogether overridden by tractability constraints. While visiting MIT in 1961, Tinbergen’s former student Sukhamoy Chakraborty penned a review of solution to the optimal saving problem. Frustration dripped from the conclusion, where he remarked that both the bliss criterion and the intertemporal maximization of discounted utility “impose some ordering on the utility space, but they do it in completely arbitrary ways. Since our interest lies primarily in the meaningfulness of the order introduced…one cannot avoid feeling that such formulations have very little significance apart from ensuring solvability of the mathematical problem of maximizing a functional.”

But remember, deriving the optimal interest rate wasn’t the endgame of these models: those were aimed at understanding actual and optimal growth path, the role of population, savings, technological progress and more. Guess who, then, brought these models to bear on the pragmatic question outlined in the previous post: how to pick a discount rate in the cost-benefit analysis of public investments?

Next (S1E4). The equation “implicit in Ramsey”: back to public investment