Population Dynamics

When Samuelson left Harvard, he had a reputation as a “cooperator.”^a He reported that this was because he had once co-authored an article with one of his friends, but his reputation rested on more than that.

His starting point was the notion of a function, which he stressed did not need to be numerical. He illustrated nonlinear functions using a simple phys­ics example—the motion of a freely falling body. He used this to explain the idea of limits. Algebra, diagrams, and numerical examples were all used. The same techniques were then applied to cost curves and the supply curve of the profit-maximizing company, as well as the effect on output of imposing five different types of tax on a company. These were all essentially exercises in using differential calculus to analyze maximum positions.¹ Having started with just two variables, Samuelson then went through similar exercises with three variables—output was a function of land and labor. Deriving the con­dition for minimum cost, he showed how the second-order conditions could

a. See chapter 15 this volume. be represented as a two-by-two determinant. He stated that the equivalent maximum and minimum conditions for three and four variables involved additional conditions relating to the signs of larger determinants.

These lectures show not only the trouble Samuelson took to assist his friends but also the gap that existed between his own mastery of these mathematical techniques and the much more limited mathematics of his colleagues—even those who, like Stolper, had successfully completed Harvard's graduate pro­gram.

The example of Bergson shows that Samuelson could be generous in cred­iting his friends and in minimizing his own role, refusing to be acknowl­edged as a co-author of work with which he had been involved. One of his most important co-authors was Marion, with whom, at some point in 1938, he began work on the dynamics of population growth. This was a problem of interest both to Wilson (as professor of vital statistics) and, on account of its implications for the business cycle, to Hansen.^b In February 1939, Samuelson sent a copy of a paper he and Marion had written to the statistician Alfred Lotka (1880—1949).² Lotka, born in what is now Poland and trained in math­ematics and physics in Birmingham, England, applied thermodynamic ideas to biological evolution, seeing evolution as a physical law. Wilson had been one of those who reviewed Lotka's Elements of Physical Biology (1925), and the two remained in contact. Like Wilson, Lotka harbored a certain skepti­cism of sociology, once remarking to Wilson that while listening to a long- winded and platitudinous paper, he had drafted a definition: “Sociology is a pseudo-science which develops the faculty of speech at the expense of that of thought,” though hoping that the field as a whole did not merit such an opinion.³ By the time Samuelson got in contact with him, he was a statisti­cian for the Metropolitan Life Insurance Company and was working on vari­ous aspects of population dynamics. Lotka drew Samuelson's attention to a substantial literature on the problem of population dynamics, including his own, and sent him a draft of a paper on “self-renewing aggregates” and the problem of “industrial replacement.”⁴

b.

The paper that Paul and Marion wrote together was titled “A Fundamental Function in Population Analysis.”⁵ Its starting point was the observation that a population was the sum of all births, weighted by the proportion of the population surviving to each age. Thus in 1938, the total population would comprise all births in 1900 multiplied by the proportion surviving to age 38, plus the number of births in 1901 multiplied by the proportion surviving to age 37, plus births in 1902 multiplied by the proportion surviving to age 36, and so on. The problem was that it would be useful to turn the problem round and, starting from knowledge of what the population was in each year, deduce how many births there must have been in each preceding year. Lotka had solved this problem, making special assumptions about the time path of the population, but his result was not completely general. Using some com­plicated mathematics, Paul and Marion derived what they called a replace­ment function: the pattern of births over time that would keep a population constant. That their inspiration may have come from physics is shown by a sentence deleted in the manuscript, “The analogy with the Heaviside step function of electric circuit theory obviously suggests itself.”⁶

Paul and Marion stated that the replacement functions they were calcu­lating could apply to either human populations or to stocks (populations) of industrial equipment. However, rather than discuss specific numbers, they derived two general theorems. The first was that if population growth were exponential (i.e., grew at a constant percentage rate) after a certain date, the number of births must asymptotically approach an exponential form (a con­stant growth rate), and the age distribution of the population would eventu­ally stabilize. The second theorem generalized this result to the case where there were cyclical fluctuations in the population.

Samuelson’s correspondence with Lotka continued, and in March 1939, Lotka said that he thought Samuelson had not provided a good motivation for the statistical problem he was trying to solve.⁷ In looking for the birth rate that would keep the population constant, Samuelson was assuming that when someone died, the person was immediately replaced by a new­born baby. This was not what happened with human populations where the birth rate depended on the age distribution of potential mothers. However, Samuelson’s problem was relevant to the problem of industrial investment, where a worn-out capital good might immediately be replaced with a new one. This comment might have been the motivation for an unfinished paper Samuelson wrote, “A Note on the Net Reproductive Ratio and the Intrinsic Rate of Population Growth,” in which he argued in terms of female births per female of any given age.⁸ In this note he argued in terms of limits rather than precise values, one of the reasons being that the length of a genera­tion was uncertain. Biology determined minimum and maximum ages for childbearing, and the length of a generation could be anywhere between those limits.

In the fall, Samuelson submitted a paper under his sole authorship, “The Structure of a Population Growing According to Any Prescribed Law,”⁹ to the Journal of the American Statistical Association. This paper began by noting that while it was straightforward to deduce the behav­ior of a population from knowledge of births (and other assumptions), it was much harder to go in the other direction. Up until then, he claimed, the problem of finding the birth rate when one knew about the growth of the population had been solved only for certain special cases, such as when a population was growing exponentially (a constant percentage growth rate) or according to a logistic curve (in which the growth rate rises and falls in a specific pattern).

Samuelson's method was to consider a simple case in which population began at zero and then suddenly rose to one, calculating the number of births (a fraction) needed to keep population at this new level.^c It was then possible to analyze any time path of the population: “A population growing according to any law whatsoever can be regarded as made up of a sum of step functions, and the number of births at any time is equal to the sum of the replacements computed for each such step function.”¹⁰ No doubt responding to Lotka's criticisms of his earlier paper, Samuelson explained that his results assumed a simplified model of reproduction, in which all births occurred at the average age of confinement, assumed to be thirty. Whereas conventional estimates showed a slow decline in the rate of population growth, his own methods suggested that population growth was slowing down rapidly and that after i960 it would begin falling.

The editor, Frederick Stephan, showed Samuelson's paper to Lotka, who then contacted Samuelson. Lotka pointed out that he had presented a paper on the same subject at the American Statistical Association

c. This is the Heaviside step function, taken from electrical circuit theory, mentioned previously.

meeting in December 1938, which had been published in June.¹¹ Lotka suggested that Paul might present a slightly different paper, drawing on his own (Lotka's) equation, to the forthcoming meeting of the American Statistical Association.¹² Samuelson was unable to take up Lotka's invi­tation, as he was not attending the AEA/ASA meeting that year, but he explained the motivation for his and Marion's work.¹³ The reason they could not use Lotka's statistical methods was that they would not work when there were periodic fluctuations, as in business cycle data.¹⁴ They had used a different technique, and “much to our gratification” this had led to the same type of equation as used by Lotka.¹5^,d This letter shows the connection that Paul and Marion were trying to make between popula­tion growth and business cycle theory, using a type of mathematics very different from that involved in his more familiar multiplier-accelerator model.^e