Population Dynamics

Alongside this work on statistics, Samuelson continued to work on the prob­lem of population growth, on which he had begun to work with Marion while he was still at Harvard.^h At the beginning of 1942, Alfred Lotka wrote to say that he had just read Samuelson’s latest paper on stability, and that he presumed Samuelson realized that his system of equations was similar to one he had used in Elements of Physical Biology (1925).³⁵ Lotka also claimed that he had already discussed in print the relationship between the Le Chatelier Principle and stability.³⁶ He also noted not only that he had analyzed similar problems but also that Alfred North Whitehead, whom Samuelson knew in the Society of Fellows, had many years before asked him whether such sys­tems might not be applicable to economics.

Later that year, Samuelson became interested in the antecedents of his work on population dynamics. He had been reading the work of Robert Kuczynski, the author of a large number of works on demography in the 1930s, and had become convinced that one of Kuczynski’s historical claims was wrong. This was the claim that the Russian statistician, Ladislaus Bortkiewicz, had derived a theorem showing that a population subject to

h. See chapter ii this volume. constant fertility and survival conditions would eventually approach a stable age distribution with an exponential rate of growth. Samuelson conjectured, in a letter to Lotka in July 1942, that what Bortkiewicz had actually done was to show that this was true of a specific numerical example.³⁷ Lotka was delighted that Samuelson had asked him about this, because his view was that Bortkiewicz had shown nothing of the sort: indeed, there was nothing new in his calculations, and he (Lotka) had been the first person to show that a population with given birth and mortality rates would converge on a fixed age distribution.³⁸ Familiar with developments in the 1920s, Kuczynski had, Lotka claimed, unconsciously read into Bortkiewicz things that were not there.

This letter made Samuelson believe that he was finally becoming clear about the confusions in the literature, and that he had been “thrown off by the fact that Kuczynski had misinterpreted Bortkiewicz more than [he] had realized.”⁴⁰ However, Samuelson argued that though Kuczynski had misunderstood Bortkiewicz, he had stumbled on a valid theorem that was independent of assumptions about fertility. To clarify this, Samuelson listed four theorems, all based on constant survival functions. The first two were straightforward: that a population in which births grow exponentially will achieve a stable age distribution in which population is also growing expo­nentially; and that if the age distribution is stable, the population must be growing exponentially. The third theorem was the important one that Lotka had derived in 1911—that any population with constant mortality and age­specific fertility would approach exponential growth. This left the fourth theorem, which Samuelson believed he was the first to have proved: that if population is growing exponentially, and mortality is constant, births must eventually grow exponentially. Samuelson then summarized the confusions in the literature in terms of these four theorems, asking Lotka whether he was correct.

Lotka replied with a list of his publications, explaining to Samuelson where he could find the first two theorems stated, repeating the point that neither Kuczynski nor Bortkiewicz had discovered anything new (he obvi­ously agreed with Samuelson’s statement about the third theorem). Then, rather than challenging Samuelson’s claim about his own theorem, Lotka explained why he thought it was of limited practical value, “at least as applied to human communities.”⁴¹ It would be necessary to have just the right birth rate.

This exchange with Lotka reveals Samuelson in pursuit of a problem that must have seemed to him to be an interesting piece of mathemat­ics. Population growth had been one of Wilson's concerns, and it was also an important factor behind Hansen's explanation of stagnation. However, Samuelson's continued interest in the problem after the United States had joined the war shows how, even when busy with teaching at MIT, improving his knowledge of mathematical statistics, preparing a thesis for publication, working on the theory of fiscal policy, and organizing a major project in Washington (see chapter 19 this volume), he could not let this mathematical problem go.

Samuelson invested much time in statistics; he came to grips with the latest developments relating to the application of statistical methods to economic problems, and he used such methods to analyze macroeconomic data. These actions represented a major shift in the focus of his work, for at Harvard, despite his emphasis on generating testable hypotheses, he had mostly confined his attention to mathematical economic theory. However, now he chose not to venture further into formal statistical testing of eco­nomic theory. Instead, his research went in two directions. He continued to revise his thesis for publication, and he became involved in the very active debates over Keynesian economics and fiscal policy that were increasingly taking place in Washington, as economists were drawn into wartime govern­ment service. These economists were developing what rapidly came to be called the New Economics, or, as Samuelson was to call it in his textbook, the modern theory of income determination.⁴³ The theoretical foundations of this theory had been laid in the 1930s, but there was much that remained to be sorted out.