# US Gross Domestic Product (GDP) per capita

Real US Gross Domestic Product(GDP) per capita (average annual income in the United States adjusted for inflation) appears in Figure 1. This article offers some interpretation of that plot. An Appendix includes an R Markdown vignette, which provides a discussion of how to download these data and produce this and related plots using R (programming language) in a way that should be completely reproducible, even for people who start without knowing R.
As of 2020-11-21 the Wikipedia article on "Measuring economic worth over time" mentioned only one online data source: MeasuringWorth.com They provide real GDP per capita for the US since 1790, for the UK since 1270, Spain since 1850, and Australia since 1789. We focus here on the US, but it should be relatively easy to modify the vignette in the appendix to consider the UK, Spain and Australia or to consider other models.
The Discussion section at the end of this article suggests that the spectacular performance of the US economy during the administration of Franklin Roosevelt (the steep rise at roughly 60 percent of the way through the image in Figure 1) was due primarily to their willingness and ability to ignore budget deficits and spend whatever it took to put people to work and to win World War II. This was combined with effective measures to limit inflation, which has accompanied other wars of comparable magnitude but not World War II. In particular we suggest that it was not the special circumstances of the Great Depression and World War II that created this spectacular growth with relatively little inflation: No other period in US history has been accompanied by such spectacular growth with such small inflation. Rather we claim that those special circumstances gave the Roosevelt administration the political support they needed to run large deficits while aggressively preventing price gauging by major companies, in spite of the traditional wisdom that such deficits yield financial disasters.
Advocates of Modern Monetary Theory insist we can do this again: They claim that the US or any other nation with its own sovereign currency can have faster, more stable, and more broadly shared economic growth with a job guarantee. The 1933-1944 Public Works Administration (PWA), the 1933-1942 Civilian Conservation Corps (CCC), and the 1935-1943 Works Progress Administration (WPA) provide successful models. Efforts to control inflation today would likely be substantially different from the Office of Price Administration that seems to have made major contributions to the productivity improvements seen during the war.

## Data checking

The simplest naive assumptions with a time series like that in Figure 1 is to assume that the changes from one year to the next on a log scale are normally distributed with a constant mean and standard deviation. To check those assumptions, we plot those changes; see Figure 2.
Figure 2 has two vertical scales. The axis on the left gives changes in log(real GDP per capita). The axis on the right gives the same numbers as proportion (percentage) changes. The two are very similar, because by Taylor's theorem, $\log(1+p)=p+O(p^{2})$; the difference between the two scales shows the extent to which this log transformation is nonlinear over this range.
This plot seems quite interesting. It suggests that the standard deviation may have been gradually increasing from 1791 to 1946, after which the standard deviation may have been much smaller, while the growth rate may have been slowing down slightly.
Are these annual growth numbers normally distributed?

## Normal probability plot

One of the most sensitive ways to evaluate the possible normality of a distribution is by creating normal probability plots: Normally distributed observations look very close to a straight line with the slope proportional to the standard deviation. Mixtures of normal distributions look like line segments with different slopes. Figure 3 presents two normal probability plots of the growth numbers of Figure 2 with the numbers since 1946 in red and the numbers to 1946 in black. The red line look fairly straight. The black line suggests a possible mixture with 7 percent of the slowest growth (less than -0.05) and 16 percent of the fastest growing numbers (above 0.05) having a larger standard deviation. The structural break analysis below suggests 5 epochs, which would make this a mixture of 5 components. However, the post-World War II recession lasted only 3 years (1945-47), and two other epochs (1930-33 and 1934-44) lasted only 4 and 11 years, nearly matching the Herbert Hoover (1929-32) and Franklin Roosevelt (1933-44) administrations, respectively. We could redo this analysis with 3 or 5 components, not just 2. However, we won't bother with that now.
Another standard assumption is that the economic growth in one year is statistically independent or at least uncorrelated with the growth increments in later years. Let's check that.

## Autocorrelation

Figure 4 displays the autocorrelation function of the annual growth in real US GDP per capita since 1790 [first differences in log(GDP per capita)]. This uses the same data plotted in Figure 2 and analyzed in Figure 3. The correlation between the growth rates in two successive years in 0.29. That's highly significant. The analyses below assume this correlation is 0. There are procedures for considering this excessive correlation, but they are not as easy to use. We will focus here on doing what is simpler and easier first, recognizing their deficiencies. More careful modeling will be postponed for future work.

## A data-driven structural break analysis

Literature on data-driven structural break analysis is conveniently summarized in the strucchange package for R. Figure 5 shows the Bayesian Information Criterion (BIC) and the residual sum of squares for models assuming the mean change was constant within epochs for between 1 and 11 epochs (0-10 breakpoints). The best model (i.e., minimum BIC) gives 4 breakpoints: 1929 ,1933, 1945, and 1947. Figure 6 adds the means for the resulting 5 epochs to the growth numbers in Figure 2.

## Epochs

A review of Figure 1 suggests five different epochs, consistent with the breakpoint analysis with Figure 5:
* Up to 1929.
* The Herbert Hoover administration, 1929 to 1933.
* The Franklin D. Roosevelt period, 1933 to 1945, including the Great Depression and World War II.
* The post-war recession, 1945 to 1947, during which the economy declined at essentially the same rate as it had during the first two years of the Hoover administration.
* Post-war growth continuing to the most recent data available as this is written (2019).
Two other conceptualizations may be consistent with Figure 1:
* A four-epoch model that ignores the post-war recession, combining it with the growth that has dominated the US macroeconomy since.
* A six-epoch model that ends the post-war growth in 2008, followed by an epoch we might call "the Great Recession and since".
Saez and Zucman (2020) suggest an alternative six-epoch model, splitting the post-war period in 1980: "From 1946 to 1980, average per adult national income rose 2 percent a year, one of the highest growth rates recorded over a generation in a country at the world’s technological frontier. Moreover, this growth was widely shared, with only the income of the top 1 percent growing a bit less than average. ... From 1980 to 2018, average annual growth in per adult national income falls to 1.4 percent a year." The Measuring Worth data plotted in Figure 1 show GDP per capita, i.e., per person. It's not obvious how to get data on average per adult national income, but it seems unlikely that its behavior would be dramatically different from Figure 1.
For present purposes, we assume that the changes in log(real GDP per capita) from one year to the next are all statistically independent with mean and standard deviation that are constant within epoch but change between epochs.
Table 1 summarizes statistics computed from these different epochs. For each period of years, we give the mean and standard deviation of the changes in the natural logarithms of real GDP per capita from one year to the next. The change in the logarithms is essentially equivalent to the percentage change. We consider changes in logarithms or percentage changes rather than changes in dollars, because the percentage changes are more comparable between the 1790s and the 2000s with the average incomes in the latter period being almost 50 times those of the 1790s. We also give the log(normal likelihood) assuming the variance is either the same across epochs (constant variance or homoscedastic) or changes between epochs (is heterscedastic). We do this, because standard statistical theory says that that the "most powerful" tests of additional parameters in more complicated models generally come from the likelihood ratio, which is equivalent to computing the differences in the log(likelihoods), and because log(likelihood ratio) is generally approximately distributed as chi-square with degrees of freedom equal to the difference in the number of parameters estimated.
Table 1. Alternative epochal periods considered :
| | epoch | epoch | change in log(real GDP per capita) | change in log(real GDP per capita) | log(normal likelihood) | log(normal likelihood) |
| model | start | end | mean | stdev | homoscedastic | heteroscedastic |
| (constant) | 1790 | 2019 | 0.017 | 0.042 | 1674.38 | 1674.38 |
| (all) | 1790 | 1929 | 0.015 | 0.039 | 254.67 | 255.70 |
| (all) | 1929 | 1933 | -0.084 | 0.053 | 6.65 | 6.68 |
| (all) | 1933 | 1945 | 0.078 | 0.063 | 14.50 | 16.60 |
| 4 | 1945 | 2019 | 0.017 | 0.029 | 149.64 | 158.53 |
| 5, 6, 6s | 1945 | 1947 | -0.082 | 0.073 | 3.00 | 3.09 |
| 5 | 1947 | 2019 | 0.02 | 0.022 | 152.36 | 173.3 |
| 6 | 1947 | 2008 | 0.021 | 0.023 | 128.58 | 144.99 |
| 6 | 2008 | 2019 | 0.011 | 0.016 | 24.04 | 30.42 |
| 6s | 1947 | 1980 | 0.023 | 0.026 | 67.98 | 73.67 |
| 6s | 1980 | 2019 | 0.017 | 0.017 | 84.53 | 103.21 |
Table 2 summarizes Table 1 by model: 4-, 5-, or 6-epoch model plus the Saez and Zucman 6-epoch model.
Table 2. Model summaries :
| | log(normal likelihood) | log(normal likelihood) | p(heterscedastic) |
| model | homoscedastic | heteroscedastic | p(heterscedastic) |
| 4 | 425.46 | 437.51 | 2.4E-05 |
| 5 | 431.18 | 455.37 | 7.8E-10 |
| 6 | 431.44 | 457.48 | 5.0E-10 |
| 6s | 431.33 | 458.95 | 1.1E-10 |
The estimated standard deviations in Table 1 range by a factor of 4 between 0.016 and 0.063. In addition, the significance probabilities of the tests for whether these differences can be attributed to chance are all quite small; the largest one being 0.000024 (written 2.4E-5 in the table). That tells us that if we don't get the same answers in comparing the constant variance and the heterscedastic models, we should generally consider the heteroscedastic comparisons more credible.
Table 3 summarizes the comparisons between models.
Table 3. Model comparisons :
| model | model | log(likelihood ratio) | log(likelihood ratio) | significance probability | significance probability |
| smaller | larger | homoscedastic | heteroscedastic | homoscedastic | heteroscedastic |
| (constant) | 4 | 1248.92 | 1236.87 | 0 | 0 |
| 4 | 5 | 5.72 | 17.86 | 7.2E-04 | 1.7E-08 |
| 5 | 6 | 0.26 | 2.11 | 0.48 | 0.12 |
| 5 | 6s | 0.15 | 3.58 | 0.59 | 0.028 |
This says that the 5-epoch model fits the data substantially better than the 4-epoch model, but neither 6-epoch model provides a substantively better explanation than random variability.
Figure 7 adds the average annual percentage growth in real GDP per capita within epoch to Figure 1.

## Discussion
### The Great Depression and World War II

The standard interpretation of the spectacular growth of real US GDP per capita of the US economy during the presidency of Franklin Roosevelt is that it was due to the special circumstances of the Great Depression and World War II and therefore does not include lessons that can be replicated.
The proponents of Modern Monetary Theory (MMT) insist that the phenomena portrayed in Figures 1 and 2 were due to two things:
* The willingness of the federal government to ignore budget deficits and spend however much money that was required to provide productive meaningful jobs to everyone willing to work.
* The Office of Price Administration established by Executive Order 8875, 1941-08-28, to combat inflation responding to the increase in demand from the war raging in Europe and Asia, even though the US was not officially in the war until the day after the Japanese attack on Pearl Harbor, 1941-12-07.
The special circumstances of the depression and the war gave the Roosevelt administration the political support it needed to implement these two policies. This is important, because it supports the MMT claims that any nation with its own sovereign currency, like the US, can have faster, more stable, and more broadly shared economic growth with a job guarantee.
To make that work, we also need solid procedures to limit inflation. That's a critical issue in the US today, because the US in recent decades has allowed mergers and acquisitions beyond reason.
Excessive mergers and acquisitions are problems, because they increase the risk of inflation. This is particularly true if increased employment led to increased demand. When there is only one major supplier, that monopolist can easily increase prices, claiming their costs have gone up when the opposite may actually be the case. The same occurs with an oligopoly, i.e., a market dominated by a very small number of producers: Each producer knowns that they can increase prices without a substantive loss of market share and will likely not gain enough incremental market share to justify maintaining prices lower than their their major competitor(s).
This is less likely to happen with more competition in the market. And that, of course, is a major reason companies want to merge.
Meanwhile, the mainstream media have a conflict of interest in honestly reporting on any merger involving major advertisers. As a result, major mergers may be mentioned in the media without sufficient information that would allow consumers to understand why they would likely be harmed by the merger. Thus, a job guarantee would likely be more effective if it were accompanied or preceded by substantive media reform.

### The Post-War Recession

A web search for literature on the post-war recession visible in Figures 1 and 2 produced nothing. It looks huge relative to other features in these plots and surely must have been discussed among leading economists at that time. One might expect that it would have been part of the justification for the Marshall Plan, recommended by then-US Secretary of State George Marshall in June 1947 and enacted 1948-04-03. However, it's not mentioned in the Wikipedia article on Marshall Plan as of 2020-11-21.

## Appendix. Companion R Markdown vignette

Statistical details that make the research in article reproducible are provided in an R Markdown vignette on "US Gross Domestic Product (GDP) per capita/Measuring Worth".