Political Calculations
Unexpectedly Intriguing!
December 4, 2020
Movers in Brooklyn - Photo by Handiwork NYC via Unsplash: https://unsplash.com/photos/x6pnKtPZ-8s

Writing in 1996, future Nobel prize-winning economist Paul Krugman confronted the mystery of urban hierarchy:

The size distribution of cities in the United States is startlingly well described by a simpler power law: the number of cities whose population exceeds S is proportional to 1/S. This simple regularity is puzzling; even more puzzling is the fact that it has apparently remained true for at least the past century. Standard models of urban systems offer no explanation of the power law. A random growth model proposed by Herbert Simon 40 years ago is the best try to date — but while it can explain a power law, it cannot reproduce one with the right exponent. At this point we are in the frustrating position of having a striking empirical regularity with no good theory to account for it.

The "simpler power law" to which Krugman refers is Zipf's Law, which was originally developed to describe a power law relationship that was found to exist between the rank of a word and how often it appears in use in language. A similar pattern was observed in the ranking and relative size of cities by population in the U.S., which was described by Xavier Gabaix in a much-cited 1999 paper.

But that exercise was largely based on comparing the relative ranks and populations of cities at a given snapshot in time. Does a similar pattern hold as cities evolve over time?

The answer to that question is "sort of, but not really". If Zipf's Law worked reliably, it could be use to accurately forecast the growth of population in cities over time. It can in a limited sense, but only to the extent that the growth over time is steady and not subject to chaotic conditions.

That's where a new study by physicists Vincent Verbavatz and Marc Barthelemy makes a potentially significant advance in modeling the growth of cities over time. Here's the abstract from their paper (boldface emphasis ours):

The science of cities seeks to understand and explain regularities observed in the world’s major urban systems. Modelling the population evolution of cities is at the core of this science and of all urban studies. Quantitatively, the most fundamental problem is to understand the hierarchical organization of city population and the statistical occurrence of megacities. This was first thought to be described by a universal principle known as Zipf’s law; however, the validity of this model has been challenged by recent empirical studies. A theoretical model must also be able to explain the relatively frequent rises and falls of cities and civilizations, but despite many attempts these fundamental questions have not yet been satisfactorily answered. Here we introduce a stochastic equation for modelling population growth in cities, constructed from an empirical analysis of recent datasets (for Canada, France, the UK and the USA). This model reveals how rare, but large, interurban migratory shocks dominate city growth. This equation predicts a complex shape for the distribution of city populations and shows that, owing to finite-time effects, Zipf’s law does not hold in general, implying a more complex organization of cities. It also predicts the existence of multiple temporal variations in the city hierarchy, in agreement with observations. Our result underlines the importance of rare events in the evolution of complex systems and, at a more practical level, in urban planning.

Verbavatz and Barthelemy find changes in migration flows to be especially significant in their new formulation, where they find a Lévy stable law provides a much better fit with empirical data from France, Canada, the United Kingdom, and the United States. The following figure from the paper illustrates these differences with results that would be expected from Gaussian-based models of growth (based on normal distributions of migration flows):

Verbavatz and Barthelemy (2020), Extended Figure 4 - Migration Flow Analysis

They also note that the Zipf-Gabaix-Gibrat model is not capable of accounting for scenarios where a city's population rank might significantly change within a short period of time. By contrast, their new model can accommodate these kinds of rapid transitions with reasonable accuracy, predicting how a city's population might change in response to shock changes in migration flows.

That's significant because 2020 is providing several potential case studies for the Verbavatz-Barthelemy model, in the form of the rapid out-migrations of people and businesses from cities experiencing break downs in public order as a result of their implementation of progressive political policies, such as New York, San Francisco, and Portland. These are examples of the kinds of turbulent shocks that Zipf's Law cannot handle but which the new general model for population growth in cities would appear to be well suited.


Verbavatz, Vincent and Barthelemy, Marc. The Growth Equation of Cities. Nature, 587, 397-401(2020). [Ungated PDF document]. DOI: 10.1038/s41586-020-2900-x. 18 November 2020.

Gabaix, Xavier. Zipf's Law for Cities: An Explanation. Quarterly Journal of Economics, Volume 114, Issue 3, August 1999, pp 739-767. [Ungated PDF Document]. DOI: 10.1162/003355399556133. 1 August 1999.

Yirka, Bob. A stochastic equation for modeling population growth in cities. Phys.org. [PDF Document]. 20 November 2020.

Krugman, Paul. Confronting the Mystery of Urban Hierarchy. Journal of the Japanese and International Economies, Volume 10, Issue 4, pp 339-418. [Ungated PDF Preprint Document]. DOI: 10.1006/jjie.1996.0023. December 1996.

Newitz, Annalee. A mysterious law that predicts the size of the biggest cities. Gizmodo. [Online article]. 9 December 2013.

Image Credit: Photo by Handiwork NYC on Unsplash

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December 3, 2020

Like the month that preceded it, November 2020 was a strong month for dividend paying stocks in the U.S. stock market.

In the year of the coronavirus recession however, that strength is a relative measure. Compared to previous months, the market appears strong. Compared to a year ago, the market looks like its been whacked, which in truth, it has.

Both observations for November 2020 can be seen in a chart tracking the number of dividend increases and reductions announced in each month since January 2004.

Number of Publicly-Traded U.S. Companies Either Increasing or Decreasing Their Dividends Each Month, January 2004 - November 2020

Here's our tally of the U.S. stock market's dividend metadata for November 2020:

  • A total of 3,125 U.S. firms declared dividends in November 2020, an increase of 93 over the 3,032 recorded in October 2020. That figure is 248 lower than what was recorded a year ago in November 2019.
  • 78 U.S. firms announced they would pay a special (or extra) dividend to their shareholders in November 2020, an increase of 45 over the number recorded in October 2020 and an increase of one over the number recorded a year ago in November 2019. This figure typically peaks in December each year, but that peak is often conditional on investor expectations for the U.S. government's future tax policy.
  • 125 U.S. firms announced they would boost cash dividend payments to shareholders in November 2020, a decrease of 60 from the 185 recorded in October 2020, and a decrease of 25 from the 150 dividend rises declared back in November 2019.
  • A total of 15 publicly traded companies cut their dividends in November 2020, an increase of 2 over the number recorded in October 2020 and also a decrease of 9 from the 24 recorded in November 2019.
  • Just one U.S. firm omitted paying their dividends in November 2020, the same as the number recorded in October 2020. That figure is also a decrease of three from the total of four firms that omitted paying dividends back in November 2019.

Our near real time sampling of dividends, taken from Seeking Alpha's Market News (filtered for Dividends) and the Wall Street Journal's Dividend Declarations database, identified 14 of the 15 reductions during November 2020. Here is the list of firms that announced dividend cuts during the month:

Six of these firms are from the oil & gas sector, three are retail-oriented REITs, which shows the ongoing influence of the coronavirus pandemic on retail-oriented firms in 2020. The remaining firms in our sampling are made up of one firm each from the materials, utility, financial services, insurance and consumer products industries.

Overall, the pace of dividend cuts during the fourth quarter of 2020 to date appears more healthy than the same quarter in 2017, 2018, and 2019, which can be seen in our chart showing the cumulative dividend cuts announced by day of quarter for the current and three previous years.

Cumulative Dividend Cuts and Suspensions Announced by Day of Quarter, 2017-Q4 vs 2018-Q4 vs 2019-Q4 vs 2020-Q4 to Date, Snapshot on 30 November 2020

Previously on Political Calculations

We've been listing the firms that have announced dividend cuts or suspensions from our near real-time sampling of these declarations in our previous editions. Please follow these links to see all the dividend cuts and suspensions we've tracked during the coronavirus recession.


Standard and Poor. S&P Market Attributes Web File. [Excel Spreadsheet]. 30 November 2020.

Seeking Alpha Market Currents. Filtered for Dividends. [Online Database].

Wall Street Journal. Dividend Declarations. [Online Database when searched on the Internet Archive].

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December 2, 2020

Since we last looked at Arizona's experience with COVID-19, the number of coronavirus-related cases, hospital admissions, ICU bed usage, and deaths has continued to rise at an increased pace.

Currently, the available data does not yet show any change related to social mixing during the Thanksgiving holiday. But the data does continue to show the impact of social mixing the back calculation method indicates coincided with political campaign events that occurred within Arizona during the period from Friday, 23 October 2020 through Sunday 25 October 2020.

Hospital Admissions

Arizona's new hospital admissions datacontinues to provide the some of the clearest data for determining when events occurred to change the incidence of exposure to the SARS-CoV-2 coronavirus. Our first chart shows the available data from 3 March 2020 through 30 November 2020, with the rolling 7-day average shown through 23 November 2020.

Daily COVID-19 New Hospital Admissions in Arizona, 3 March 2020 - 30 November 2020

In the chart, we've added a linear trend line to indicate the trajectory that new hospital admissions were taken after what we've identified as Event H, which corresponds to when most high exposure-risk businesses reopened in the state. The trajectory of new hospital admissions (shown as the heavy purple line for the 7-day rolling average of this data) turns upward after the expected 11-13 day lag following the political campaign events that centered around the 24 October 2020 "National Vote Early Day" events in the state some two-and-a-half weeks before election day.

ICU Bed Usage

ICU bed usage data in Arizona has nearly the same lag from initial exposure to hospitalization, but is more complete because there is a limited supply of such facilities making it easier to track serious COVID-19 hospitalizations in real time. Our second chart shows the available data for ICU bed usage in Arizona from 3 March 2020 through 30 November 2020, with the rolling 7-day average shown through 30 November 2020.

Daily COVID-19 ICU Bed Usage in Arizona, 3 March 2020 - 30 November 2020

While this data is current through 30 November 2020, that is still several days too early to tell what impact the Thanksgiving holiday had on the incidence of coronavirus infections within Arizona. We anticipate that if the holiday had an effect on the current trend for hospitalizations, it will start showing up in this data in the period from 4 through 7 December 2020.

Newly Confirmed Cases

Unlike many states, Arizona makes its data on confirmed cases by date of test sample collection available. While this data indicates the shortest lag between virus exposure to positive COVID-19 test results, it takes several weeks for a given day's result to become relatively finalized, although it is mostly complete after two weeks.

Our third chart shows the available data for newly confirmed COVID-9 cases in Arizona from 3 March 2020 through 30 November 2020, with the rolling 7-day average shown through 23 November 2020.

Daily COVID-19 Newly Confirmed Cases in Arizona, 3 March 2020 - 30 November 2020

Like the new hospital admissions chart, this chart confirms the increased pace of new cases being recorded some nine to eleven days after the political campaign events of 23-25 October 2020.

At the same time, the chart confirms the rolling 7-day average number of newly confirmed cases is nearing the peak of what was recorded during Arizona's first surge in cases during in the early summer of 2020.


The data for coronavirus-related deaths in Arizona is reported with the greatest lag, but it is now starting to confirm the effect of the 23-25 October 2020 political campaign events had on changing the incidence of COVID-19 infections. Our fourth and final chart shows the available data for deaths attributed to COVID-19 in Arizona from 3 March 2020 through 30 November 2020, with the rolling 7-day average shown through 19 November 2020.

Daily COVID-19 Deaths in Arizona, 3 March 2020 - 30 November 2020

Using dates for which all data is relatively finalized, we can compare 15 June 2020 with 6 November 2020. For both dates, the rolling 7-day average for the number of newly confirmed cases in Arizona first rose above 2,000 cases per day after having previously been below that level. On 15 June 2020, 7-day rolling average for COVID-19 deaths reached 30 per day in Arizona, but on 6 November 2020, the equivalent figure was 17 per day.

That change suggests the SARS-CoV-2 coronavirus has become less deadly. We think that difference is most likely attributable to improvements in available medical treatment since the summer.

Since we won't start seeing the impact of Thanksgiving in the data for another week, we'll close this edition of our series on that bit of good news.

Previously on Political Calculations

Here's our previous Arizona coronavirus coverage, with a sampling of some of our other COVID analysis!


Arizona Department of Health Services. COVID-19 Data Dashboard. [Online Application/Database].

Maricopa County Coronavirus Disease (COVID-19). COVID-19 Data Archive. Maricopa County Daily Data Reports. [PDF Document Directory, Daily Dashboard].

Stephen A. Lauer, Kyra H. Grantz, Qifang Bi, Forrest K. Jones, Qulu Zheng, Hannah R. Meredith, Andrew S. Azman, Nicholas G. Reich, Justin Lessler. The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application. Annals of Internal Medicine, 5 May 2020. https://doi.org/10.7326/M20-0504.

U.S. Centers for Disease Control and Prevention. COVID-19 Pandemic Planning Scenarios. [PDF Document]. Updated 10 September 2020.

COVID Tracking Project. Most Recent Data. [Online Database]. Accessed 10 November 2020.

More or Less: Behind the Stats. Ethnic minority deaths, climate change and lockdown. Interview with Kit Yates discussing back calculation. BBC Radio 4. [Podcast: 8:18 to 14:07]. 29 April 2020.

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December 1, 2020

Political Calculations' initial estimate of median household income of in October 2020 is $66,066, an increase of 0.6% above the initial estimate of $65,630 for September 2020.

The following chart shows the nominal (red) and inflation-adjusted (blue) trends for median household income in the United States from January 2000 through October 2020. The inflation-adjusted figures are presented in terms of constant October 2020 U.S. dollars.

Median Household Income in the 21st Century: Nominal and Real Modeled Estimates, January 2000 to October 2020

The chart confirms an upward trajectory for median household income in both nominal and inflation-adjusted terms, point to the ongoing recovery from the bottom of the coronavirus recession.

Analyst's Notes

Significant upward revisions were made to the BEA's aggregate personal wage and salary income data from April 2020 (+0.4%) through September 2020 (+1.4%). The changes were such that we're now showing July 2020 as the bottom for the coronavirus recession-related decline of median household income, which now we now estimate to be $65,857 based on the BEA's revised data. This is a 1.2% decline from the pre-recession peak of $66,654 recorded for March 2020.

Other Analyst's Notes

Sentier Research suspended reporting its monthly Current Population Survey-based estimates of median household income, concluding their series with data for December 2019 before ceasing to operate in early 2020, as its principals would appear to have permanently retired. In their absence, we are providing the estimates from our alternate methodology for estimating median household income on a monthly basis. Our data sources are presented in the following section.


Sentier Research. Household Income Trends: January 2000 through December 2019. [Excel Spreadsheet with Nominal Median Household Incomes for January 2000 through January 2013 courtesy of Doug Short]. [PDF Document]. Accessed 6 February 2020. [Note: We've converted all data to be in terms of current (nominal) U.S. dollars.]

U.S. Department of Labor Bureau of Labor Statistics. Consumer Price Index, All Urban Consumers - (CPI-U), U.S. City Average, All Items, 1982-84=100. [Online Database (via Federal Reserve Economic Data)]. Last Updated: 12 November 2020. Accessed: 12 November 2020.

U.S. Bureau of Economic Analysis. Table 2.6. Personal Income and Its Disposition, Monthly, Personal Income and Outlays, Not Seasonally Adjusted, Monthly, Middle of Month. Population. [Online Database (via Federal Reserve Economic Data)]. Last Updated: 25 November 2020. Accessed: 25 November 2020.

U.S. Bureau of Economic Analysis. Table 2.6. Personal Income and Its Disposition, Monthly, Personal Income and Outlays, Not Seasonally Adjusted, Monthly, Middle of Month. Compensation of Employees, Received: Wage and Salary Disbursements. [Online Database (via Federal Reserve Economic Data)]. Last Updated: 25 November 2020. Accessed: 25 November 2020.


November 30, 2020

2020's Thanksgiving holiday-shortened trading week came and went, leaving the S&P 500 (Index: SPX) sitting atop a new record high for the index before disappearing into market history.

For those who checked out early for the holiday, we'll repeat how we closed the previous edition of the S&P 500 chaos series, since it helps explain this week's update to the alternative futures chart.

On a final note, sharp-eyed readers will note that over the projected future of next two weeks, the alternative futures chart is showing a relatively short duration "echo" of the volatility that struck the market about a month ago. This is an artifact from the model's use of historic stock prices in setting the base reference points from which it projects the future. For longer duration events, we will often add a redzone forecast range to account for the echo effect, but since this upcoming echo is comparatively short, we'll simply note that the trajectory of the S&P 500 will likely appear to "run hot" with respect to the model's projections over these weeks.

Here is this week's alternative futures chart, which shows what running hot looks like in the context of what we just described.

Alternative Futures - S&P 500 - 2020Q4 - Standard Model (m=+1.5 from 22 September 2020) - Snapshot on 27 Nov 2020

Although we're not adding a redzone forecast to the chart, all that exercise involves is connecting the dots of the projected forecast for a given future quarter investors are focusing upon from a point before the echo effect skews the projection to a more distant point in the future after the echoes of past stock price volatility have dissipated. If we were to add one, we would start by assuming investors would remain focused on 2021-Q3 and connect points for this alternative trajectory on opposite sides of the echo. We would then indicate a range of plus-or-minus three percent of the value of the index to account for typical day-to-day trading volatility.

The current echo, which traces back to the volatility of a month earlier, will run out before the end of this week. But since you now know how that particular magic works, we'll leave it as an exercise for you to either amaze or dismay your friends by making your own redzone forecast with one of our spaghetti forecast charts!

Monday, 23 November 2020
Tuesday, 24 November 2020
Wednesday, 25 November 2020
Friday, 27 November 2020

A succinct summary of the positives and negatives Barry Ritholtz found in the holiday-shortened trading week's economics and markets news is available over at The Big Picture.

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