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29 October 2021

Are you afraid of trolls? Vampires? Matrices?

For your Halloween pleasure, Matt Parker takes on all three fearsome concepts in the following 12 minute video:

If you're looking for more math to be scared by, since it was referenced in the video, here's Numberphile's five-and-a-half minute video on Vampire Numbers.

It's amazing how dated all the featured vampires other than Count von Count look in the videos....

Previously on Political Calculations


28 October 2021

The Biden administration has taken a strange view that the rising prices many American households have faced throughout 2021 are a good thing because more people are buying goods.

Watch the following exchange between CNN's Jake Tapper and White House Press Secretary Jen Psaki in this short video clip from 15 October 2021:

If prices are rising because more people are buying goods, the economic law of supply says we should also be seeing the supply of those goods increase. That's because rising prices, all things being equal, gives producers an incentive to sell more of the things they produce. More importantly, rising prices give marginal producers who can dial up their production a very strong incentive to produce more. Otherwise, they're losing out on easy profits.

For much of 2021, you can see the Law of Supply at work, with some very noticeable exceptions, in the following diagram charting the weekly price and quantities of crude oil being processed at U.S. refineries throughout 2021.

U.S. Crude Oil Price vs Quantity, 1 January 2021 through 22 October 2021

Those very noticeable exceptions include:

In the case of the February 2021 Cold Weather Event, oil production in the U.S. fully recovered, with rising prices coinciding with rising supplies. The recovery from Hurricane Ida looks similar, up to a point. We see that something went very awry after 1 October 2021. Instead of the supply of crude oil rising with rising prices as it did earlier in the year, the supply of crude oil going into U.S. refineries has fallen, while prices have continued to escalate. Not only that, refineries never recovered to the levels they were producing before Hurricane Ida disrupted production.

Since 1 October 2021, something very different is at work in the U.S. economy. To confirm that's the case, here's our tool for determining whether supply or demand is behind changes in price. We've set the default entries to apply to the situation with oil being processed at U.S. refineries since 1 October 2021.

Price and Available Quantity Data
Input Data Values
How has the price of the item changed over a given period of time?
How has the available quantity of the item changed over that same time period?

What's Behind the Change in Price?

We confirm that oil prices are rising now because the supply of crude oil being processed at U.S. refineries has fallen. That is something that's now showing up across the U.S. in the form of rising gasoline prices.

The U.S. Energy Information Administration is claiming that demand is rising faster than supply, but their own data says otherwise. We also looked at the weekly number of barrels of crude oil from U.S. field production and being imported into the U.S., and find they have recovered to pre-Hurricane Ida levels.

Weekly U.S. Imports and Field Production of Crude Oil, 1 January 2021 through 22 October 2021

The only explanation we've found that correlates with the price and quantity data points to refineries temporarily shutting down to change over to approved winter blends of petroleum products. If that's indeed all it is, that environmental regulation-driven change has a negative economic effect similar to a sustained, small-scale supply disruption caused by a natural disaster.


U.S. Energy Information Administration. Petroleum & Other Liquids: Weekly Crude Oil Spot Prices (West Texas Intermediate). [Online Database]. 27 October 2021.

U.S. Energy Information Administration. Petroleum & Other Liquids: Weekly U.S. Refiner Net Input of Crude Oil. [Online Database]. 27 October 2021.

U.S. Energy Information Administration. Petroleum & Other Liquids: Weekly U.S. Field Production of Crude Oil. [Online Database]. 27 October 2021.

U.S. Energy Information Administration. Petroleum & Other Liquids: Weekly U.S. Imports of Crude Oil. [Online Database]. 27 October 2021.

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27 October 2021

The U.S. new homes market is showing signs of stagnation. That can be seen in the trend for the market capitalization of new homes sold in the U.S., which moved sideways from August to September 2021. The initial estimate of the market cap itself for September 2021 is $29.35 billion.

Trailing Twelve Month Average New Homes Sales Market Capitalization, January 1976-September 2021

By this measure, the U.S. new home market last peaked in December 2020 at $30.12 billion. Since then, the market for new homes in the U.S. has shrunk in both real and nominal terms.

Speaking of real shrinkage, the number of new homes being sold in the U.S. has continued to decline from its November 2020 peak.

Trailing Twelve Month Average U.S. New Homes Sales, January 1976-September 2021

The market cap for new homes however is only managing to tread water because the average prices of new homes sold is escalating at its fastest pace on record.

Trailing Twelve Month Average Mean New Home Sale Price in U.S., January 1976-September 2021

The new home market has become a source of headwinds for the U.S. economy in 2021.


U.S. Census Bureau. New Residential Sales Historical Data. Houses Sold. [Excel Spreadsheet]. Accessed 26 September 2021. 

U.S. Census Bureau. New Residential Sales Historical Data. Median and Average Sale Price of Houses Sold. [Excel Spreadsheet]. Accessed 26 September 2021. 

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26 October 2021

When Joe Biden was sworn in as President of the United States on 20 January 2021, the U.S. national debt had reached $27.8 trillion. Through the end of the U.S. government's fiscal year on 30 September 2021, the total public debt outstanding increased to $28.4 trillion.

The following chart identifies the entities who have loaned the most money to the U.S. government. The percentage shown for each indicates how big each entity's share of the U.S. national debt is as of 30 September 2021.

20 January 2021: To Whom Does the U.S. Government Owe Money?

The values for foreign nations shown on the chart represent a first estimate because the U.S. Treasury Department's data for the amount of U.S. government-issued debt held by foreign entities only reflects its estimates through August 2021. Data through September 2021 will become available next month, which will be subject to revision before being finalized sometime in 2022.

Once again, the U.S. Federal Reserve is Uncle Sam's largest single entity creditor, outranking its former top creditor, Social Security's Old Age and Survivors Insurance Trust Fund, by a widening margin. That margin is widening because Social Security is running in the red, which means it has to cash in its holdings of U.S. Treasuries to keep paying benefits to Social Security beneficiaries at promised levels.

But what is really remarkable is the extent to which the U.S. Federal Reserve is funding the U.S. government's spending above and beyond what it collects in taxes that has taken place during Joe Biden's tenure in office. Since 20 January 2021, the U.S. national debt has increased by $669.3 billion, but the U.S. Federal Reserve's holdings of U.S. government-issued debt securities has increased by $687.5 billion.

That's possible because the U.S. Federal Reserve has more than offset a net reduction of $520.4 billion in the amount of money other U.S. entities have loaned to the U.S. government during this time. At the same time, foreign entities have boosted the amount of money they've loaned to the U.S. government by $502.2 billion, which when combined with the other figures, accounts for the overall net change since 20 January 2021. Here's a visual rundown of the net national debt change math as presented using a waterfall chart:

20 January 2021: To Whom Does the U.S. Government Owe Money?

As of the end of the U.S. government's 2021 fiscal year, the Federal Reserve had loaned nearly one out of every five dollars the U.S. government owed through the end of September 2021.

About the Data

These figures represent the most current information available as of 30 September 2021, which for the total public debt outstanding and data on U.S. government entity holdings is current through that date. The Federal Reserve's holdings is fully current through 29 September 2021. Data for foreign holdings is based on estimates through August 2020 that were published on 18 October 2021.

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25 October 2021

Given all the issues that are dominating the news, would you have guessed the S&P 500 (Index: SPX) would reach a new record high?

The index closed at a record high of 4,549.78 on Thursday, 21 October 2021, which puts the trajectory of the index into the upper half of the latest redzone forecast range:

Alternative Futures - S&P 500 - 2021Q3 - Standard Model (m=-2.5 from 16 June 2021) - Snapshot on 22 Oct 2021

Sharp-eyed readers will catch that the trajectory of the range has itself shifted upward, which is a result of the 'dynamic' method we use to set it. When we bridge across periods where the echoes of past volatility in stock prices affect the dividend futures-based model's projections, the past end of the range is fixed while the future end 'floats' with changes in expectations.

For the chart, the rising expectations for the future now means parts of the S&P 500's actual trajectory that were once within the redzone forecast range now fall outside of it. That's visible by design in this period because we set the total width of this forecast range to be plus-or-minus three percent of the historic typical level of volatity for stock prices.

Under typical volatility levels, the trajectory of the index should generally fall well within that statistically-determined range. But as the chart visually confirms, the market is experiencing greater than typical levels of volatility.

For us, what that means is that when we get around to projecting the S&P 500's future trajectory for 2021-Q4, we'll need to generate a new redzone forecast, since today's stock prices will become the base reference points from which we project the future for the index in that period. The echoes of today's volatility will affect the accuracy of the dividend futures-based model's projections a year from now.

The market moving headlines of the week point to several contributing factors for what new information has contributed to improving the outlook for investors. Pay particular interest to the signs and portents for the U.S. economy in the latter half of the week that was:

Monday, 18 October 2021
Tuesday, 19 October 2021
Wednesday, 20 October 2021
Thursday, 21 October 2021
Friday, 22 October 2021

If you haven’t already guessed it by clicking through the headlines, Reuters is a terrific resource for business and markets news, with a higher-than-average signal to noise ratio for picking up on market-moving news items.

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22 October 2021

The Inventions in Everything team has seen a lot of really strange inventions over the years, but today's featured invention is the only one that truly creeps us out. Which is why we're featuring it just ahead of Halloween!

We're referring of course to U.S. Patent 112,550, issued to Robert J. Clay on 14 March 1871. If you read the patent, the description of the invention sounds rather innocent:

This invention relates to a new doll, which is provided with a concealed clock-work and mechanism, whereby its limbs are moved in imitation of those of a creeping baby while the doll itself is being propelled on the floor by concealed rotating wheels which support it. A very amusing toy is thus produced at small cost.

What could possibly be wrong with that? Well, the title of the invention is "Improvement in Creeping Dolls", which should start sending up warning flags on its own, but wait, there's more. Prior to 1880, in order to be issued a patent, inventors were required to submit a physical model of their invention as part of the patent examination process. The model Clay submitted of an "Improved Creeping Baby-Doll" still exists and is in the possession of the cryptkeepers at the Smithsonian Institute.

Think of every horror movie you've ever seen and let us know what your reaction would be if you suddenly saw this thing in action on the screen. Or better still, what if you saw it in your home, coming toward you?

U.S. Patent 112,550 Physical Model

"Pure nightmare fuel" is how the IIE team has come to describe this invention, where the patent's illustrations fail to do justice to what the real-life thing looks like.

Within a decade after it was delivered to the U.S. Patent Office, the U.S. Patent Office discontinued its requirement for inventors to provide models of their inventions, perhaps motivated by the possibility they might receive more things like the physical model of U.S. Patent 112,550.

We're being a bit facetious in saying that. In truth, the U.S. Patent Office started running out of space to store all the stuff they were getting, so they dropped the requirement except in the case of flying machines and perpetual motion machines, preposterous and impossible inventions that demanded a higher level of examination to be considered for a patent.

After the Wright brothers earned U.S. Patent 821,393 on 22 May 1906, the requirement for inventors of flying machines to deliver working models of their inventions to the U.S. Patent Office was dropped. But should you invent a perpetual motion machine, you will need to provide your working model to the U.S. Patent Office when you submit your patent application.

The IIE team is aware of other creepy inventions, but subjectively, we rank this one the creepiest. Pleasant dreams, and happy Halloween!


21 October 2021

How does your household compare to the average American household when it comes to spending for food at home?

Answering questions like that takes data, and for that, the Consumer Expenditure Survey has detailed the average amount of money spent by American household "consumer units" in each year from 1984 through 2020. In the following chart, we've extracted the survey's data for expenditures by food category, showing the data for each year to show the trends for each.

Average Annual Expenditures for Food at Home by Major Food Categories, 1984-2020

In 2020, the average total amount of food-at-home expenditures for American households was $4,942. Here's how that breaks down into major food categories:

  • Meats, poultry, fish, and eggs: $1,075
  • Fruits and vegetables: $977
  • Miscellaneous foods: $973
  • Cereals and bakery products: $640
  • Dairy products: $474
  • Nonalcoholic beverages: $455
  • Other food at home: $348

This list brings up a good question: what's the difference between "Miscellaneous foods" and "Other food at home". Here's how the Bureau of Labor Statistics defines both of them (we've added the additional examples in parentheses):

Miscellaneous foods includes frozen prepared meals and other foods; canned and packaged soups; potato chips, nuts and other snacks; condiments and seasonings, such as olives, pickles, relishes, sauces and gravies, baking needs and other specified condiments; and other canned and packaged prepared foods, such as salads, desserts, baby foods, and vitamin supplements. Other food at home primarily represents sugar and other sweets (such as sugar, candy and chewing gum; artificial sweeteners; and jams, jellies, preserves, fruit butters, syrup, fudge mixes, icings, and other sweets), and also fats and oils (including margarine, shortening, and salad dressings, vegetable oils, nondairy cream substitutes and imitation milk, and peanut butter).

Since food prices have been rising, you can use these average 2020 expenditures to benchmark how much more money you're having to spend on these items in 2021.


U.S. Bureau of Labor Statistics. Consumer Expenditure Survey. Multiyear Tables. [PDF Documents: 1984-1991, 1992-1999, 2000-2005, 2006-2012, 2013-2020]. Accessed 9 September 2021. 

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20 October 2021

We've updated our chart tracking the periods of order and chaos that have characterized the S&P 500 (Index: SPX) over the past 30 years.

S&P 500 Average Monthly Index Value versus Trailing Year Dividends per Share, December 1991 - September 2021

It won't be a surprise, but the index entered into a period of chaos with the arrival of the coronavirus pandemic in March 2020. Through eighteen full months later, the index is still characterized by chaos.

The big question: Is the U.S. stock market in an economic bubble?

If you want to know where we stand on the question, we wrote the definition of an economic bubble.

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19 October 2021

We are seeing an unexpected development in Arizona's experience with the coronavirus pandemic. While the trends for the number of cases and deaths in the state are showing a slow, steady decline, the number of new hospital admissions in the state have plunged by a much larger amount by comparison.

The following chart shows the state's data for all three of these COVID-19 data series from 15 March 2020 through 15 October 2021, or rather, their rolling seven-day moving averages for which the data is at least 95% complete:

Arizona's Coronavirus Pandemic Experience, 15 March 2020 - 15 October 2021, Snapshot on 16 October 2021

According to the state's data, the number of new hospital admissions has fallen by over 57% from their third-wave peak during the week of 14 through 22 August 2021. That figure applies for 20 September 2021, which meets the 95% complete threshold at this writing. By contrast, Arizona had only seen a 23% decline in cases at the same point in time, which has grown to a nearly 33% reduction through 3 October 2021. The data for deaths through 20 September 2021 will be too incomplete to provide an additional point of comparison for another two weeks.

The Arizona Department of Health indicates their COVID dashboard is undergoing scheduled database maintenance, so this is the most current information available for the state until it might update it later today. We're curious to see if the state's data for COVID hospital admissions might be subject to revision when that work is complete.

If the hospital admission data is unchanged or only sees minor revisions, it suggests Arizona's experience with the coronavirus pandemic as its third wave recedes is very different from what it saw during its first two waves.

But if there's been a reporting glitch for new COVID admissions from Arizona's hospitals to Arizona's DHS, we would expect the states datatrend for COVID hospitalizations to follow the same basic pattern for as for cases. We'll update the original article on our site with what we find after the state's next COVID data update.

Update 19 October 2021, 11:15 PM EDT: The patterns we've described above have held through the latest daily update, which covers available data through 18 October 2021.

Previously on Political Calculations

Here is our previous coverage of Arizona's experience with the coronavirus pandemic, presented in reverse chronological order.


Political Calculations has been following Arizona's experience with the coronavirus experience from almost the beginning, because the state makes its high quality data publicly available. Specifically, the state's Departent of Health Services reports the number of cases by date of test sample collection, the number of hospitalizations by date of hospital admission, and the number of deaths by date recorded on death certificates.

This data, combined with what we know of the typical time it takes to progress to each of these milestones, makes it possible to track the state's daily rate of incidence of initial exposure to the variants of the SARS-CoV-2 coronavirus using back calculation methods. Links to that data and information about how the back calculation method works are presented below:

Arizona Department of Health Services. COVID-19 Data Dashboard: Vaccine Administration. [Online Database]. Accessed 16 October 2021.

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]. 10 September 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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18 October 2021

The arrival of earnings season with a strong showing for several large banks sparked a small rally for the S&P 500 (Index: SPX). The index' trajectory has risen back to the middle of the redzone forecast range:

Alternative Futures - S&P 500 - 2021Q3 - Standard Model (m=-2.5 from 16 June 2021) - Snapshot on 15 Oct 2021

The S&P 500's trajectory is consistent with the projections for investors focusing on either 2021-Q4 or 2022-Q1, which have largely merged over the last several weeks. Between the two, the information provided by the week's news stream points to 2021-Q4 having the edge in drawing the focus of investors' forward-looking attention, thanks to the arrival of its earnings season and the confirmation of the Federal Reserve's plan to start tapering its purchases of U.S. government-issued securities in the next month.

Here's the week's market-moving news:

Monday, 11 October 2021
Tuesday, 12 October 2021
Wednesday, 13 October 2021
Thursday, 14 October 2021
Friday, 15 October 2021

The Federal Reserve Bank of St. Louis has a marvelous resource for economic and business data. If you haven't yet met FRED (Federal Reserve Economic Data), allow us to make the introduction!

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15 October 2021

How can starting with a guess and using Isaac Newton's iterative solving method to find the roots of polynomial equations produce fractals?

3Blue1Brown's Grant Sanderson explains in the following 26 minute video. That may sound like a lot, but it is one of the more visually captivating maths explainers we've come across in quite a while, which really takes off at about the halfway point when chaotic boundaries emerge.

Newton's method is also known as the Newton-Raphson method, which is one of the tools that make it possible to come up with approximate numerical solutions to equations where direct, exact solution aren't an option. Here's a nice written introduction to how the method works, where step one involves making your best guess at the answer!

Update 16 October 2021: The sequel, Where Newton meets Mandelbrot (or more fancifully, “holomorphic dynamics”), is out!


14 October 2021

The median sale prices of new homes sold in the U.S. rose faster than median household income in August 2021. That combination has put the median new home sale price at near record low levels of affordability.

Relative Affordability of New Home Prices | Annual: 1967-2020 | Monthly: December 2020 - August 2021

Median household income in August 2021 is 18.4% of the median new home sale price. The record low for this measure is 18.3%, so we may soon be breaking records for unaffordability.

That's a way to measure the raw affordability of new homes, but what if we take today's near record-low mortgage rates into account? In the next chart, we've calculated the percentage of median household income that would be consumed by mortgage payments from January 2000 through August 2021.

Mortgage Payment for a Median New Home as a Percentage of Median Household Income, January 2000 - August 2021

This chart suggests the mortgage payment for a median new home is generally more affordable than in the past, which is largely attributable to falling mortgage rates over time.

If you look closely at recent trends however, a somewhat different picture is emerging. A mortgage payment for a median new home has been rising faster than median household income since the Coronavirus Recession bottomed in April 2020. That reverses the pattern from October 2018 to February 2020, when household incomes generally rose faster than new home prices, a rare period of improving relative affordability outside periods of recession. It has also occurred at a time when record low mortgage rates should be reducing the portion of household income taken up by mortgage payments.

It's not and we suspect that will be a problem going forward.


13 October 2021

On 6 October 2021, we took snapshots of the sale prices of the iconic 10.75 oz can of Campbell's Condensed Tomato Soup at several major grocers around the United States. Shoppers are seeing upward price pressure in the first week of soup season in the U.S.

For the latest in our coverage of Campbell's Tomato Soup prices, follow this link!

That pressure is most visible at Amazon, which we've presented in the following screen shot along with Microsoft Edge's records of its recent price history at the online retailer:

Amazon: Campbell's Condensed Tomato Soup (10.75 oz) on 6 October 2021

Here's a summary of prices at other retailers, ranked from high to low:

If you happen to live near one of the 248 Meijer grocery stores in Wisconsin, Illinois, Indiana, Kentucky, Michigan, or Ohio, you should move quickly this week to take advantage of the chain's sale on Campbell's Condensed Tomato Soup, where they've discounted the price from the $1.09 we indicated above to just $0.79 per can from Sunday, 10 October 2021 through Saturday, 16 October 2021.

You should also know it's not just Campbell Soup's prices that are seeing upward price pressure. Consumer prices for food products will be rising across many producers, including ConAgra, Kraft Heinz, and PepsiCo, to name just three major food producers in the U.S. who have announced more price hikes are coming for their products in since the start of October 2021.

Update: As of this morning, Amazon has dropped their price of Campbell's Condensed Tomato Soup to $0.95 per can. Amazing what a little visibility can do! But really, the question is how long will that price last?

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About Political Calculations

Welcome to the blogosphere's toolchest! Here, unlike other blogs dedicated to analyzing current events, we create easy-to-use, simple tools to do the math related to them so you can get in on the action too! If you would like to learn more about these tools, or if you would like to contribute ideas to develop for this blog, please e-mail us at:

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