Back of the Envelope Class


Note: 10^3 means 10 to the third power = 10x10x10 = 1000

~ means approximately

1. When there is a water shortage, it is common for restaurants to inform their customers, sometimes at the request of their local government, that drinking water will only be provided if the customer requests it. Suppose you want to know whether this is a serious attempt to help deal with the problem or mere window dressing. One way of finding out is to calculate how large a fraction of water consumption goes for the water put out in restaurants for the customers to drink.

Do the drinking water calculation in at least two different ways.

 A1. Assume the average person eats out every other day. Assume a class holds 1 cup of water. The policy then saves ½ c of water/day.

A2. (how I did it in class): Assume a cup holds 2 cups (16 oz), the average person eats out every other day. Half the people will ask for water, so there is only a savings for the other half. The policy again saves ½ c of water/day

B1. Calculation from individual use:

Flushing a toilet: 1.6 gallons for a low flow toilet = ~30 cups, since 1 gallon = 16 cups.

Suppose you flush the toilet 4 times a day. That consumes 120 cups, or 240 times the amount of water being saved by the rule.

This greatly understates individual water usage, since water is also used for watering lawns, agricultural purposes, industrial purposes, …  . But it is enough to show that the rule does not have a large effect on water consumption.

B2. For a more elaborate version of the calculation, there is an online calculator at:

Judging by that, individual use is probably about ten times the figure I have just calculated, when you allow for baths, showers, laundry, dishwashing and such.

C. Calculation for average use, including agricultural and industrial (requires online research):

Gives about 150 gallons/person/day as the consumption in a particular region of Colordo. says:

3.9 trillion gallons of water are consumed in the United States per month.”

3.9x10^12 gallons/month/30 days =

~10^11 gallons/day/3x10^8 people

~300 gallons/person/day

If we take the lower figure of 150 gallons/day =2400 c/day,  the saving from the rule is ~1/5000 of consumption.

2. One concern sometimes expressed with regard to population growth is that we will run out of room—that with most of the earth covered by roads and houses there will be insufficient space to grow crops, not to mention space for wildlife to survive. A relevant question is how much area housing actually covers.
Produce an estimate, for the U.S., of what fraction of its land area is currently covered by housing. By buildings of all sorts. By roads.

U.S. Area: ~3000miles wide x~1000 miles high  =~3x10^6

Looking it up online, it’s 3.79 million square miles =3.79x10^6

Houses: A large house has 3000 square feet of floor space and might contain four people; most people don't have that much space.

So figure that 500 square feet per person is a high estimate, with multiple stories call it 300 square feet of land covered per person. Multiply by 3x10^8 people to give a total area of land covered by  housing in the U.S. of ~10^11 square feet:

Square mile =~5000 ft x~5000 ft=25x10^6 square feet

Divide 10^11 square feet by 25 x 10^6 square feet in a mile to get

~4x10^3 square miles covered by housing.

Or about 1/1000th of the land area.

 I will leave the road and stores calculation to you—but it’s hard to believe that the total could be as much as ten times the amount of land covered by housing, visualizing the city you live in. So all land used for all these purposes is almost certainly less than one percent of land area, and probably much less.

3. “Only a cantankerous man like Henry Ford, with dictatorial power over his business, would dare to create a mass market for automobiled by arbitrarily setting his prices low enough and his wages high enough that his workers could afford to buy his product.”

(Freeman Dyson in The Scientist as Rebel)

Is this explanation of Ford’s success—that he created a market for his cars by paying high enough wages so that his workers could buy them—plausible? Possible? How would you decide?

Checking the Wikipedia article, by 1920 Ford was producing more than a million cars a year. In 1921, total employment in producing "Vehicles for land transportation" (1922 Statistical Abstracts) was 281,000. If we assume that half those people worked for Ford and that each of them bought a new car from Ford every ten years, that would be 18,000 cars/year, or less than 2% of total sales.


So purchases by his well paid workers could not be a significant factor in his total sales.

4. "Now it is worth remembering, and the cold figures of finance prove it, that during that time there was little or no drop in the prices that the consumer had to pay, although those same figures proved that the cost of production fell very greatly; corporate profit resulting from this period was enormous; at the same time little of that profit was devoted to the reduction of prices. The consumer was forgotten. Very little of it went into increased wages; the worker was forgotten, and by no means an adequate proportion was even paid out in dividends—the stockholder was forgotten.


What was the result? Enormous corporate surpluses piled up—the most stupendous in history. ..."

(FDR's description of the events of the twenties in his 1932 nomination address)

The quote used lots of quantitative language but no numbers. Suppose we take "cost of production fell very greatly" as meaning that it fell at least in half and assume that the "very little" going to a variety of things can be approximated by zero.

Can you calculate any testable implications of Roosevelt's claims? How might you test them?

Profit=Revenue-Cost > 0 before the drop in costs. For simplicity, make it zero, since this will underestimate accumulated profits during the decade, and we are trying to figure out whether, on FDR's account, they were impossibly large.

Cost falls in half, everything else stays the same, so after the change:

Profit=2 x cost = revenue/2.


Over ten years, accumulated profit = 20 x cost = 5 x revenue


I find that implausible, but actually testing it is hard, due to the shortage of economic data that early. I did come up with the following very rough argument:


1929 non-agricultural employment 37 million

Per capita income about $700, so non-agricultural wages about $37x7x10^8=~$3x10^10

Wages are part of cost, so retained profit of 20 times cost should be >6x10^11 dollars over ten years.
~$600 billion.


(This assumes that all non-agricultural employment was in whatever sort of production FDR is talking about, which is surely an overestimate. Employees in manufacturing numbered 10,000,000, which would presumably give a lower bound on the number he was talking about, and would cut my calculation here roughly  by a factor of four.)


In 1952, total holdings of equities =$169 billion


The Consumer Price Index was 5.83 in 1929, 3.765 in 1952, so prices had gone up by about 50%. So $600 billion in 1929 corresponds to  $900 billion in 1952, which was more than 5 times total equity values in 1952. It seems unlikely that by 1929 firms had accumulated profits equivalent to more than five times the total value of all publicly traded firms in 1952.


But it would be nice to find a cleaner argument. From Historical Statistics we have:


1926 Dividend/price ratio=4.94%, earnings/price=9.19%


So dividends were more than half of total earnings. It is hard to see how that could be true if earnings had increased a lot and dividends hadn’t changed.