Back of the Envelope


The ability to do approximate numerical calculations using readily available information is a useful skill. Suppose, for example, that in the course of litigation you are considering searching some body of documents for relevant material. It does not matter very much whether the search will take one week or two. But it matters very much whether it will take a week, a year, or a century. Or suppose the plaintiff’s attorney in a civil suit argues that the defendant ought to have taken some simple precaution to guard against risks inherent in the use of its product, and is negligent for not doing so. If the injury being sued over is a broken arm, the argument becomes much less plausible if the precaution would have required thousands of hours of employee time than if it would have taken fifteen minutes.

Such calculations are useful in many other contexts. 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. An exact calculation would be difficult, but using what you know about the amount of water in a glass and how often people in your area go out to dinner at the sort of restaurant that is affected by such a policy, you should be able to estimate how much water a day is saved—not precisely, but to within a factor of ten or so. You can then go online and find an estimate of how much water is consumed in the U.S. per capita per day. If the first number turns out to be one ten thousandth of the second, you can be pretty sure that the rule is not doing much to solve the problem—and the conclusion would still hold if the right number was one thousandth or one hundred thousandth.

One could argue that going online to find estimates of water consumption goes beyond “readily available information.” But there are lots of alternative approaches. Your toilet may well say on it how much water is used each time you flush it; if not, you can measure the dimensions of the tank and estimate from that. You know about how many times a day you flush a toilet. That’s only one small part of water consumption—but if that small part is a hundred times the savings from the restaurant policy, it is unlikely that the policy is doing much good.

For evidence of the risks, in the legal context, of not being competent at such calculations consider a real case (as described in my Law’s Order):

About one person in a million who was innoculated with the Sabin live virus vaccine got polio as a result. Since there was no known way of immunizing against polio without that risk, courts held that the vaccine was “inherently dangerous,” hence not a defective product. 

While there is no way of making an inherently dangerous product safe, one can still avoid the danger by not using the product. Hence the question arose of whether the producers of the vaccine were negligent, not for producing the vaccine but for failing adequately to warn those who used it. The producers provided warning information to physicians and others who dispensed the vaccine but did not take precautions to make sure that everyone who got the vaccine also got an individualized warning.
    Whether that was negligent, at least in the economist’s sense, depends on whether the warning would have made a difference, whether there was a significant chance that someone who was informed of the risk would conclude as a result that it was not in his interest to be vaccinated. In Davis v Wyeth Laboratories, Inc., the court accepted that argument and attempted to calculate the relevant costs and benefits:


The Surgeon General’s report ... predicted that for the 1962 season only .9 persons over 20 years of age out of a million would contract polio from natural sources. ... Thus appellant’s risk of contracting the disease without immunization was about as great (or small) as his risk of contracting it from the vaccine. Under these circumstances we cannot agree ... that the choice to take the vaccine was clear.


If the mistake is not obvious, read it again. The court is comparing the risk to Davis from being immunized—about one chance in a million of contracting polio—with the risk of not being immunized. To estimate the latter it uses the fraction of the adult population that could be expected to contract polio from natural sources in one year. But immunization lasts for life, so the relevant comparison is to his lifetime chance of ever contracting polio, perhaps with a weighting factor to take account of the fact that the later he contracts it the smaller the fraction of his life affected.  

If we assume that the risk was constant over time and measure the cost of getting polio by number of years of normal life lost, the court’s calculation was off by about a factor of twenty-five, making the benefit of the vaccine more than twenty times the cost. The mistake is one that a bright high school student should have caught. If judges, like pharmaceutical companies, were legally liable for the consequences of their negligence, the judge who wrote that opinion and those who joined in it would have owed Wyeth a very large amount of money. 

Orders of Magnitude


A useful tool in such calculations is the idea of an order of magnitude. One order of magnitude is a factor of ten. The numbers six and nine are the same order of magnitude. Fifteen and two, on the other hand, differ by about an order of magnitude.

I once came across the claim that, in the Elizabethan period, a peppercorn was worth as much as a horse, an error probably based on the legal doctrine that contract requires consideration but not adequate consideration, hence that the exchange of a horse for a single peppercorn would be a legally binding contract. A little online research on prices, along with weighing some peppercorns, established that the claim was off by more than four orders of magnitude.

An approach to calculations which students with a scientific background are probably familiar with, but others may not be, is to put everything in terms of powers of ten. Thus, for instance, using invented numbers because I am too lazy to check the real ones, if there are 1340 peppercorns in an ounce, an ounce of pepper in 1600 sold for two pennies, and a horse in 1600 sold for between 20 and 120 pennies, we might calculate (using the lower value for a horse):


1.34x103 peppercorns = 2 pennies

1 peppercorn = 2/1.34 x 10-3 pennies 10-3 pennies

1 horse costs 20 pennies  10 pennies.

1 horse  104 peppercorns

Here we are ignoring small differences, such as between 1/1.34 and 1, or between 2 and 1, in order to focus on differing orders of magnitude.

One convenient feature of powers of ten is that you can multiply them by simply adding the powers:


103x102=105   i.e. 1000x100 = 100,000


105x10-3=102   i.e. 100,000 x 1/1000 = 100




Adding small numbers is much easier than multiplying large numbers, so doing everything in terms of orders of magnitude, when you don’t need more accuracy than that, makes calculations much easier.

Easy enough to be done on the back of an envelope.


Exercises (need not be handed in, will be discussed in class):


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


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.


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?

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?

If you want to look at some possibly relevant numbers, Historical Statistics is webbed.