Mesopotamia’s standard commercial interest rate from around 2500 BC through the Neo-Babylonian epoch in the first millennium was high — the equivalent of 20 percent annually. This rate was not reached in modern times until the prime loan rate by U.S. banks peaked at 20 percent in 1980, causing a crisis. Yet this rate remained stable for more than two thousand years for contracts between financial backers and commercial traders or other entrepreneurs. The rate did not vary to reflect profit levels or the ability to pay. It was not set by “market” supply and demand, but was an administered price set as a matter of mathematical convenience by the initial creditors: the Sumerian temples and, after around 2750 BC, the palaces that gained dominance.
A mina-weight of silver was set as equal in value to as “bushel” of grain. And just as the bushel was divided into 60 “quarts,” so a mina-weight of silver was divided into 60 shekels. It was on this sexagesimal basis that temples set the rate of interest simply for ease of calculation — at 1 shekel per month, 12 shekels in a year, 60 shekels in five years.
The exponential doubling and redoubling of debt
Any rate of interest implies a doubling time — the time it takes for interest payments to grow as large as the original principal. A Babylonian scribal exercise circa 2000 BC asked the student to calculate how long it will take for a mina of silver to double at the normal simple interest rate of one shekel per mina per month. The answer is five years, the typical time period for backers to lend money to traders embarking on voyages. Contracts for consignments to be traded for silver or other imports typically were for five years (60 months), so a mina lent out at this rate would produce 60 shekels in five years, doubling the original principal. Assyrian loan contracts of the period typically called for investors to advance 2 minas of gold, getting back 4 in five years.
The idea of such exponential growth is expressed in an Egyptian proverb: “If wealth is placed where it bears interest, it comes back to you redoubled.” A Babylonian image compared making a loan to having a baby. This analogy reflects the fact that the word for “interest” in every ancient language meant a newborn: a goat-kind (mash) in Sumerian, or a young calf: tokos in Greek or foenus in Latin. The “newborn” paid as interest was born of silver or gold, not from borrowed cattle (as some economists once believed, missing the metaphor at work). What was born was the “baby” fraction of the principal, 1/60th each month. (In Greece, interest was due on the new moon.) The growth was purely mathematical with a “gestation period” for doubling dependent on the interest rate.
The concept goes back to Sumer in the third millennium BC, which already had a term mashmash, “interest (mash) on the interest.” Students were asked to calculate how long it will take for one mina to multiply 64 times, that is, 26 — in other words, six doubling times of five years each. The solution involves calculating powers of 2 (22 = 4, 23 = 8 and so forth). A mina multiplies fourfold in 10 years (two gestation periods), eightfold in 15 years (three periods), sixteenfold in 20 years (four periods), and 64 times in 30 years. The 30-year span consisted of six fiveyear doubling periods.
Such rates of growth are impossible to sustain over time. Automatic compounding of arrears owed on debts was not allowed, so investors had to find a new venture at the end of each typical five-year loan period, or else draw up a new contract. With the passage of time it must have become harder to find ventures to keep on doubling their savings.
Martin Luther depicted usurers scheming “to amass wealth and get rich, to be lazy and idle and live in luxury on the labor of others.” The growing mass of usurious claims was depicted graphically as a “great huge monster … who lays waste all … Cacus.” Imbuing victims with an insatiable desire for money, Cacus encouraged an insatiable greed that “would eat up the world in a few years.” A “usurer and money-glutton … would have the whole world perish of hunger and thirst, misery and want, so far as in him lies, so that he may have all to himself, and every one may receive from him as from a God, and be his serf for ever. … For Cacus means the villain that is a pious usurer, and steals, robs, eats everything.”
The mathematical calculation of interest-bearing debt growing in this way over long periods was greatly simplified in 1614 by the Scottish mathematician John Napier’s invention of logarithms (literally “the arithmetic of ratios,” logos in Greek). Describing the exponential growth of debt in his second book, Robdologia (1617), Napier illustrated his principle by means of a chessboard on which each square doubled the number assigned to the preceding one, until all sixty-four squares were doubled — that is, 263 after the first doubling.
Three centuries later the 19th century German economist, Michael Flürscheim, cast this exponential doubling and redoubling principle into the form of a Persian proverb telling of a Shah who wished to reward a subject who had invented chess, and asked what he would like. The man asked only “that the Shah would give him a single grain of corn, which was to be put on the first square of the chess-board, and to be doubled on each successive square,” until all sixty-four squares were filled with grain. Upon calculating 64 doublings of each square from the preceding, starting from the first gain and proceeding 1, 2, 4, 8, 16, 32, 64 and so on.
At first the compounding of grain remained well within the physical ability of the kingdom to pay, even after twenty squares were passed. But by the time the hypothetical chessboard was filled halfway, the compounding was growing by leaps and bounds. The Shah realized that this he had promised “an amount larger than what the treasures of his whole kingdom could buy.”
The moral is that no matter how much technology increases humanity’s productive powers, the revenue it produces will be overtaken by the growth of debt multiplying at compound interest. The major source of loanable funds is repayments on existing loans, re-lent to finance yet new debts — often on an increasingly risky basis as the repertory of “sound projects” is exhausted.
Strictly speaking, it is savings that compound, not debts themselves. Each individual debt is settled one way or another, but creditors recycle their interest and amortization into new interest-bearing loans. The only problem for savers is to find enough debtors to take on new obligations.
A mathematical principle called the “Rule of 72” provides a quick way to calculate such doubling times: Divide 72 by any given rate of interest, and you have the doubling time. To double money at 8 percent annual interest, divide 72 by 8.
The answer is 9 years. In another 9 years the original principal will have multiplied fourfold, and in 27 years it will have grown to eight times the original sum. A loan at 6 percent doubles in 12 years, and at 4 percent in 18 years. This rule provides a quick way to approximate the number of years needed for savings accounts or prices to double at a given compound rate of increase.
The exponential growth of savings (= other peoples’ debts)
One of Adam Smith’s contemporaries, the Anglican minister and actuarial mathematician Richard Price, graphically explained the seemingly magical nature of how debts multiplied exponentially. As he described in his 1772 Appeal to the Public on the Subject of the National Debt: