The beauty of tautologies - by Scott Sumner
The Pursuit of Happiness
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The beauty of tautologies<br>Seeing the world in a new way
Scott Sumner<br>May 06, 2026
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Tautologies have a bad reputation. They are often dismissed as mere definitions. People get scolded for trying to draw causal implications from tautologies. They are viewed as being simplistic.<br>It is true that tautologies are (implicit) definitions. It is true that they are simplistic. It is true that they have no direct causal implication. Nonetheless, tautologies are one of the most important parts of economics. They promote clear thinking and thus make it easier to see how the economy functions. They do not establish causal relationships, but they make it easier to see which causal relationships are plausible and which are not.<br>The Pursuit of Happiness is a reader-supported publication. To receive new posts and support my work, consider becoming a free or paid subscriber.
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Consider this rather banal tautology:<br>The number of shares of stock sold equals the number of shares of stock purchased.<br>I don’t know how many times I’ve heard the news media attribute a sharply decline in stock market indices to a “selling wave” hitting Wall Street: The Dow fell 800 points as investors sold 15.4 billion shares of stock. Yes, but investors also purchased 15.4 billion shares of stock. Two sides of the same coin:
At one time, the stock market was closed at night and yet market indices often changed dramatically, even without a single share being traded. A hundred years ago, the Dow might close one day at 243 and open the following morning at 227, reflecting bearish overnight news. In that case, it is fairly obvious that the market moves on new information, not trading activity. To the extent that trading activity has any impact on prices, it is due to what the trading reveals about information held by various participants in the market.<br>Here I’ll look at six tautologies:<br>M*V = P*Y = NGDP
M = k*P*Y = NGDP k*NGDP
Saving = Investment
Aggregate quantity demanded = aggregate quantity supplied
GDP = GDI (gross domestic income)
[Domestic] Saving - investment = Current account balance
Proponents of various policies often use tautologies as a sort of intuition pump—a way of making their model seem more plausible. I will show that the first (and most famous) tautology listed above is actually the least useful, whereas each of the other five offer valuable insights into macroeconomics<br>Part 1: Monetarism<br>Tautologies #1 and #2 show the relationship between the money supply and nominal GDP (P*Y). In a mathematical sense the two equations are exactly equivalent, as V = 1/k. But the Equation of Exchange (M*V=P*Y) is less useful than the Cambridge Equation (M=k*P*Y), as the latter has a common-sense explanation that is intuitively appealing, while the former does not.<br>The variable V in the Equation of Exchange is often described as “velocity”, the average number of times a unit of money is spent in a given year. But that is not actually what V represents, as money is frequently spent on goods that are not a part of NGDP, and some purchases of goods do not involve money.<br>The Cambridge equation says that if k is the share of gross income that people hold in the form of cash balances, then the level of nominal GDP is the ratio of the money supply to k. Unlike V, the variable k really does represent the variable described in the textbooks. The k ratio is a variable that you can visualize. You can imagine holding a larger or smaller share of your income in the form of cash balances. And you can also imagine the central bank directly changing the money supply through policies like open market operations.<br>The Cambridge equation tautology tells us that anything that changes NGDP does so by influencing one of two variables; either the stock of base money, or the share of gross income held in the form of base money. You can model NGDP by modeling each of those two variables.<br>I believe that M = k*P*Y is an especially illuminating tautology because most people have no idea that these two variables (the stock of base money and the share of income held as money) determine total national income. I doubt if one person in a hundred could explain this fact. And the reason is simple—most people don’t understand the distinction between nominal and real variables.<br>Think about the fact that for any economic variable X, it is true that:<br>X = k*P*Y, where k is defined as X/P*Y<br>Thus, X might be the total stock of gold, measured in dollars. Why is this not an interesting tautology?<br>If we were on the gold standard, then that sort of equation would be quite interesting. Suppose the public held a stock of gold equal to 3.5% of GDP. If the stock of gold rose by 20% due to a new discovery, and if the public’s demand for gold as a share of GDP were unchanged, then NGDP would rise by 20%. That’s an interesting fact to know, even if in the real world the public’s...