The beauty of tautologies

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. 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.Consider this rather banal tautology:The number of shares of stock sold equals the number of shares of stock purchased.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. Here I’ll look at six tautologies:M*V = P*Y = NGDPM = k*P*Y = NGDP k*NGDPSaving = InvestmentAggregate quantity demanded = aggregate quantity suppliedGDP = GDI (gross domestic income)[Domestic] Saving - investment = Current account balanceProponents 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 macroeconomicsPart 1: MonetarismTautologies #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. 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. 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. 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. 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. Think about the fact that for any economic variable X, it is true that:X = k*P*Y, where k is defined as X/P*YThus, X might be the total stock of gold, measured in dollars.

Why is this not an interesting tautology?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 demand for gold as a share of GDP is not exactly constant.But suppose that gold is not money. Now consider the same example, a 20% rise in the physical stock of gold due to a new discovery. If we are not on a gold standard, then, the nominal gold price of gold might change. If so, then even if the public continued to hold exactly 3.5% of gold as a share of GDP, the enlarged stock of physical gold need not have any impact on its value in monetary terms, and hence NGDP. Thus, the Cambridge Equation is enlightening due to three critical assumptions:Nominal GDP is priced in money terms.The stock of base money is directly controlled by the central bank.It seems plausible that the public’s demand for base money as a share of income is at least partly determined by factors that are independent of central bank policy, especially in the long run. After all, do you let Jay Powell determine the amount of base money that you hold as a share of your income?Part 2: S = I and the Paradox of ThriftThe savings = investment relationship is often misunderstood. At an aggregate level, saving is defined as the funds used to finance investment—the construction of capital goods. Don’t try to construct counterexamples. If society decides to redefine a good formally viewed as investment as now being a consumer good (let’s say something like pickup trucks), then the funds used to pay for the good get redefined from saving to consumption. If I save money by lending it to a neighbor who blows it all in Vegas, then the neighbor’s actions are treated as negative saving, exactly offsetting my positive saving.Don’t bother trying to find real world examples of where saving doesn’t equal investment; by definition they are equal. Saving is the portion of income used to finance investment.

Period, end of story.So, what are we to make of the often-stated fear that increased saving might cause a depression. In fact, no depression has ever been caused by increased saving. That’s because saving equals investment, and if saving increases then investment increases. By definition. And depressions generally feature declining investment, often sharply declining investment. So stop worrying about too much saving.In The General Theory, Keynes worried about a slightly different problem. He worried about an increase in the propensity to save, i.e., the intention to save, which is a radically different concept from an increase in actual saving. Keynes worried that if people intended to save more money, it would lead to less nominal spending and a fall in national income. Keynes argued that because of the fall in aggregate income, the public would not actually save more. National income would adjust (downward), rather than S and I adjusting upward.Unfortunately, many people misinterpreted Keynes’s “paradox of thrift” as a claim that more saving is bad for the economy. That’s not what Keynes said! Keynes argued that a decline in aggregate demand (basically nominal spending or NGDP) is bad for the economy, and that this sort of decline might be caused by an increase in the public’s desire to save. But Keynes never said that more saving is itself a bad thing, as he understood that in equilibrium there is an equality between saving and investment. And since Keynes was very much pro-investment, that means he was also very much pro-saving. By combining the Cambridge Equation with the S = I identity, we can finally begin to understand Keynes’s paradox of thrift. In plain English, Keynes was saying that if the public tried to save a higher share of their income, it was likely to lead to an increase in the Cambridge k ratio. And the Cambridge Equation tells us that if the k ratio increases at a time when the money supply is constant, then NGDP must decline. That’s why Keynes worried about a higher propensity to save. But that fear has nothing to do with an actual increase in saving, which is generally associated with rising investment and a strong economy. What Keynes called the “paradox of thrift” should be called the problem of money hoarding.

When we combine the Cambridge Equation with the S = I identity we can also see why Keynes’s paradox of thrift fell out of style after the 1960s. When we moved to a fiat money regime, it became clear that the central bank could offset any decline in the k ratio by increasing the monetary base and thus prevent a fall in nominal GDP. Now there was no longer any reason to worry about the public becoming too thrifty. Saving is good, actually. Moving from a gold standard to a fiat money system removed the worry that saving was bad for the economy. (A point I should have made in my previous post, which critiqued the gold standard.)Part 3: AS = AD and Say’s LawHere’s AI Overview:Say's Law, often summarized as "supply creates its own demand," posits that the production of goods generates the necessary income (wages, rent, profit) to purchase that total output. It implies that general overproduction or widespread "gluts" are impossible in a market economy, as total demand always equals total supply.When defined this way, Say’s Law is true. The Great Depression was not caused by overproduction (as Franklin Roosevelt believed), rather it was a case of too little production. On the other hand, Say’s Law does not mean that we need not fear a situation where nominal spending is falling. Even if aggregate supply equals aggregate demand in a Depression, the equilibrium may occur at a undesirably low level of output and employment:[Graph courtesy of ChatGPT]Much of the confusion is due to the use of the term “aggregate demand”. I wish the model were called the nominal spending/real output model, not the AS/AD model. It has nothing to do with “demand” in the ordinary sense of the term as used in microeconomics.Classical economists understood that if the government were to pass a law setting a very high minimum wage rate—say $40/hour, it would lead to less employment and output. Most economists would call this sort of policy an “adverse supply shock”. Now consider a big drop in “aggregate demand” such as occurred during 1929-33, when NGDP fell by roughly 50%.