The beauty of tautologies

Seeing the world in a new way

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:

1. M*V = P*Y = NGDP

2. M = k*P*Y = NGDP k*NGDP

3. Saving = Investment

4. Aggregate quantity demanded = aggregate quantity supplied

5. GDP = GDI (gross domestic income)

6. [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

Part 1: Monetarism

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.

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*Y

Thus, 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:

1. Nominal GDP is priced in money terms.

2. The stock of base money is directly controlled by the central bank.

3. 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 Thrift

The 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 Law

Here’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%. Because nominal wages are sticky, this drop in nominal spending did not result in a 50% fall in wages and prices, leaving output unchanged. Instead, nominal wages were sticky and remained above equilibrium. Firms responded much as they would to a higher minimum wage rate, by reducing employment and output.

From the perspective of Say’s Law, these sticky wages create an aggregate supply problem, analogous to higher legal minimum wage rates. Thus, the fact that a fall in nominal spending in an economy featuring sticky wages leads to high unemployment in no way contradicts Say’s Law; indeed, it is a prediction of classical economics.

Keynes was wrong when he suggested that perfectly flexible wages would not solve the problem of falling nominal spending. But he won the intellectual battle anyway, as there is no way to make nominal wages perfectly flexible, so the sort of nominal spending shortfall that Keynes worried about really is a problem, a problem that must be addressed public policies that stabilize nominal spending.

I once wrote a paper entitled The Real Problem is Nominal, where the term “real” meant “actual”. Say’s Law teaches us that all economic problems are real, meaning supply side problems, even if the ultimate source of the problem is nominal wage rates that are too high relative to falling NGDP, triggered by a shortfall in “demand”.

Part 4: GDP = GDI (Gross output equals gross income)

The production/income identity is useful in a couple of different areas. First, it provides the intuition for one popular argument for NGDP targeting. In the previous section I alluded to the fact that a stable path of NGDP helps to stabilize the labor market. That’s because NGDP is the nominal revenue that firms have to pay nominal wages. The production/income equality suggests that stabilizing NGDP will also help to stabilize the financial system, as nominal income (NGDI) is the resource that individuals, firms and governments have available to pay nominal debts. Falling NGDI often leads to a financial crisis (as in 2008-09 and 1931-33.)

This identity is also useful when thinking about the impact of AI on the economy. Some fear that intelligent robots will eventually provide almost unlimited labor, leading to mass unemployment. But this hypothetical outcome would also imply almost unlimited output, meaning the economy would experience extremely high levels of gross income, despite near 100% unemployment rates. That’s not to deny the fact that AI might present important challenges for public policymakers, but it does suggest that policymakers would have an extremely powerful tool should they use to utilize it—the fiscal space to easily provide every single citizen a UBI equal to at least an upper middle-class lifestyle, if not higher.

When the public envisions mass unemployment, they also envision mass poverty. But that’s because technological progress over the past two centuries has not resulted in mass unemployment. Rather, occasional periods of high unemployment have been due to the interaction of falling NGDP and sticky wages, which really does impoverish a nation. In contrast, a mass unemployment produced by near limitless technological progress would enrich a nation. The GDP = GDI tautology helps us to understand that fact.

Part 5: Saving, investment and the current account balance

Do trade policies cause a change in the gap between saving and investment, or do policies that influence domestic saving and investment cause shifts in the trade balance? By itself, the S - I = CA balance tautology doesn’t prove anything about causation. But it is highly suggestive.

[Update: In the original post I should have indicated domestic saving minus domestic investment, as total saving equals total investment.]

Suppose you asked a protectionist to explain why they thought that trade barriers would improve the current account balance. How would they explain this outcome in terms of saving and investment?

There are only two ways that a country can increase its current account balance, either by reducing domestic investment or by increasing domestic saving. Protectionists would almost never point to lower domestic investment as the preferred explanation, for obvious reasons. Higher investment is almost universally viewed as a good thing, even among protectionists. The argument that protectionism will increase the trade balance almost always hinges on the question of whether it will increase domestic saving. And it might!

For instance, a tariff might increase government revenue, leading to higher government saving (or at least less dissaving) and this could increase the current account balance (i.e., reduce the deficit.) I’ve generally been skeptical about the impact of the Trump tariffs, but that’s not for purely theoretical reasons. Rather it is because Trump has aggressively cut taxes in other areas, thus the net effect of his fiscal policies has not been a boost in government saving. (BTW, quotas don’t even bring in government revenue, so they are especially unlikely to boost domestic saving.)

Of course, an extremely protectionist trade policy with prohibitive tariffs would almost certainly reduce America’s current account deficit by dramatically reducing both imports and exports. In that case, however, the most likely mechanism would be lower domestic investment, hardly the desired outcome.

Tautologies are some of my favorite economic tools. They force us to see connections that otherwise might be hidden from view. They allow us to see why certain causal relationships are likely to be true, or why they might be highly implausible. The Fisher Equation tautology (i = r + inflation) doesn’t prove anything about causality. But it points us in the right direction—it helps us to organize our thinking about the likely impact of inflation on interest rates. That’s because we know something about the world. Our views regarding the real interest rate are not a blank slate, just as our intuition about the public’s demand for cash balances as a share of income is not a blank slate.

Some tautologies are obvious in one sense, but not in another. The amount of money that people pay in interest is exactly equal to the amount of money that people receive in interest. The amount of money that people pay to buy homes, is exactly equal to the amount of money that people receive when they sell homes. Duh!! But keeping these truisms in mind helps to prevent the fallacy of reasoning from a price change.

And while it is true that tautologies, by themselves, do not prove any causal relationships to be true, isn’t that true of all social science models?

PS. To see some examples of tautologies being misused, check out my two posts on MMT (here and here.)

PPS. Slightly off topic, but I’d like to point readers to one of my favorite weekly summaries of the blogosphere, provided in David Levey’s substack.