S=I: The most misunderstood equation in economics

The saving equals investment identity (S=I) is probably the most misunderstood equation in economics. Achieving the top rank in this highly competitive category requires an extraordinary level of miscomprehension, and the S=I identity does not disappoint. PhD students and many professors frequently misunderstand it, and misrepresent when teaching their students. In fact, it is sometimes difficult to find an explanation of it that is actually correct. The identity puzzled when I first encountered it, and it took me years to finally begin to understand it.

Many of the explanations people offer of the S=I identity are hampered by a lack of understanding of what an identity is. An identity is a mathematical fact that is true in any possible world, with any possible set of behaviours. Often, people’s explanations of the S=I identity are not correct because they are not true under all specifications of consumer and investor behaviour.

For example, one incorrect explanation of the S=I identity I have encountered says that saving equals investment because any saving is automatically invested. According to this view, if I save 200\$, banks will lend 200\$ more to investors, and thus investment will increase by 200\$. However, the above view cannot be correct. Even if banks and investors operate according to the above behavioural rules in practice, an equation must be true for all possible sets of behaviours. It is very possible that banks do not lend 200\$ more dollars when someone deposits money in them, and so this understanding of the S=I identity is incorrect.

What helped me to finally understand S=I was working through examples of very simple economies and seeing how the S=I identity must hold. For example, consider an economy consisting of two people, person A and person B, each who starts off with 100\$. We assume that neither person in the economy invests, and that both of them run businesses that produce consumption goods when those consumption goods are ordered. Person A spends 100\$ buying goods from person B, and person B spends 50\$ buying goods from person A.

At the end of the period person B has 150\$. Their income was 100\$ and they spent 50\$, meaning they saved 50\$. Yet there is no investment in the economy! It appears at first as if the S=I identity is violated. However, person A has actually spent 50\$ more than their income. Their saving is negative 50\$. So total saving in the economy is actually zero. The key thing that is often missed in explanations of S=I is that saving refers to net saving ie. saving minus borrowing. Since in everyday terms we use borrowing to refer to negative saving that point is often missed.

What would happen if both person A and person B decided they wanted to save 50\$ in the above example? Then the income of both of them would drop to 50\$, and neither would save anything or go into debt. Saving in financial assets by one party in the economy is only possible if another party in the economy decides to dissave. Otherwise all that occurs is that aggregate income in the economy goes down.

There is nothing magical about saving that causes investment, what is occurring is that saving in the form of financial assets requires someone else to divest themselves of financial assets. Every financial asset is a liability of some other party: the only way financial assets can be created is if someone goes into debt. If I end up with 100\$ more than I started with at the end of the year that means the sum of the deficits and surpluses of all other agents in the economy must be -100\$.

What happens if we add investment to the picture? Let’s say that instead of saving 50\$ in his bank account person B instead decides to buy 50\$ worth of machinery for his store from person A, in addition to consuming 50\$ worth of goods. At the end of the period both person A and person B have 100\$ in their bank account, but now person B has 50\$ worth of investment goods. In order for saving to be equal to investment we must include saving in terms of investment goods in our definition of saving. Sometimes, in everyday life, we refer to saving solely to refer to saving of financial assets, but that is not how it is defined for the purposes of the S=I accounting identity.

It is not always clear what exactly should be defined as saving. For example, if I buy a durable good, for example a car, that is typically defined as consumption in macroeconomics, while if a business buys a car that is typically defined as investment. However, it does not matter exactly how we define investment: the S=I identity will always hold as long as we make sure to add the amount invested to our saving identity. If I decide that buying a car as a consumer is investment then I must add that amount to saving, if I decide not to count it as investment then I don’t.

In essence, what the S=I identity is saying is that the only net saving that is possible in the economy is saving in real goods. Because financial transactions always sum to zero, the only way to increase wealth is by increasing the amount of physical goods in the economy. Whatever physical goods we decide to include in our measure of wealth that will always be true.

Now that we understand the S=I identity we can dispel some of its common misuses. Sometimes people infer from the identity that if we encourage people to save more the stock of productive capital will increase. Our first example showed this to be false: if people save in financial assets their saving will simply lead to dissaving by someone else, or their saving will be unsuccessful and lead to a decline in income. However, if people want to save in terms of physical assets the wealth of others in the economy will not decrease.

Keynesians argue that investment is typically what causes net saving because people typically want to save financial assets more than they want to save in physical goods. If, when people wanted to save, they simply created more physical assets, then an increased desire to save would not slow down the economy, in the absense of debt accumulation. Keynes actually discusses in detail why people typically want to save financial assets as opposed to real capital goods in Chapter 17 of the general theory.

Keynes argued that every asset has a holding cost, a rate of return, and a liquidity premium. The holding cost of the asset is the cost to store it, the rate of return is the return the asset generates, and the liquidity premium is how easy the asset is to sell or exchange for something else. Financial assets have a very low holding cost and a very high liquidity premium. If a capital good generates a very high return then people might prefer to save in this form rather than in money. Keynes argued that the return on every capital good decreases as more of the capital good is acquired, and therefore eventually people will prefer saving in financial terms.

In an economy with many productive investment opportunities, an increased desire to save would therefore only lead to increased investment. Eventually, the return on those investments would not be enough to justify the cost of holding the capital goods and the difficulty selling those assets, so people will save in money terms. This fits with what we see in everyday life: in general, if we want to save we will do so by accumulating money in our bank account. We will only invest if we see an investment we think will have a suitably high return.

A final point that should be clarified is that investment in the S=I identity does not need to be productive investment. While you could argue that in practice people will not invest unless the return on that investment is positive, the identity holds even if the investment we are talking about has a negative return. For example, if a business buys equipment they do not use that will increase investment in the S=I identity, but it will not increase the productive capacity of the economy.

Probably the most egregiou example of an economist who gets the S=I identity wrong is Gregory Mankiw. Mankiw is commonly regarded as one of the most important macroeconomists, and his macro and micro textbooks are among the most commonly used. In the third edition of his microeconomics textbook he explains the S=I entirely wrong.

Mankiw writes

Consider an example. Suppose that Larry earns more than he spends and deposits his unspent income in a bank or uses it to buy a bond or some stock from a
corporation. Because Larry’s income exceeds his consumption, he adds to the nation’s saving. Larry might think of himself as “investing” his money, but a macroeconomist would call Larry’s act saving rather than investment.

This is entirely wrong. In order for Larry to save a portion of his income in the form of a financial asset the rest of the economy must be running a deficit. Larry’s saving does not add to the national wealth.

Mankiw then embarks on an elaborate rationalization where he discusses how the additional saving in Larry’s bank account causes additional investment. This entire discussion is wrong. An identity must be true in all possible economies, so you cannot appeal to some particular fact of the banking system when justifying it. In particular, the identity must hold in a world where, for example, people’s decisions about saving and investment have nothing to do with the interest rate. It is beyond embarrassing for the economics profession that someone who is so prominent within the field can make such a basic error, and it can remain in macroeconomics textbooks unchallenged for three editions.

Economists sometimes claim that macroeconomics is hard because people have feelings. I would submit that any field would be difficult if the leading researchers in it made elementary errors in the textbooks they present to the public. Why should anyone trust the complicated models macroeconomists use when the leading proponents of such models can’t even get basic accounting correct? I am sure every year thousands of students give up economics because it makes no sense to them after they read material that is simply incorrect on a basic level.

9 thoughts on “S=I: The most misunderstood equation in economics”

1. James says:

Good post. I have enjoyed your earlier posts too. I agree with most of the principles of what you have written here although I disagree with some of the detail.

I am not an economist, but I have been reading about economics since 2008. My career was in business and government and it often involved resolving disputes between very smart people with different perspectives.

Academic economics is undoubtedly the most dysfunction profession I have ever come across. For example, the economy IS an accounting system. However, most economists do not understand accounting and talk about “mere accounting” in the same way that mathematicians might talk about “mere arithmetic”. The difference is that mathematicians have studied and absorbed arithmetic and then moved on to more advanced subjects, whereas most economists have not studied basic accounting and see it as irrelevant even when discussing accounting identities that are based on … accounting. That is absurd but academic economics is extremely insular and self-regarding.

Keynes first talked about S = I about 80 years ago. Accounting identities are included in introductory courses. Yet, there are endless debates about these identities between the people who teach these courses and write the textbooks. That suggests that no academic economist fully understands these identities. Otherwise, an enterprising academic economist could write a definitive paper on identities and put the endless debates to rest.

MG: “An identity must be true in all possible economies”

Yes. This is a crucial point. Assume an identity is true in all possible economies at a macro level. It must also be true for all possible individual transactions at a micro level. Otherwise, we can start with a macro economy where the identity holds and add precisely one occurrence of a micro-level transaction where the identity does not hold. That would lead to a new macroeconomy where the identity does not hold – which contradicts our initial assumption. Another way of saying this is that the rules of accounting are the micro-foundations of economics

MG: “What helped me to finally understand S=I was working through examples of very simple economies and seeing how the S=I identity must hold”

Yes. If an identity holds at a micro transaction level, we can play with a toy economy of specific micro-level transactions to work out the accounting logic that must apply for the identity to hold. That is what you are starting to do here. For some reason, hardly anyone seems to do this. We could have a useful conversation about such examples to clarify how and where we agree or disagree on the detail.

MG: “it does not matter exactly how we define investment”

No. I do not agree with that. Either economists have clear and consistent definitions of terms like investment and saving or they do not. No sensible discussion is possible if we do not agree basic terminology. Imagine if chemists did not agree on terms such as Oxygen or atom!

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1. Excellent comment. A lot of the points you make are points I was thinking of making in a more general post about accounting identities in economics and how economists talk about them.

When I said it doesn’t matter how you define investment, I meant for the purposes of the S=I identity being true. You are of course correct that it is important to have a consistent definition within the field to facilitate good communication.

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2. Learning Stuff says:

Thanks for this very clear explanation.

In table 3 (with investment), Total Income (A+B) should be 100+100 = 200 and not 150?

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3. ericson2314 says:

A friend pointed me to this, and I’ve done a lot of thinking about it. I think there is an ecological economics angle here that is also useful to consider.

So, one thing that struck me as funny at first was that the machinery was accounted for *after*, but not *before* the sale: i.e. there was no “initial capital stock”.

> …both of them run businesses that produce consumption goods when those consumption goods are ordered.

I think you meant goods in general?

Because it seems the statement applies to the consumable goods and machinery alike. In this case, person A made the machinery “on demand” just like the consumable goods, and that’s why there is no initial capital stock accounted for: it was 0. (All that said, maybe you might want to still add that row for completeness and clarity?)

So taking a step back, before the investment we had *net* saving = 0, which makes sense because this is a closed monetary system. And net investment = net saving = 0 makes mathematician me that’s new to economics much more at ease. Consumable good were also being created and consumed in equal parts in those earlier scenarios, another nice conservation law. Finally, had the machinery been stocked at the beginning by person A, we would have also had another -50 for them, and thus more net 0. (And would that be -50 saving or -50 investment? I’m not quite sure.)

The salient thing here is *extraction*, and specifically the conversion of unaccounted for goods into capital which *is* assessed and given a nominal value.

Absent extraction, all saving and investment is net 0, and without modeling the uncertainly in assessing durable goods, depreciation/appreciation, etc. etc., as done in these simple examples, the difference in monetary and non-monetary capital also hasn’t been motivated. What *does* distinguish them in this simple model of the economy / it’s accounting that is that supply money is fixed while the supply of durable goods isn’t, and so we are back to extraction.

Finally, net saving = net extraction is just as correct and sounds a hell of lot more intuitive.

So, putting it all together, the ecological economics deplore not accounting for externalities as a moral failure, but I think this topic shows that even in the short term it’s also a pedagogical failure. Econ freshmen’s “closed economy” is not a real closed system in the sense their physicist classmates would learn, and so it would seem the former are told that it’s the easy simple case without it supporting the nice reasoning and good intuitions a bonafide closed system would.

The neoclassicists will probably insist those externalities are too hard to assess, more so than productive physical capital and warehoused goods, or that the ecological economist’s arguments are flawed, but maybe they that could at least be persuaded to save the adjective “closed” for actual closed systems!

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4. This is a fascinating post, but I think it is incorrect — in that it sees the S==I identity as based on some kind of “wealth” balance sheet for the economy, when the national accounts are mainly used to track income (flows), not wealth (stocks). Recall, too, that NIPA GDP figures are built from surveys, by measuring flows of economic activity per period and then by bucketing the results into “accounts.” The income accounts are not truly maintained by double entry. Rather, the bucketing is subjected to a set of definitions that attempt to make the results consistent ex post.

Now, the consistency requirement leads to some arguably strange definitions. Your point that ‘S’ (in the restricted sense of personal income consumers do not spend) does not equal ‘I’ (in the restricted sense of investment in income-producing things) is no doubt correct –and Keynes spent much ink on what to do, in certain circumstances, about the difference between these numbers, but the ‘S’ and ‘I’ buckets in the NIPA accounts are not this restrictive. For example, “I” (or Private Investment) includes new home purchases, net changes in inventory, etc.

You pick on Mankiw for oversimplifying this issue (fair enough, but in this passage you quote, he’s thinking of savings as a stock). I keep a shelf of intermediate macro textbooks. Most do not mention the S==I identity in the context of national accounts. Robert Gordon is an exception; he mentions the identity in passing. Dornbusch, Fischer, and Starz are another exception – but they point to the excess of S over I (using your definitions) right in the NIPA discussion.

For all of this, I think most macroeconomists would agree that that S==I identity is often a useful simplification: in the attempt, for example, to build simple models of economic growth. The Solow Model, one of the foundational models of modern macro, is a case in point, and it’s an example (I would hold) of a valid use of S==I in that the model’s results are not sensitive to the simplification.

All of my objections to your point presume some sort of beyond-Econ-101 sophistication. I agree that introductory or journalistic discussions of the economy or of economics do contain many oversimplifications and distortions (of which S==I is far from the worst!).

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In this particular case it does not really matter whether we are using stocks or flows. Stock vs flow issues are important if you are comparing a stock to a flow, but comparing flows to flows or stocks to stocks is usually equivalent. After all, in order for two things to always be equal their rates of change must be equal.

I think you are misunderstanding accounting identities. They are not created in an ad hoc way to justify the accounts, they are true by definition. Ie unless we are measuring things incorrectly they must always hold. This means that we can’t appeal to measurement error to attempt to explain differences between S and I if we use incorrect definitions. The definitions must be such that the identities always hold. The way measurement errors effect NIPA accounts is an interesting topic, but it isn’t really relevant to this post.

>For all of this, I think most macroeconomists would agree that that S==I identity is often a useful simplification: in the attempt, for example, to build simple models of economic growth. The Solow Model, one of the foundational models of modern macro, is a case in point, and it’s an example (I would hold) of a valid use of S==I in that the model’s results are not sensitive to the simplification.

You are misunderstanding what an identity is. An identity must always hold true. Mainstream economists misunderstand the S=I identity and appeal to some sort of behavioural growth model to justify it, but that is a basic level error. It isn’t an oversimplification, it is flat out wrong.

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5. Steve Roth says:

“An identity is a mathematical fact that is true in any possible world, with any possible set of behaviours.”

I disagree. An accounting identity is a definition of terms. “In this coherent, closed-loop web of accounting relationships that we’re working within, X labeled measure has these relationship to A, B, C, D, E labeled measures.” X only has meaning (is defined) relative to those other measures.

Now sure: within that complete accounting construct the identity is always true. But that construct is an inevitably stylized model of the world/economy, necessarily based on economic assumptions that include…the constituent accounting identities.

Click to access Borges.pdf

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