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# Explaining S=I: Inventories vs Adding up Individuals

Summary:
It's easy to teach students the arithmetic showing that actual saving must equal actual investment (S=I). But many students (quite rightly) want more than the arithmetic. Because S=I is not intuitive, and good students want to understand the intuition. I think that most profs will try to explain the intuition by talking about inventories of unsold goods. But I think it would be better to explain the intuition behind S=I by talking about the difference between individual saving and aggregate saving; some types of individual saving don't add up to aggregate saving. A student's email is what prompted me to write this post. He's read three textbooks on the subject, and has read my old post, and understands the arithmetic, but still doesn't understand the intuition. Here's one of his

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It's easy to teach students the arithmetic showing that actual saving must equal actual investment (S=I). But many students (quite rightly) want more than the arithmetic. Because S=I is not intuitive, and good students want to understand the intuition. I think that most profs will try to explain the intuition by talking about inventories of unsold goods. But I think it would be better to explain the intuition behind S=I by talking about the difference between individual saving and aggregate saving; some types of individual saving don't add up to aggregate saving.

A student's email is what prompted me to write this post. He's read three textbooks on the subject, and has read my old post, and understands the arithmetic, but still doesn't understand the intuition. Here's one of his questions: "Or, suppose I buy a used car. Is it saving? How will in this example saving be equal to investment?". Here's how I would answer his question: "Yes, your buying a used car is saving, because you spent part of your income on something that is not a newly-produced consumption good. Which is how we define "saving" in National Income Accounting. But someone else must have sold you that same used car, so if you were saving (buying an asset), that someone else was dissaving (selling an asset). It is impossible for everyone to save by buying used cars."

My Twitter poll suggests the student is not alone. And I suspect that many of those who answered "Easy" did so because they were taught the "Inventory" intuition: "If saving exceeds desired investment, then firms produce more goods than they sell, so unsold goods pile up as undesired inventory accumulation, which counts as actual investment. The orchard sold the unsold apples to itself!"

Here's why I don't like that "Inventory" way of explaining the intuition. S=I is an accounting identity, and accounting identities are true by definition. So it's wrong to try to explain an accounting identity by smuggling in an assumption about the world: that firms decide on the level of production before knowing how many goods they will sell. S=I is still true in an economy where firms produce goods to order, so never find themselves unable to sell goods they have produced. And S=I is still true in an economy where all goods are pure services, like haircuts (consumption) and teaching haircutting (investment), that can't be produced then stored in inventory waiting for a buyer. And, I think that what students have difficulty understanding is not about inventories but about the difference between individuals vs aggregates. And Macro is supposed to be about the difference between individuals and aggregates (fallacies of composition etc.) anyway.

Here's how I think about "saving":

I earn \$1,000 income this month (or year or whatever) from producing goods. What can I do with that \$1,000?

1. I can spend it on newly-produced consumption goods.

All the remaining categories count as "saving":

2. I can spend it on newly-produced investment goods.

3. I can spend it on anything that is not a newly-produced consumption good or newly-produced investment good. The list is endless, but would include used cars, old paintings, land, bonds (when I lend money I am buying an IOU or bond), etc.

4. I can not spend the money at all, but leave it sitting in my pocket or chequing account.

When we add up across individuals, category 3 above disappears. If I buy your used car, then you sold that used car. If I buy your old painting, then you sold that old painting. If I buy your land, then you sold that land. If I buy your IOU (lend you money), then you sold your IOU (borrowed money). It all cancels out in aggregate.

If it weren't for Category 4, I could stop here. Since Category 3 saving cancels out in aggregate, that leaves only Category 2 saving, so aggregate saving must equal aggregate investment. The only way we can all accumulate more assets (saving) is if we produce more assets (investment). Anything else is just one individual transferring an asset to another individual.

Category 4 is where the fun starts.

You might say that Category 4 is like Category 3, and in a way it is. If I accumulate money, then you or someone else must deccumulate money, if there's a fixed stock of money. Or if a bank creates money, so I can accumulate money, it does so by buying an IOU, so you must have sold an IOU, and the money the bank creates is the bank's IOU, so all the IOUs cancel out, so categories 3 and 4 together cancel out.

But there's a difference between Category 3 and Category 4.

Category 3 saving is voluntary exchange. I can't buy your used car, and you can't sell your used car to me, unless both of us agree to the deal. I want to save, and you want to dissave. Category 3 saving is a bilateral decision with desired saving by me and desired dissaving by you. Category 4 saving is a unilateral decision. If I want to save by leaving the money I have earned sitting in my pocket, I just do it. I don't need to ask your permission and persuade you to dissave.

But since my expenditure is your income, if I spend \$100 less money on anything, your money income is \$100 less. And it's the same for me if you spend \$100 less money on anything. Each of us as individuals can save by reducing the flow of money out of our pockets, without it affecting the flow of money into our pockets. But collectively the flow in equals the flow out, so collectively we can't save that way. Our individual attempts to save by accumulating money will collectively fail, but will cause our money income to fall until we stop trying.

Addendum: Here's the arithmetic of S=I:

Define Y as market value of newly-produced final goods (and services). In a closed economy (no net exports NX) with no government spending G or taxes T we divide Y into consumption goods C and investment goods I, so Y=C+I. And we define saving S as S=Y-C. Substitute the first equation into the second to get S=Y-C=C+I-C=I, so S=I. If we bring government into the picture it changes to Y=C+I+G. And private saving is now Sp=Y-T-C, government saving is Sg=T-G, and national saving is S=Sp+Sg=Y-C-G=C+I+G-C-G=I. If we bring foreigners into the picture it changes to Y=C+I+G+NX, so S=I+NX (saving that is not invested domestically is lent to foreigners to finance net exports).