by *Keynesian of sorts*

I see a lot of confusion concerning why the private sector balance is represented by (S-I) in the sectoral balances equation. This is something that confused me for a long time when starting to learn about macroeconomics, so I thought it might be a useful resource to quickly explain what (S-I) means and why it represents the balance of the private sector.

Firstly lets derive the sectoral balances equation for a closed economy with no government:

We have the expenditure approach to GDP: Y=C+I

We then consider the uses approach to GDP: Y=C+S

Where Y=GDP, C=Consumption, I=Investment and S=Savings

As these are both equal to Y, we can write C+I=C+S, simplifying we’re left with the identity S=I. In a private sector without flows from a government or external sector, savings are equal to investment.

There are two main arguments that attempt to explain the causation that makes this identity hold. The neoclassical approach emphasises the concept of a market for loanable funds. The loanable funds market is conceptualised as a market that brings savers and borrowers together; savers store their deposits at a bank and then the bank lends these savings to borrowers who then invest the funds. The interest rate then acts as the adjustment mechanism that brings the quantity of savings into equality with the level of investment. In the neoclassical narrative, savings and investment are only equal at the point of equilibrium. The other view is the Keynesian argument, employed by Post-Keynesians and MMTers; this argument claims that savings are always equal to investment ex post due to adjustments in income. Let’s suppose investment increases. This grows income for the firms that supplied the capital goods. The increase in income will then grow the level of savings for a given marginal propensity to save as the individuals will now have a higher income from which to save. This process will adjust savings until they are again equal to investment.

It’s important to note here that the definition of investment used in the GDP identity differs from that used colloquially. In macroeconomics, investment specifically refers to firms purchasing final capital goods and firms’ inventories. The purchase of financial assets or the trading of existing assets are not counted as investment in the GDP identity.

So we now know that for a closed economy with no government S=I, but what happens when we add external and government sectors? To find out, we need to derive the full sectoral balances equation.

We have the expenditure approach to GDP: Y=C+I+G+(X-M)

We have the uses approach to GDP: Y=C+S+T

Where G=Government spending, X=Exports, M=Imports and T=Taxation.

The uses approach says that once funds are all distributed to the household level (whether they are workers or capitalists) they can use their funds to consume, save and pay their taxes. Equating the two equations and simplifying gives us:

S+T=I+G+(X-M), isolating S to give us a definition of savings S=I+(G-T)+(X-M).

This says that savings are equal to investment plus the balance of the external and government sectors. Since we know that in a closed economy with no government S=I, then if S>I we know that this must be due to a net debit/deficit (flow) from the government sector, from the external sector, or both sectors. We can then rearrange to get the classic sectoral balances equation:

(S-I)=(G-T)+(X-M)

As we have shown above, as long as (G-T)+(X-M) ≠0, S≠I as the balance of the other sectors will either add to or subtract from the savings of the private sector bringing S and I out of equality. Going through an example let’s assume that G-T=2 (a government deficit) and X-M=0, then we know that S-I=2. Rearranging we get S=I+2, in other words savings are equal to investment plus the 2 that is a flow from the government sector due to the government deficit. Another way to think about this is to say that the net savings created in the private sector by the other sectors is equal to 2.

This therefore means that the amount by which savings and investment differ can be seen as a measure of the extent that the balance of the private sector is being influenced by the balance of the government and external sectors at any one time. If S>I we know that a government deficit, a trade surplus or both are adding net savings to the private sector; and if S<I we know that a government surplus, trade deficit or both are draining net savings from the private sector. This is therefore why we consider (S-I) to be a measure of the net balance of the private sector.