Recently I have come across many people saying S = I is not correct or that it is valid only for a purely private economy.
People who come across the sectoral balances identity
S − I = G − T + CAB
sometimes mix up with another macroeconomic identity
S = I
They are unsure of the correctness of each or the kind of setup one or the other is valid. Or – how can both be valid?
The reason both are valid is that the former is a sectoral equation and the S refers to the private sector saving and in the latter, S refers to the sum of savings of all sectors of the world*.
Remember good authors always specify the terminology used and you should too always.
But the first equation doesn’t make the second equation invalid. It is just that the contexts are slightly different.
So in the first equation, S is the private sector saving, I is private sector investment and G is the government expenditure, T is tax receipts of the government and CAB is the current account balance of payments.
So, in case you want to use the two simultaneously, use different notations.
Take an example of a closed economy.
The government’s expenditure is comprised of current and capital expenditures.
G = Gcurrent + Igovt
Government’s saving is T − Gcurrent
Spriv − Ipriv = G − T = Igovt − (T − Gcurrent)
Spriv − Ipriv = Igovt − Sgovt
Spriv + Sgovt = Ipriv + Igovt
which is written as S= I in short by authors – which is fair because they assume a context.
There is source for further confusions. This is because of mixing of the same set of symbols for planned investment and planned saving with the recorded ones. Clearly, planned investment needn’t be equal to planned saving. However, it is always good when discussing ex-ante and ex-post variables to use different symbols.
Some authors mix notation everywhere in their analysis, and call others confused!
If one doesn’t keep track of notations, a more complicated analysis will produce nonsensical results.
If someone doesn’t understand this, his confusions are his problems, not others’.
*Savings used as a plural of saving and as a flow instead of informal terminology referring to a stock