Wednesday, 17 December 2014

(S – I) = (G – T) + (X – M)?

Frances Coppola describes the above as her “favourite equation”. I think the equation has problems. Re the letters / symbols:
S = Private sector saving.
I= Private sector investment.
G= Government spending.
T=Government income, i.e. tax.
X =Exports.

One problem with the equation is that there is no sharp dividing line between “I” (investment) and current spending. To illustrate, if I buy something that will last a year, is that an “investment”? Or does it have to last two years, three years or what?

And if the dividing line is one year, that implies that if I buy something that will last 360 days that’s not an investment, so (S-I) remains unchanged, whereas if I buy something that will last 370 days, then (S-I) is reduced by the amount of the “investment”. That is clearly a nonsense.

Worse still, if someone buys bonds in some domestic (i.e. not foreign) corporation, that counts as an “investment” as per normal use of the English language, but the flow of money there remains entirely within the domestic private sector, so there’s no change to G-T or X-M. Yet if that purchase of bonds counts as an investment, then S –I changes. A self-contradiction.

I suggest the equation is better written as S=(G-T)+(X-M), where all symbols refer simply to movements of cash. E.g. S is simply “domestic private sector” savings or accumulations of cash. Also, (X-M) is not “trade balance” as is often suggested: it’s movements of cash in payment for exports and imports (not quite the same thing).


P.S. (1st Jan 2015). If you think I’ve got my knickers in a twist over this, you’d be right. But getting all the definitions (of S,I,G,T etc) right is not easy: JKH devotes 17,000 words to the above simple equation. I’ll try to sort it out in about one hundredth that number of words. Here goes.

As JKH rightly points out, “(S – I) is the saving delivered from outside the private sector; (I) corresponds to saving delivered from within it.” Thus as far as sectoral balances go, “I” is irrelevant: it’s a movement of assets WITHIN a particular sector – the domestic private sector. So in that case, the equation might as well be written S=(G-T)+(X-M), where S is defined as something like “net accumulation of assets by the domestic private sector from the government  and foreign sectors during some given period”.



  1. "if I buy something that will last 370 days, then (S-I) is reduced by the amount of the “investment” "
    Nope. Reclassifying part of C as I increases both S and I by the same amount. So S-I is unchanged.

    1. Is this a better answer to my 360/370 day point?

      “I” does not represent investment quite as the word is normally understood (machinery, buildings, etc). It represents “net accumulation of goods during the relevant period” and that will obviously include investment in the “machinery etc” sense, but it also represents CONSUMABLE items (e.g. oil for central heating) purchased during the relevant period, but not used.

      Thus my 360/370 day point is irrelevant because the relevant number of days is the length of the period under consideration, which can be anything: one week, one month, two years, etc.

      If the equation is re-arranged like this S=I+(G-T)+(X-M), it makes more sense. It’s saying that private sector saving over some period equals, 1, net accumulation of goods / assets over the period (i.e. goods bought less goods consumed), plus 2, flows of cash from government to the private sector net of tax, plus 3, flows of cash from the foreign sector net of imports.


Post a comment.