“There can be no doubt whatsoever that a problem which has not yet been solved in all its aspects under its simplest conditions will be still more difficult to tackle if other, 'more realistic' assumptions are being made.” (Morgenstern, 1941, p. 373)
So, let us take at first the special case M=X and G=T in your expanded formula then we are back to I=S. Now, in my comment of Nov 30 above I have given the proof that this formula is logically indefensible (see alternative link below). The deeper analysis shows that — in the most elementary case — the correct result is not saving equals investment but saving equals loss.
Clearly, when in the most elementary case saving is not equal to investment then your expanded formula does not hold either. So there is no need to consider it further.
The logical mistake in Wren-Lewis's formal argument is the following:
He says: “In the most simple model of a closed economy without government, income (Y) = consumption (C) + saving (S), but also expenditure (Y) = consumption (C) + investment (I). So S=I by definition. But here investment includes what is called ‘stockbuilding’ or ‘inventory accumulation’, which includes goods that firms wanted to sell but could not.”
Note first that Wren-Lewis pulls inventory accumulation out of the empty hat. Change of inventory presupposes a difference of quantity produced O and quantity bought/sold X, that is, O-X>0 or O-X<0. These real magnitudes do not appear in Wren-Lewis's formalism at all. And from the nominal magnitudes that do appear does not follow that the market is not cleared.
The statement 'investment includes what is called ‘stockbuilding’ or ‘inventory accumulation’' neither has a logical nor a real justification.
Note also that the real variables O and X appear in my formalism. Therefore, my formalism is complete while Wren-Lewis's formalism is incomplete. That is, his argument is unfounded.
From household sector saving does not follow that the market is not cleared and that there is an unsold quantity. Under the condition of market-clearing O=X follows that the market-clearing price falls and that the business sector makes a loss which is exactly equal to saving, i.e. -Qm=S.
Therefore, all I=S-models are false. This applies, of course, also to Keynes's General Theory (1973, p. 63) and to IS-LM (2014).
The deeper reason is that neither Keynes nor Wren-Lewis nor the representative economist can tell the difference between profit/loss and income. And this is why economics is a failed science.
Kakarot-Handtke, E. (2014). Mr. Keynes, Prof. Krugman, IS-LM, and the End of
Economics as We Know It. SSRN Working Paper Series, 2392856: 1–19. URL
Keynes, J. M. (1973). The General Theory of Employment Interest and Money.
The Collected Writings of John Maynard Keynes Vol. VII. London, Basingstoke:
Morgenstern, O. (1941). Professor Hicks on Value and Capital. Journal of Political
Economy, 49(3): 361–393. URL
Alternative link, i.e., full version, no space restriction
For cross-references see here.