S=I+(I-S)?

Commenter Greg points me to this post: More on Savings and Investment

“It is perfectly possible to hold the international balance constant, have the government reduce debt, and have “people” save more.

“People’s” financial savings consists of claims on firms and claims on government. If I perform some work for a firm that (however infinitessimally) increases the firm’s real economic value, and I accept as payment a share of that firm’s stock, I have performed the economic act of saving, and increased the net saving of “people” — of the household sector. Net private sector financial assets have not increased: my “savings” is the firms’ obligation, the household sector’s surplus is offset by the business sector’s deficit.

But much of what we call saving is exchanging real resources for claims on the private business sector. And as long as the private business sector doesn’t entirely squander those real resources, that act contributes to macroeconomic S. If the private business sector does squander the resources, then while I still perceive my contribution as “saving”, the value of macroeconomic S = I does not increase, and my claim amounts to a transfer from other shareholders of the firm.

It seems similar to this same theme raised in Interfluidity (here and here). At any rate, I think they are all related.

Honestly, I'm having difficulty making head or tail of the discussion. I'm not sure what distinction they are making, and I don't know why the distinction I think they are making is important. It seems to be that, the usual sector de-composition is between the Government and non-Government sector, where Government is the currency issuer (consolidated Treasury and Federal Reserve function) and non-Government is everyone else (all currency users, includes foreign Governments). I think they are saying that within the non-Government sector, distinguishing the household from commercial sector is important, but I don't know why.

Anyone care to enlighten me?

Thanks