Headscratcher Trilogy (3) — A Different Answer (Concerning a Question on Sectoral Balances)

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[Mich beschäftigt die Frage, wie es möglich ist, das nach einer Darstellungsweise der drei Sektorsalden für Privatwirtschaft, Staat und Außenwirtschaft, alle drei Sektoren gleichzeitig einen Überschuss ausweisen können, während dies nach einer anderen Darstellungsweise nicht sein kann, und vielmehr gilt, dass ein Überschuss in einem Sektor ein Defizit in mindestens einem der anderen beiden Sektoren zwingend erfordert.]

A brief, first sketch — in statu nascendi, I'm detained by other duties — of my tack to this question.

The confusion arises because the third sector — the external sector — is being defined from differing angles, one accommodating the possibility of simultaneous surpluses in all three sectors, and one demanding the "Wray condition", whereby a surplus in one of the three sectors must be balanced by a deficit in at least one of the other two sectors.

(1.)

There is a genuine three sector model - strictly defined from the perspective of one economy,  say the US economy. It comprises the (1) private domestic sector of the USA, (2) the government sector of the USA, and (3) the external sector of the USA.

The external sector of the USA may be defined as all activities that affect national income by exports from the US economy to non-US-economies (enhancing income in the US economy) or imports into the US economy from non-US-economies (reducing income in the US economy). 

This is captured in

 (S – I) = (G – T) + (X – M),

reflecting what I call the "genuine" three sector model — "genuine" in the sense that all three sectors belong to one and the same economy.

Within this model, it is possible for all three sectors to be in surplus. For it implies that there is a sector that goes into deficit to balance the three domestic surpluses — the rest of the world.

(2.)

In describing three sectors that affect the income of the US economy, I may replace the domestic external sector, (X-M), by "the rest of the world". Or, in fact, define a surplus of the domestic external sector as a deficit of the rest of the world, and a deficit of the domestic external sector as a surplus of the rest of the world. I am changing my perspective and look at trade-related deficits and surpluses from the point of view of the rest of the world, rather than from the point of view of the domestic economy.

The rest of the world is implied in the "domestic external sector". If the domestic external sector of the US economy is in surplus vis-à-vis the rest of the world, the rest of the world must be in deficit vis-à- vis the US economy).

So there is a second, alternative, and, if you wish, "impure" three sector model, one that takes two sectors of the domestic economy plus the rest of the world.

In this second and alternative account of income changes to a given economy, it is true what Wray contents, namely that a surplus in one sector must be accompanied by a deficit in at least one of the other two sectors.

So, depending on whether you describe the change to the income of an economy induced by transactions with foreign economies 

• as an inflow into/outflow out of that economy or

• as an inflow into/outflow out of the rest of the world, 

you end up with models 

• that both accurately capture income changes to the economy, 

however, 

• one of which allowing for three simultaneous surpluses (and thus violating the Wray condition), while the other does satisfy the Wray condition.

When the rest of the world (ROW) is needed to balance the surpluses of the domestic economy, the rest-of-the-world-model brings that deficit sector directly into the picture, while the genuine-three-sector-model only implies the ROW as the deficit pendant to the surplus of the domestic external sector, explicitly featuring only the latter.