ASD is right. My maths are bad. The fault is entirely mine in trying to explain the concept, not of the concept itself. Here is economist Bill Mitchell’s explanation.
We can view the basic income-expenditure model in macroeconomics in two ways: (a) from the perspective of the sources of spending; and (b) from the perspective of the uses of the income produced. Bringing these two perspectives (of the same thing) together generates the sectoral balances.
From the sources perspective we write:
GDP = C + I + G + (X – M)
which says that total national income (GDP) is the sum of total final consumption spending (C), total private investment (I), total government spending (G) and net exports (X – M).
From the uses perspective, national income (GDP) can be used for:
GDP = C + S + T
which says that GDP (income) ultimately comes back to households who consume (C), save (S) or pay taxes (T) with it once all the distributions are made.
Equating these two perspectives we get:
C + S + T = GDP = C + I + G + (X – M)
So after simplification (but obeying the equation) we get the sectoral balances view of the national accounts.
(I – S) + (G – T) + (X – M) = 0
That is the three balances have to sum to zero. The sectoral balances derived are:
The private domestic balance (I – S) – positive if in deficit, negative if in surplus.
The Budget Deficit (G – T) – negative if in surplus, positive if in deficit.
The Current Account balance (X – M) – positive if in surplus, negative if in deficit.
These balances are usually expressed as a per cent of GDP but that doesn’t alter the accounting rules that they sum to zero, it just means the balance to GDP ratios sum to zero.
A simplification is to add (I – S) + (X – M) and call it the non-government sector. Then you get the basic result that the government balance equals exactly $-for-$ (absolutely or as a per cent of GDP) the non-government balance (the sum of the private domestic and external balances). This is also a basic rule derived from the national accounts and has to apply at all times.
The private domestic sector is only one part of the non-government sector – the other being the external sector.
Most countries currently run external deficits. This means that if the government sector is in surplus the private domestic sector has to be in deficit.
However, some countries have to run external surpluses if there is at least one country running an external deficit. That country can depending on the relative magnitudes of the external balance and private domestic balance, run a public surplus while maintaining strong economic growth. For example, Norway.
In this case an increasing desire to save by the private domestic sector in the face of fiscal drag coming from the budget surplus can be offset by a rising external surplus with growth unimpaired. So the decline in domestic spending is compensated for by a rise in net export income.
So if all governments (in all nations) are running public surpluses and some nations are running external deficits (the majority), public surpluses have to be associated (given the underlying behaviour that generates these aggregates) with private domestic deficits.
Even if the external sector balance was zero, the proposition would still be true. At least one private domestic sector would be unable to save overall.
These deficits can keep spending going for a time but the increasing indebtedness ultimately unwinds and households and firms (whoever is carrying the debt) start to reduce their spending growth to try to manage the debt exposure. The consequence is a widening spending gap which pushes the economy into recession and, ultimately, pushes the budget into deficit via the automatic stabilisers.
So you can sustain economic growth with a private domestic surplus and government surplus if the external surplus is large enough. So a growth strategy can still be consistent with a public surplus. Clearly not every country can adopt this strategy given that the external positions net out to zero themselves across all trading nations. So for every external surplus recorded there has to be equal deficits spread across other nations.