I have written a bit about multipliers, particularly of the balanced budget kind, but judging by comments some recap and elaboration may be useful. So here is why, for all government spending multipliers, one is the number to start from. To make it a bit of a challenge (for me), I’ll not use any algebra.
Any discussion has to be context specific. Imagine a two period world. The first period is demand deficient because interest rates are stuck at the zero lower bound, but in the (longer) second period monetary policy ensures output is fixed at some level independent of aggregate demand (i.e. its supply determined). Government spending increases in period 1 only. That is the context when these multipliers are likely to be important as a policy tool.
1) Balanced budget multiplier
To recap, for a balanced budget multiplier (BBM), here is a simple proof in terms of sector balances for a closed economy. A BBM by definition does not change the public sector’s finance balance (FB). It seems very reasonable to assume that consumers consume a proportion less than one of any change to their first period post-tax income. So if higher taxes reduced income, their consumption falls by less, so their FB moves into deficit. But as the sum of the public and private sector’s FB sums to zero, it cannot do this. So post-tax income cannot fall. Hence pre-tax income must rise to just offset the impact of higher taxes. The BBM is one.
The nice thing about this result is that it holds whatever fraction of current income is consumed (as long as it’s less than one), so it is independent of the degree of consumption smoothing. What about lower consumption in the second period? No need to worry, as monetary policy ensures demand is adequate in the second period.
Although a good place to start, allowing for an impact on expected inflation and therefore real interest rates will raise this number above one. In addition, as DeLong and Summers discuss, hysteresis effects will also raise period 2 output and income from the supply side, some of which consumers will consume in period 1. We would get similar effects if the higher government spending was in the form of useful intrastructure investment. So in this case one is the place to start, but it looks like a lower bound.
2) BBM in an open economy
I’m still seeing people claim that the BBM in an open economy is small. It could be, if the government acts foolishly. Suppose the government increases its spending entirely on defence, which in turn consists of buying a new fighter jet from an overseas country. The impact on the demand for domestic output is zero. But consumers are paying for this through higher taxes, so their spending decreases – we get a negative multiplier.
Now consider the opposite: the additional government spending involves no imported goods whatsoever. The multiplier is one. You can do the maths, but it is easy to show that this is a solution by thinking about the BBM in a closed economy. There consumption does not change, because a BBM=1 raises pre-tax income to offset higher taxes. But if consumption does not change, neither will imports, so this is also the solution in the open economy case.
What the textbooks do is apply a marginal propensity to import to total output, which implicitly assumes that the same proportion of government spending is imported as consumption spending. For most economies that is not the case, as the ‘home bias’ for government spending is much larger. Furthermore, if the government is increasing its spending with the aim of raising output, it can choose to spend it on domestically produced output rather than imports. So, a multiplier of one is again a good place to start. Allowing some import leakage will reduce the multiplier, but this could easily be offset by the real interest rate effects discussed above, particular as these would in an open economy depreciate the real exchange rate.
3) Debt financed government spending with future tax increases
Although this is the standard case, from a pedagogical point of view I think it’s better to start with the BBM, and note that it’s all the same with Ricardian Equivalence. We can then have a discussion about which are the quantitatively important reasons why Ricardian Equivalence does not hold. All these go to raise the multiplier above one. You have to add, however, some discussion about the impact that distortionary tax increases will have on output in the second period, which reduces second period output and, through consumption smoothing, the size of the first period multiplier.
4) Debt financed government spending without tax increases
In an earlier post I queried why arguments for the expansionary impact of government spending increases always involved raising taxes at some point. For debt finance, why not assume lower government spending in the future rather than higher taxes. The advantage is that you do not need to worry about supply side tax effects. Monetary policy ensures there is no impact on output of lower government spending in the second period. Now, unlike the BBM case, we do need to make some assumptions about the degree of consumption smoothing. If you think the first period is short enough, and consumers smooth enough, such that the impact of higher income on consumption in the first period is negligible, then we have a multiplier of one again.
 I assume Quantitative Easing cannot negate the ZLB problem, and that inflation targets are in place and fixed. This is not about fiscal stimulus versus NGDP targeting, but just about macro theory.