I finally got round to reading this paper by Iván Werning - Managing a Liquidity Trap: Monetary and Fiscal Policy. It takes the canonical New Keynesian model, puts it into continuous time, and looks at optimal monetary and fiscal policy when there is a liquidity trap. (To be precise: a period where real interest rates are above their natural level because nominal interest rates cannot be negative). I would say it clarifies rather than overturns what we already know, but I found some of the clarifications rather interesting. Here are just two.
1) Monetary policy alone. The optimum commitment (Krugman/Eggertsson and Woodford)  policy of creating a boom after the liquidity trap period might (or might not) generate a path for inflation where inflation is always above target (taken as zero). Here is a picture from the paper, where the output gap is on the vertical axis and inflation the horizontal, and we plot the economy through time. The black dots are the economy under optimal discretionary policy, and the blue under commitment, and in both cases the economy ends up at the bliss point of a zero gap and zero inflation.
In this experiment real interest rates are above their natural level (i.e. the liquidity trap lasts) for T periods, and everything after this shock is known. Under discretionary policy, both output and inflation are too low for as long as the liquidity trap lasts. In this case output starts off 11% below its natural level, and inflation about 5% below. The optimal commitment policy creates a positive output gap after the liquidity trap period (after T). Inflation in the NK Phillips curve is just the integral of future output gaps, so inflation could be positive immediately after the shock: here it happens to be zero. As we move forward in time some of the negative output gaps disappear from the integral, and so inflation rises.
It makes sense, as Werning suggests, to focus on the output gap. Think of the causality involved, which goes: real rates - output gap (with forward integration) - inflation (with forward integration), which then feedback on to real rates. Optimum policy must involve an initial negative output gap for sure, followed by a positive output gap, but inflation need not necessarily be negative at any point.
There are other consequences. Although the optimal commitment policy involves creating a positive output gap in the future, which implies keeping real interest rates below their natural level for a period after T, as inflation is higher so could nominal rates be higher. As a result, at any point in time the nominal rate on a sufficiently long term bond could also be higher (page 16).
2) Adding fiscal policy. The paper considers adding government spending as a fiscal instrument. It makes an interesting distinction between ‘opportunistic’ and ‘stimulus’ changes in government spending, but I do not think I need that for what follows, so hopefully it will be for a later post. What I had not taken on board is that the optimal path for government spending might involve a prolonged period where government spending is lower (below its natural level). Here is another picture from the paper.
The blue line is the optimal commitment policy without any fiscal action: the same pattern as in the previous figure. The red line is the path for output and inflation with optimal government spending, and the green line is the path for the consumption gap rather than the output gap in that second case. The vertical difference between red and green is what is happening to government spending.
The first point is that using fiscal policy leads to a distinct improvement. We need much less excess inflation, and the output gap is always smaller. The second is that although initially government spending is positive, it becomes negative when the output gap is itself positive i.e. beyond T. Why is this?
Our initial intuition might be that government spending should just ‘plug the gap’ generated by the liquidity trap, giving us a zero output gap throughout. Then there would be no need for an expansionary monetary policy after the gap - fiscal policy could completely stabilise the economy during the liquidity gap period. This will give us declining government spending, because the gap itself declines. (Even if the real interest rate is too high by a constant amount in the liquidity trap, consumption cumulates this forward.)
This intuition is not correct partly because using the government spending instrument has costs: we move away from the optimal allocation of public goods. So fiscal policy does not dominate (eliminate the need for) the Krugman/ Eggertsson and Woodford monetary policy, and optimal policy will involve a mixture of the two. That in turn means we will still get, under an optimal commitment policy, a period after the liquidity trap when there will be a positive consumption gap.
The benefit of the positive consumption gap after the liquidity trap, and the associated lower real rate, is that it raises consumption in the liquidity gap period compared to what it might otherwise have been. The cost is higher inflation in the post liquidity trap period. But inflation depends on the output gap, not just the consumption gap. So we can improve the trade-off by lowering government spending in the post liquidity trap period.
Two final points on what the paper reaffirms. First, even with the most optimistic (commitment) monetary policy, fiscal policy has an important role in a liquidity trap. Those who still believe that monetary activism is all you need in a liquidity trap must be using a different framework. Second, the gains to trying to implement something like the commitment policy are large. Yet everywhere monetary policy seems to be trying to follow the discretionary rather than commitment policy: there is no discussion of allowing the output gap to become positive once the liquidity trap is over, and rules that might mimic the commitment policy are off the table.  I wonder if macroeconomists in 20 years time will look back on this period with the same bewilderment that we now look back on monetary policy in the early 1930s or 1970s?
 Krugman, Paul. 1998. “It’s Baaack! Japan’s Slump and the Return of the Liquidity Trap.” BPEA, 2:1998, 137–87. Gauti B. Eggertsson & Michael Woodford, 2003. "The Zero Bound on Interest Rates and Optimal Monetary Policy,"Brookings Papers on Economic Activity, Economic Studies Program, The Brookings Institution, vol. 34(1), pages 139-235.
 Allowing inflation to rise a little bit above target while the output gap is still negative is quite consistent with following a discretionary policy. I think some people believe that monetary policy in the US might be secretly intending to follow the Krugman/Eggertsson and Woodford strategy, but as the whole point about this strategy is to influence expectations, keeping it secret would be worse than pointless.