Take a standard New Keynesian macroeconomic model, where the central bank sets the one-period interest rate. Now let's hit it with an unexpected shock. For simplicity and concreteness, let the shock be a reduction in government spending that will last for n periods, after which government spending will return to normal. So government spending was 100, will now be 90 for n periods, then will be 100 again in period n+1 and thereafter.
If n=1, the model says the central bank should cut the real interest rate for one period, then raise it again to its original level.
If n=2, the model says the central bank should leave the real interest rate unchanged for one period, then cut it in the second period, then raise it again to its original level in the third period.
If n=3, the model says the central bank should leave the real interest rate unchanged for two periods, then cut it in the third period, then raise it again to its original level in the fourth period.
If n=infinity, the New Keynesian model says the central bank should leave the real interest rate unchanged.
My hunch is that most students of New Keynesian macroeconomics knew about the results for n=1 and n=infinity, but did not know about the results for n=2 and n=3 etc., and had to think about it. Am I right?
For the case where n=1, the New Keynesian model gives exactly the same policy advice as an Old Keynesian model. But for all the other cases it gives very different policy advice. That's because the Old Keynesian IS curve says that the level of private demand in period t is a negative function of the real interest rate in period t. But the New Keynesian IS curve says that the expected growth rate of private demand between periods t and period t+1 is a positive function of the real interest rate in period t.
How long is the period?
In the real world, central banks like the Bank of Canada set an overnight rate of interest, and they typically set it every six weeks. "Overnight" and "six weeks" are both very short periods. Most shocks last for longer than six weeks.
If the New Keynesian model is correct, it would be impossible for central banks to compensate for the effects of most shocks by doing what they do now. If a shock hit that was expected to last for 3 years, the central bank should use "forward guidance" and announce that it would lower the rate of interest temporarily 3 years from now. Or it could get to work on the term-structure of interest rates, by lowering the 3 year interest rate, while leaving all interest rates with less than 3 year terms unchanged.
I don't observe central banks actually doing anything like that.
The only sort of shocks where current policy would work would be shocks that have their maximal impact immediately, and then immediately start to decay at some geometric rate back to zero. In those special cases New Keynesian models would give the same advice, qualitatively, as Old Keynesian models.
If instead we had shocks that steadily grew over time, then started to decay back to zero, New Keynesian models would give the exact opposite advice to Old Keynesian models in the initial periods. For example, if the government announced that government spending would start to decline, stop declining after two years, then start to rise again, the New Keynesian model says the central bank should immediately raise real interest rates, slowly lower them for two years, then slowly start raising them again.
Do New Keynesians understand their own models? Do New Keynesians rig their models by only allowing shocks that make New Keynesian models behave like Old Keynesian models? Why aren't New Keynesian macroeconomists the ones arguing that central banks must operate on the whole term structure of interest rates and not just on the overnight rate of interest for the next 6 weeks?
Mathematic appendix: let C(t) be private demand, G(t) be government demand, and Y* be potential output, at time t. The Euler equation IS curve in New Keynesian models says that C(t)/Et[C(t+1)] = alpha(1+z)/(1+r(t)) where Et is expectations at time t, r(t) is the real interest rate, z is the rate of time preference proper, and alpha is a positive elasticity parameter. The central bank's job is to set r(t) such that C(t)+G(t)=Y*. Which means (Y*-G(t))/Et[Y*-G(t+1)] = alpha(1+z)/(1+r(t)). Assume C(n+1)+G(n+1)=Y* because that's what New Keynesians assume. Then solve backwards from there.