Update: see update at very end.
Some people aren't as rational and future-oriented as agents in New Keynesian models. Some people aren't as present-oriented as agents in Old Keynesian models. A hybrid Old/New Keynesian model, with both types of agents, looks attractive.
But other people's hybrid models looked too complicated, so I decided to build my own. All you really need is one equation. (OK, plus a couple more little ones to help explain it.)
I discuss the first-best solution. Then the second-best solution, if the Zero Lower Bound on nominal interest rates is a binding constraint.
The results are not what you might expect. If the ZLB is a binding constraint, the government should reduce the growth rate of government spending, and/or increase the growth rate of taxes, to ensure the economy remains at potential output.
(Paul Krugman again misses this point. "If some of them are instead liquidity-constrained, the increase in income from the rise in G will lead to some increase in C as well, and we have a story that is even closer to the old Keynesian version." It is not the rise in G; it is the fall in the growth rate of G that does the work, which is far away from the Old Keynesian version.)
I explain this weird result, and reconcile it with the Old Keynesian intuition.
The Old Keynesian agents are straight out of the textbook. C(t)=A(t)+B(Y(t)-T(t)), where A is autonomous consumption, B is the marginal propensity to consume, and (Y-T) is current disposable income. I've written it A(t) to allow for shocks to autonomous consumption.
The New Keynesian agents are also straight out of the (different) textbook. If the real rate of interest r(t) exceeds their rate of time preference n(t), their current consumption will be less than their permanent income, and will be growing over time. Assume it's Cdot(t)=R(r(t)-n(t)), where R(.) is some function representing the Euler equation with the properties R(0)=0 and R' > 0, where Cdot(t) is consumption growth. I've written it n(t) to allow for shocks to the rate of time preference.
What we are looking for is a fiscal and monetary policy (I look at both together) that will keep output at potential at all times. Ignoring investment and net exports to keep it simple, that means we want C(t)+G(t)=Y*(t) for all t, where Y*(t) is potential output. Or, taking the derivative with respect to time, we want Cdot(t)+Gdot(t)=Y*dot(t) for all t. Consumption growth plus government expenditure growth needs to equal potential output growth.
Assuming some fraction k of agents are Old Keynesian, and (1-k) are New Keynesian, we want a fiscal/monetary policy that ensures:
k[Adot(t)+B(Y*dot(t)-Tdot(t))] + (1-k)[R(r(t)-n(t))] + Gdot(t) = Y*dot(t)
That's the model.
If k=1 and you integrate with respect to time (delete the "dot"s) you get the first-year Keynesian Cross model's solution for fiscal policy G(t) and T(t) to keep output at potential.
If k=0 you get the New Keynesian model, where Tdot(t) disappears because Ricardian Equivalence holds.
(I've cheated a little because strictly speaking k should represent expenditure shares rather than population shares, but I don't think this matters much.)
The government/central bank has three instruments: G(t), T(t) and r(t), and it only needs one instrument to keep the economy at potential. But if the government wants to maximise agents' utility, it will need to use all three instruments.
Philosophically this is a bit tricky, as Frances Woolley explains, because the government is acting in loco parentis for the non-rational Old Keynesian agents. If A(t) increases, for example, does this represent a change in their underlying preferences that should be accommodated, or is it just an irrational whim that should be suppressed?
The simplest assumption (though one that is certainly contestible) is that the true preferences of the Old Keynesian agents are exactly like the preferences of the New Keynesian agents; they just don't act on them. Given this assumption, we can solve for optimal fiscal and monetary policy.
The full solution is left as an exercise for the reader. Here are my answers:
T(t) should alone be used to offset shocks to A(t). If the Old Keynesian agents go on an irrational consumption binge, raise taxes enough to stop them, without changing G(t) or r(t).
Set G(t) so that the marginal utility of G(t) is equal to the marginal utility of C(t). (Taxes are assumed non-distorting.) This means that if Y*(t) increases, both C(t) and G(t) should increase.
Set r(t) to ensure that output is at potential, given G(t) and T(t). This means that r(t) must respond positively to n(t) and to Y*dot(t), and negatively to Gdot(t) and positively to Tdot(t). (Look at the equation!)
That is the first best solution for fiscal and monetary policy.
Now suppose that monetary policy is constrained by the Zero Lower Bound. The central bank cannot cut r(t) to implement the first-best solution. What are the second best solutions for G(t) and T(t)?
Inspect the equation above. To offset an r(t) that is too high (because of the ZLB), the government needs to reduce Gdot(t) and/or increase Tdot(t), if it wants to keep output growing at potential.
That result is very different from the Old Keynesian model, which says you should increase the level of G(t) and/or reduce the level of T(t), if you want to avoid a recession.
Here's the intuition:
Let's start in a stationary equilibrium, to keep it simple. All the dot variables are zero, and nothing else is changing. Then the rate of time preference n(t) falls. If G(t), T(t) and r(t) were unchanged, the drop in n(t) would cause the New Keynesian agents to want to cut C(t) and increase Cdot(t). (They want to save now, so they can grow their consumption over time.) But this would cause a recession. The first best optimal policy is to keep G(t) and T(t) constant, and reduce r(t), so that Cdot(t) stays at zero and C(t) stays constant. But if the ZLB prevents r(t) from being cut, the second best policy will be to reduce Gdot(t) and increase Tdot(t). Both these policies allow the New Keynesian agents to have a positive Cdot(t) without cutting C(t). By reducing the growth of government spending, and reducing the growth of consumption by the Old Keynesian agents (by growing taxes), the permanent income of the New Keynesian agents is now higher than their current income, and this offsets their reduced rate of time preference, and prevents them wanting to save part of their current income.
Why is this result counter-intuitive, if you approach this with an Old Keynesian intuition? It's because if an Old Keynesian economist hears that agents have an increased desire to save, he interprets that as a reduction in A(t), to which the first best optimal response is a cut in T(t), not a cut in r(t). A reduction in A(t) never causes the ZLB to be a binding constraint on the first-best solution in the first place, if we assume it's an irrational whim that should be suppressed.
Update: I have to confess: these results still creep me out. So I'm not surprised if you have the same reaction. Are they "right"? Well, if by "right" you mean "do they follow from the model?", then yes, I think they do. But would I implement them? No, I wouldn't. Despite my giving the intuition, there is still something weird going on here, and I don't know what it is. My hunch is that the indeterminacy problem of the New Keynesian model is at the root of all this. I'm beginning to wonder if the indeterminacy of equilibrium output (under interest rate control) mightn't be a feature, rather than a bug, of the New Keynesian/Neo-Wicksellian model. And policy recommendations that are based on solutions that assume away the indeterminacy problem (which is what NK macroeconomists do) are deeply flawed. The recession was just the indeterminacy problem telling us it refuses to be assumed away.
It's sort of weird. Macroeconomists like Roger Farmer are trying to build indeterminacy into their models to help them explain the world; while New Keynesians, who already have indeterminacy in their models to begin with, are trying to get rid of it by assuming it away.