And anyone with even an ounce of Old Keynesian blood left in his veins, if they understood what the New Keynesians are doing, would be screaming blue murder that we are teaching this New Keynesian model to our students as the main macro model, and that central banks are using this model to set monetary policy.
Here is a very simple version of a standard New Keynesian model.
Assume no investment, government spending or taxes, and no exports and imports. There is only consumption. Assume a "haircut" economy of self-employed hairdressers, cutting each other's hair, in which all goods are services, with labour the only input, so counsumption, output, and employment are all the same thing. And so prices and wages are the same thing too.
Assume no exogenous shocks, ever. And no growth either. Nothing exogenous ever changes.
Assume a constant population of very many, very small, identical, infinitely-lived agents, with logarithmic utility of consumption, and a rate of time-preference proper of n.
The individual agent's consumption-Euler equation, with r(t) as the one-period real interest rate, is therefore:
C(t)/C(t+1) = (1+n)/(1+r(t))
Ignore the Zero Lower Bound on nominal interest rates. In fact, just to make the central bank's job even simpler, ignore nominal interest rates altogether, and assume the central bank sets a real interest rate r(t).
Suppose the "full employment" (natural rate) equilibrium is (say) 100 haircuts per agent per year consumption, income, and employment. Forever and ever.
The central bank's job is to set r(t) such that C(t)=100, for all t.
Inspecting the consumption-Euler equation, we see that this requires the central bank to set r(t)=n for all t. Assume the central bank does this.
It is obvious that setting r(t)=n for all t only pins down the expected growth rate of consumption from now on. (It pins it down to zero growth.) It does not pin down the level of consumption from now on.
Suppose initially we are at full employment. C(t)=100. Then every agent has a bad case of animal spirits. There's a sunspot. Or someone forgets to sacrifice a goat. So each agent expects every other agent to consume at C(t)=50 from now on. So each agent expects his sales of haircuts to be 50 per period from now on. So each agent expects his income to be 50 per period from now on. So each agent realises that he must cut his consumption to 50 per period from now on too, otherwise he will have to borrow to finance his negative saving and will go deeper and deeper into debt, till he hits his borrowing limit and is forced to cut his consumption below 50 so he can pay at least the interest on his debt.
His optimal response to his changed expectation of other agents' cutting their consumption to 50, if he expects the central bank to continue to set r(t)=n, is to cut his own consumption immediately to 50 and keep it there.
C(t)=50, which means 50% permanent unemployment (strictly, underemployment), is also an equilibrium with r(t)=n. So is any rate of unemployment between 0% and 100%.
What can the central bank do to counter the bad animal spirits?
If it cuts r(t) below n, even temporarily, we know there exists no rational expectations equilibrium in which there is always full employment. All we know is that we must have negative equilibrium growth in consumption for as long as r(t) remains below n. It is not obvious to me how making people expect negative growth in their incomes from now on should cause everyone to expect a higher level of income right now from a higher level of everyone else's consumption right now.
Sacrificing a goat sounds more promising as a method of restoring full employment.
Did every other New Keynesian macroeconomist already know about this, and just swept it under the mathematical rug? Didn't I get the memo?
"Under the assumption that the effects of nominal rigidities vanish asymptotically [lim as T goes to infinity of the output gap at time T goes to zero]. In that case one can solve the [consumption-Euler equation] forward to yield..."
Bullshit. It's got nothing to do with the effects of nominal rigidities. What he really means is "We need to just assume the economy always approaches full employment in the limit as time goes to infinity, otherwise our Phiilips Curve tells us we will eventually get hyperinflation or hyperdeflation, and we can't have our model predicting that, can we?"
That Neo-Wicksellian/New Keynesian nonsense is what the best schools have been teaching their best students for the last decade or so. They have been teaching their students to just assume the economy eventually approaches full employment, even though there is absolutely nothing in the model to say it should.
Remember the Old Keynesian Income-Expenditure/Keynesian Cross diagram? What we have here, if the central bank sets r(t)=n, is a version of that diagram in which APC=MPC=1 for all levels of income, so the AE curve coincides with the 45 degree line. Any level of income between 0 and full employment income is an equilibrium.
New Keynesians simply must put money back into the model.