I was scared of reading Stephanie Schmitt-Grohe and Martin Uribe's paper (pdf). All I knew was that it was a technically demanding paper, and that it had the "Neo-Fisherite" result -- if the central bank increased the rate of interest, inflation would rise.
I have now read it. Sort of. I think I now understand the gist of what is going on in their model. But I do not understand all the math. It's an orthodox New Keynesian model, except for the Phillips curve. The Phillips Curve is rigged.
There exists a level of full-employment (potential) output Y*. There exists a natural rate of interest, call it n, that is compatible with output growing at the growth rate of Y*. The central bank sets a nominal interest rate i. There is a ZLB, so i cannot be negative. (There is a Taylor Rule in the model, but I will ignore that, because it doesn't matter.) There is no investment or government expenditure, so output and consumption are the same thing. There is a consumption-Euler equation IS curve in which the planned growth rate of consumption is a positive function of the actual real rate of interest r. (Yes, I do mean "positive", and Old Keynesians will find the sign surprising, but that's another story.)
Using hat notation, so that X^ means dX/dt/X (the growth rate of X), we can write that as:
Y^-Y*^ = F(r-n) where F(0)=0 and F' > 0
So far, it's a standard New Keynesian/Neo-Wicksellian model.
Now for the Phillips curve. It's definitely not New Keynesian. For starters, it's kinked at Y*:
1. For Y=Y*, P^ = i - n (The inflation rate does what the theorist wants it to do, to keep planned demand growing at potential. That's the first place where it's rigged.)
2. For Y < Y*, it's complicated. There are two things going on.
2a. The price level P is a positive function of the level of output Y. So if Y jumps down, P jumps down too.
2b. The deflation rate -P^ is a positive function of the gap between potential and actual output. So if Y jumps down, P^ jumps down too.
We can write this segment of the Phillips curve in inverse form as:
(Y*-Y)/Y* = H(-P, -P^) where H(0,0)=0 and H1 > 0 and H2 > 0 [How the hell do I write that in math? Never mind. It doesn't really matter.]
(That's the second place where it's rigged. And it looks like the only reason they put firms in the model, and put a labour market in the model, and distinguished between prices and wages in the model. To rig it so that wage deflation is a continuous function of Y but price deflation a discontinuous function of Y.)
Suppose the economy is initially in a stationary equilibrium at the ZLB. Output is below potential, but is growing at the same rate as potential output. There is a constant deflation rate, so the real interest rate stays constant, and equal to the natural rate.
Now suppose the central bank makes the nominal interest rate jump up, and pegs it at a new higher level. What happens?
Output cannot jump.
Because if output jumps up, the price level would jump up too. And if the price level jumps up, inflation would be plus infinity. And if inflation is plus infinity, the real interest rate would be minus infinity. And if the real interest rate is minus infinity, consumption would jump down. Which is a contradiction.
And because if output jumps down, the price level would jump down too. And if the price level jumps down, inflation would be minus infinity. And if inflation is minus infinity, the real interest rate would be plus infinity. And if the real interest rate is plus infinity, consumption would jump up. Which is a contradiction.
So if output cannot jump, it must either start growing or start falling.
If output starts falling, deflation jumps up, which means the real interest rate jumps up even more than the nominal rate, which means consumption starts rising. Which is a contradiction.
Therefore the only equilibrium is for output to start rising, and for inflation to start rising too.
God I hate [don't like. I've calmed down a bit now] this paper. And I hate having to plough through it to figure out what's really going on. It's like a snowball with a pebble inside. 90% snowjob, and a 10% kernel of wrongness. John Cochrane's papers are much better. He really does try to get to grips with the question of indeterterminacy and multiplicity. This one doesn't.
OK. I never was that great an economist, and I'm past my prime (such as it was), and I never could do math, and my brain is tired. But that is what I think is really going on in that paper. I might be wrong. But if I'm right, there is something seriously wrong with the state of economics today. Where did the economics intuition go? Why am I the one that has to do this?