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# Increased Price Flexibility is Destabilising in New Keynesian Models. (And a Price-Level Path Target is Stabilising)

Summary:
Start with a very simple New Keynesian "IS" (or "Aggregate Demand") equation: y(t) = E[y(t+1)] - a[r(t)-r*(t)] The "real" (inflation-adjusted) interest rate r(t) is defined as the "one period" nominal interest rate, minus expected inflation for the following "one period". In order to stabilise output y(t), relative to expected future output E[y(t+1)], the central bank needs to ensure that the real interest rate r(t) always equals some "natural rate" r*(t), which varies over time as shocks hit the economy. But the central bank can only do this if it responds to shocks instantly. And that is implausible, because it usually can't see the shocks until after they hit, and after it sees the effect of those shocks on output and inflation. Let's assume instead that there's a "one period" lag

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Start with a very simple New Keynesian "IS" (or "Aggregate Demand") equation:

y(t) = E[y(t+1)] - a[r(t)-r*(t)]

The "real" (inflation-adjusted) interest rate r(t) is defined as the "one period" nominal interest rate, minus expected inflation for the following "one period". In order to stabilise output y(t), relative to expected future output E[y(t+1)], the central bank needs to ensure that the real interest rate r(t) always equals some "natural rate" r*(t), which varies over time as shocks hit the economy.

But the central bank can only do this if it responds to shocks instantly. And that is implausible, because it usually can't see the shocks until after they hit, and after it sees the effect of those shocks on output and inflation.

Let's assume instead that there's a "one period" lag between a shock hitting the economy and the central bank observing that shock and responding to it.

If the central banks sets a real interest rate, it's simple to see what happens when a shock hits. A 1% negative shock to r* causes an a% negative shock to y(t), which lasts for one period, after which the central banks adjusts r(t), and output can return to normal (unless another unexpected shock hits). There's a recession that lasts "one period".

But central banks in New Keynesian models set a nominal interest rate, not a real interest rate. So it's the nominal rate, not the real rate, that stays constant for "one period" after the shock hits. Which complicates things, and makes the recession worse than it would be if the central bank sets a real interest rate.

When the negative shock hits, and the economy goes into a recession, actual inflation falls, and so expected inflation will presumably fall too, so the real interest rate will rise. And that rise in the real interest rate is the exact opposite of what the doctor ordered. And the more inflation falls, and the more expected inflation falls, the more the real interest rate rises, and the bigger the recession will be, and the more inflation falls. There's a nasty positive feedback loop at work.

The best case scenario is if there's a "one period" lag before inflation responds at all. So the central bank, with the same "one period" lag, can cut the nominal rate in response to the shock, before falling inflation and expected inflation worsens the effect of the shock.

In other words, increased price flexibility is destabilising in New Keynesian models.

OK. So what, if anything, can the central bank do about this problem? It can try to improve its crystal ball so it learns about the shocks more quickly, and can better distinguish permanent from temporary shocks, and demand shocks from supply shocks. But unless it finds a perfect crystal ball, some lag between shock and central bank response is inevitable.

It can target the price level (or price level path) rather than target inflation. It can make an advance commitment that if there is a recession, and if inflation falls, it will eventually (when it does respond) bring the price level back up to where it would have been without that recession. It will "repair" its past "mistakes". Which means that expected inflation, between now and "one period" ahead, won't change at all. Which eliminates that nasty positive feedback loop between lower inflation and higher real interest rates that would worsen the effect of the negative shock.

In other words, a price-level path target, relative to an inflation target, acts like an automatic stabiliser in New Keynesian models.

It's more complicated than this, of course. Because I have over-simplified the model by assuming that the central bank has a lag of "one period", and that the only real interest rate that matters is that same "one period" real interest rate.

But you get the gist.