Thursday, March 12, 2009

Can a Taylor rule lead to a liquidity trap?

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Maybe. Here is the story (it’s a long and extremely wonkish story, but hang in) – Start with the Euler condition and the Fisher equation from a standard money-in-utility (MIU) model:

Pt(1+ ίt)/Pt+1 = uc,t/βuc,t+1 (1)

Which can be rewritten as:

1+ ίt = (Pt+1/Pt) z (2)

Where z is assumed constant and summarizes the real factors that influence the real interest rate, r*. Taking logs of both sides,

ίt = pit+1 + ln(z) (3)

Further assume that the central bank follows a simple Taylor rule of the form:

ίt = r* + pi* + δ(pit – pi*) (4)

Where ίt is the nominal rate of interest and r* is the natural rate of interest. Combining (3) and (4) gives an equilibrium process for inflation that looks like:

pit+1 = pi* + δ(pit – pi*)

Finally, assume the central bank follows the so called “Taylor Principle” that the central bank reaction function should respond disproportionately, or more than one for one, to changes in inflation. Therefore, in the above equilibrium process, δ>1. Graphically, the dynamics of this equilibrium process looks like this:

Notice that the policy rule is bounded at the rate of deflation consistent with the zero nominal bound for interest rates. To see how this type of rule could lead to a liquidity trap equilibrium, observe that if inflation starts out below pi*, the stationary “good equilibrium” where inflation is equal to its target level, the central bank will cut the nominal rate in order to stimulate the economy. Under the above rule, inflation will decline leading to another rate cut, generating further expectations of lower inflation, and so on until it reaches the stable but deflationary equilibrium, pi**.

At this point you should be thinking – wait a minute, shouldn’t interest rate cuts lead to higher inflation expectations? Under normal conditions yes, but think of the above model in the present context in which inflationary expectations may have become detached from the 2% anchor and in which the economy is performing well below potential – it may be the case that rate cuts, particularly near the zero lower bound, will feed deflationary expectations. Moreover, since the only stable good equilibrium is at pi*, and because the inflation process is forward looking, the only way to get out of the deflationary equilibrium is to get expectations to jump to the good equilibrium at pi*.

I discussed how policy-makers might engineer such a jump in a previous post.

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