*For students and maybe teachers of macroeconomics. The analysis here is standard: a more general discussion can be found in Woodford's Interest and Prices for example (see pages 497-501 in particular). All this adds is a bit of intuition which I at least found helpful. If there are any mistakes in the algebra or numbers below, please let me know and I will correct them*

When monetary policy can commit
(i.e. follow a time inconsistent policy), why does the optimal response to an
anticipated cost-push shock involve bringing the price level back to its
original value? I do not think it is obvious why it should, yet the result is
an important part of the justification
for price level or nominal income targeting, so here is my attempt at some
intuition.

To make things simple, ignore
discounting in both the monetary policymakers objectives and the New Keynesian
Phillips curve (NKPC). For notational clarity, assume perfect foresight. So the
monetary policymaker tries to minimise the weighted sum of the output gap (y)
and inflation (π), both squared (the inflation target is zero), from period
zero onwards, subject to a series of NKPC constraints. The shock is a cost-push
shock (u) in period zero, which is observed at the beginning of period zero.

To start us off, assume that the
policymaker can only set period zero output and inflation. Expected inflation
in period 1 is zero (the shock is not persistent, and the central bank is
credible.) So the problem can be expressed as choosing output and inflation to minimise the Lagrangian:

This gives us two first order
conditions:

which can be combined as

Equation (1) can be thought of as a
policy rule: the combination of the output gap and inflation that optimal monetary
policy would select if it cannot achieve zero for both. So, for example, if
output has a large impact on inflation, then (1) gives a larger ‘weight’ to
inflation. If people like diagrams, we can represent the loss function by
indifference curves around the ‘bliss point’ zero, which are circles if β is
one. The monetary rule (1) is the line joining the points where these
indifference curves are tangent to the Phillips curves.

To take a concrete example, let the
cost push shock be 10, and set α=β=1. Adding (1) to the Phillips curve implies
that the central bank creates a negative output gap of 5, which gives an
inflation rate of 5. The optimal policy involves one of intratemporal
smoothing, balancing the costs of inflation against the costs of lower output.
The welfare cost is 50, compared to a cost of 100 if the policymaker allowed no
fall in output.

Suppose now that the policymaker
can make promises about period 1 only. The Lagrangian then becomes

The first order conditions always
imply that the Lagrange multiplier for any time period is equal to the output
gap for that period divided by α. In addition to the first order condition (1)
for period zero inflation, we also obtain

We can add (1) to this, to get

Equation (2) gives us the key
intuition behind the price targeting result. Suppose αβ is large, so the final
term is small. In this case (2) tells us that the sum of inflation in the two
periods will be close to zero. Higher inflation in period zero will be almost
balanced by negative inflation in period one. A moment’s thought implies that
this must mean the price level at the end of period one will be close to its
original value.

Inflation in period zero will be
positive as a result of the cost push shock. We can reduce its size by
creating negative inflation in period 1. By creating negative inflation of x in
period 1, we reduce inflation in period zero by x. With a cost push shock of 10,
creating negative inflation of 5 in period 1 balances positive inflation of 5
in period zero, which is the optimum combination. Creating less negative
inflation in period 1 will lead to a greater welfare loss, but so will reducing
inflation by more than 5 in period 1.

However, what if αβ is not large?
Specifically, suppose we return to the example where α=β=1. Combining this with
the NKPC for each period implies the optimal policy is

The optimal policy creates negative
inflation in period one, but not by enough to keep the price level unchanged.
Prices end up higher by 2, compared to 5 when we could only change period zero
values. The welfare cost is now 40, which is an improvement on 50.

Why does the case non-negligible αβ
stop approximate price level targeting in this two period case? Think about
what exact price level targeting would imply. It would involve inflation of 5
in period zero and -5 in period one. This could be achieved with an output gap
of -5 in period one, but no output gap in period zero. So although inflation
would be balanced, output gaps would not be. A more balanced output combination
involves a higher final price level.

(The policy is now time
inconsistent: at t=1 there is an incentive for the policymaker to not carry
through and reduce output, but instead set the output gap to zero.
Unfortunately if this change in policy is anticipated in period 0, inflation
will be 6 rather than 4 in period 0, and the overall welfare cost will be 52
(36+16), which is worse than the case where policy only operated in period zero.)

Suppose we now allow the
policymaker to make promises in period 0 about the output gap in period 1 and
2. Instead of just reducing output in period one, we can spread lower output
over periods one and two. The output costs become more balanced, which reduces
the extent to which we fail to achieve a balanced inflation profile. We can
then derive the following policy rule

As the fall in output in period 2
is likely to be lower than the previous fall in output in period 1, the
deviation from price level targeting is reduced.

If we allow the policymaker to make
commitments T periods ahead, then we can derive the following first order
condition:

High inflation in period 0 can now
be balanced by negative inflation in many later periods. Intuitively the output gap in period T will become very small as T becomes large. This implies that the
sum of inflation over all periods is almost zero. That means that the price
level in period T is almost the same as the original price level. Thus the
optimal policy in effect involves a long term price target, although that
target is approached gradually.

I think you mean time consistent first line

ReplyDeleteOr not

DeleteAn economic shock means the price of one commodity that has a large effect on an entire economy is unstable (such as a housing bubble bust). Relative price adjustments are then required throughout the economy. Because prices and wages (especially) are sticky downward, higher than normal inflation (especially wage inflation) is needed to allow relative prices to rapidly reset more smoothly.

ReplyDeleteIf inflation is insufficient, the pressure to reset relative prices will be mostly downward, not upward. Wages are sticky and resist deflation. Reduction in labor costs occur by increases in unemployment. Prices are sticky downward and output drops below capacity to maintain minimum price. This does not happen if wage inflation is high enough to allow relative prices to reset upward.

People worry too much about out of control inflation. Not discussed enough is the important role of inflation in allowing relative prices to reset. The problems of deflation are discussed and understood. Deflation and inflation are a continuum. Inflation that is too low is only marginally less bad than deflation. There is a misconception that higher inflation rates invariably lead to uncontrolled wage-price spirals. That is wrong and it leads to policy that keeps inflation too low and a drag on the economy.

-jonny bakho