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.