Dynamic Macroeconomics

20.2.3 Monetary Policy under a Fixed Inflation Rule
Let us now assume that, although the discounted social welfare loss is
described by the loss function 
(20.7)
, the government instructs the central
bank to follow a fixed inflation target, 
π
= 
π
*
in every period, by following a
rule that minimizes
From the first-order conditions for a minimum of 
(20.12)
, we have

---

Inflation is equal to the optimal inflation target rate in every period.
Equilibrium output still falls short of full employment output, because it is
determined as in 
(20.10)
.
The social welfare loss under the constant inflation rule is given by
where Λ
R
denotes the discounted social welfare loss under the optimal
inflation policy rule.
Thus, under a fixed optimal inflation policy rule, social welfare losses are
lower than under discretion. There is an improvement in social welfare if the
government instructs the central bank to follow the optimal inflation rule
irrespective of unemployment. Inflation is at the socially optimal level 
π
*
,
and unemployment is at its suboptimally high natural rate, which is the case
with the discretionary policy as well.
Although this is not a first-best outcome for the government, or society, it
is better than the outcome under discretion, where unemployment is at its
suboptimal natural rate anyway, and inflation is higher than optimal inflation.
In the absence of alternative policies to reduce the natural rate of
unemployment, the optimal monetary policy is the policy under the fixed
inflation rule. This rule dominates discretion in the absence of shocks.
20.2.4 Central Bank Constitutions
The question of how to constrain the government to have the central bank
follow such an optimal inflation rule (and thus not resort to the discretionary
policy) can be addressed by appropriate amendements in the constitution of
the central bank.
As Rogoff [1985] has shown, the optimal inflation rule can be
implemented by creating a politically independent central bank that is
constrained by its constitution to be concerned only with inflation and not
with full employment. This can be formally modeled by modifying the loss
function of the central bank in either of the following two ways.

---

The first is to allow the weight 
ζ
on inflation relative to output in the
preferences of the central bank to go to infinity. In this case, we end up with
the independent central bank instructed to minimize the loss function 
(20.12)
,
which only depends on inflation and results in the optimal policy rule
(20.13)
.
The second is to modify the output objective in the loss function of the
central bank. Instead of the central bank targeting deviations of output from
full employment, it can only target deviations of output from its natural rate.
The central bank does not seek to keep the economy at full employment, but
only at its natural rate. In this case, the loss function minimized by the central
bank takes the form
and is minimized subject to 
(20.5)
 rather than to 
(20.6)
.
From the first-order conditions for a minimum, we have
where
In rational expectations equilibrium, the inflation rate will be equal to
Thus, an independent central bank that does not systematically try to reduce
unemployment below its natural rate will implement the optimal monetary
policy as well. This solution can also be implemented by appropriately
modifying the constitution of an independent central bank, so that it will then
be constrained to follow the optimal policy by minimizing the objective
(20.15)
. This objective states that the central bank should aim for the optimal
inflation rate 
π
*
, unless unemployment deviates from its natural rate.

---

What is the difference between the objectives 
(20.12)
 and 
(20.15)
? In the
absence of stochastic shocks to inflation and unemployment, the two
objectives lead to the same outcome: the stabilization of inflation at its
optimal rate 
π
*
. However, as we shall see below, in the presence of shocks
that affect unemployment independently of inflation, the objective 
(20.15)
allows for more flexibility. The central bank can react to shocks that cause
deviations of unemployment from its natural rate, something that it would not
be able to do if its constitution constrained it to maintain inflation at 
π
*
under
all circumstances.