Dynamic Macroeconomics

20.3 Optimal Monetary Policy in the Presence of Stochastic
Shocks
Up to this point, our analysis of monetary policy has abstracted from the
impact of exogenous stochastic shocks. In this section, we allow for the
presence of such shocks in order to study the optimal reaction of monetary
policy.
The new Keynesian model in the presence of stochastic shocks is
described by 
(20.1)
 and 
(20.2)
. The unemployment objective of the central
bank depends on deviations of unemployment from its natural rate. The
central bank is thus assumed to choose monetary policy and the inflation rate
so as to minimize 
(20.15)
, subject to 
(20.2)
. This choice would determine the

---

optimal inflation rate. Substituting for the optimal inflation rate in 
(20.1)
, one
could then determine the optimal path for the nominal interest rate.
From the first-order conditions for a minimum of 
(20.15)
subject to
(20.2)
, the optimal inflation rate satisfies
where 
ψ
is defined by 
(20.17)
. From 
(20.24)
, it follows that
Substituting 
(20.25)
in 
(20.24)
, we have that the rational expectations
solution for optimal inflation is given by
Thus, optimal inflation in the short run is not equal to 
π
*
but also depends on
supply shocks. As a result, the optimal monetary policy rule is a contingent
inflation target, with mean 
π
*
, but is also dependent on supply shocks. It
reacts negatively (but less than one to one) to productivity (supply) shocks.
The reason is that such shocks cause a discrepancy between real wages and
productivity, because nominal wages are determined before current
productivity shocks are known. The optimal reaction of inflation implies that
real wages partly adjust to the productivity shocks, thus mitigating the impact
of such shocks on deviations of employment and output from their natural
rates. In this way, optimal monetary policy partly stabilizes fluctuations of
employment and output around their natural rates by inducing appropriate
short-run deviations of inflation around the target of the central bank.
Substituting 
(20.26)
in 
(20.2)
, under the optimal policy, deviations of
output from its natural rate are given by
Thus, optimal contingent monetary policy is 
second best
in response to
productivity shocks. It cannot completely stabilize both inflation and
employment, as it operates through aggregate demand, and productivity

---

shocks are supply shocks that cannot be counteracted directly. Optimal
monetary policy can only equate the marginal cost of deviations of inflation
from target to the marginal cost of deviations of employment and output from
their natural rates. However, in the presence of productivity shocks, this does
not completely eliminate fluctuations in employment, output, and inflation.
From 
(20.17)
, note that 
ψ
(the optimal response parameter to productivity
shocks) depends negatively on 
ζ
(the relative weight that the central bank and
society attach to deviations of inflation from target, relative to deviations of
output from its natural rate). The higher 
ζ
is, the lower will be the optimal
response of inflation to productivity shocks in 
(20.26)
, and the higher will be
the impact of productivity shocks on fluctuations of output around its natural
rate in 
(20.27)
.
Thus, in the presence of supply shocks, the optimal monetary policy rule
faces a trade-off between the stabilization of inflation at its optimal level 
π
*
and the stabilization of output and employment at their natural rates. One
instrument, the nominal interest rate or the money supply, which operates
through aggregate demand, does not suffice to achieve both objectives in the
presence of supply shocks. The central bank has to resolve this trade-off by
choosing a policy that equates the marginal social cost of deviations of
inflation from its optimal level to the marginal social cost of deviations of
output and employment from their natural rates.