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.
20.4