equilibrium or "natural" rate f.4)
Subtracting (2.6) from the labour force {s, using the approximation that the rate of
unemployment u ≈ βs — ¢, we get the following expression for the short—run determi-
nation of unemployment
ut = u - ɪɪe (pt - Et-ip + Mt) (2.7)
where u = F - {*. u can be thought of as the equilibrium or "natural" rate of unemploy-
ment in this model. Thus, (2.7) is the well-known expectations augmented Phillips curve.
Unemployment deviates from its equilibrium rate only to the extent that there are
unanticipated shocks to inflation or productivity. Anticipated shocks to inflation and
productivity are reflected in wages (equation (2.5) and do not affect unemployment. We
can now incorporate the Phillips curve into a monetary policy game. This is the subject of
the next section.
II.2. Time-Consistent Equilibrium under a "Conservative" Central Banker
As stated by Rogoff (i985, p. ii80), the adoption of central bank independence may be
viewed as an institutional response to the time-consistency problem.
Suppose, for example, that through a system of rewards and punishments the central
bank’s incentives are altered so that it places some direct weight on achieving a low rate
of growth for a nominal variable such as the price level, nominal GNP, or the money
supply. Rogoff demonstrates that society can make itself better off by selecting an agent to
head the central bank who is known to place a greater weight on inflation stabilization
(relative to unemployment stabilization) than is embodied in the social loss function Lt.
The social loss function L depends on deviations of unemployment and inflation from
their optimal (socially desired) levels
Lt = I (∆pt - ∆p*) + -2 (ut - u*)2 (2.8)
where 0 < χ < ∞ and ∆p* and u* are society’s inflation and unemployment targets. The
parameter χ is the relative weight of unemployment stabilization relative to inflation
stabilization in the preferences of society. Normalizing ∆p*, u* and pt—i at zero we
get 5)
4) Actual employment equals its natural rate when all expectations are fulfilled. Hence, the natural rate of
employment equals f.
5) Price-level targeting and inflation-rate targeting are equivalent here, since pt-i is known at the time the
central bank commits itself to achieving a target for pt - pt-i. Once monetary control errors are taken into
account it becomes important to make the distinction between a zero inflation target and a target of price