Dynamic Macroeconomics

Figure 15.10
Continuous shifts of the Phillips curve due to rising inflationary expectations.
Let us now assume, as in the case of the AD-AS model, that the
government and the monetary authorities are not concerned only about
inflation or only about unemployment, but about both. They use aggregate
demand policies to minimize a quadratic loss function that depends on both
inflation and unemployment. Assume that this function takes the form
27
where ζ is the relative weight that the government attaches to deviations of
inflation relative to unemployment from their socially desirable levels π
A
and
u
A
, respectively. The objective function 
(15.45)
 is minimized subject to the
Phillips curve 
(15.39)
, with the government taking inflationary expectations
as given.
From the first-order conditions for a minimum of 
(15.45)
, subject to the
Phillips curve 
(15.39)
, we get

---

Under the optimal discretionary aggregate demand policy, the marginal
welfare cost of deviations of unemployment from the government target
equals the marginal welfare cost of deviations of inflation from target. If both
targets could be achieved, 
(15.46)
would be automatically satisfied, but if
the government cannot achieve both targets simultaneously, optimal second-
best policy must satisfy 
(15.46)
.
Using the unemployment equation 
(15.43)
 to substitute for unemployment
in 
(15.46)
, we end up with the following equation for inflation under the
optimal policy:
Substituting 
(15.47)
 in the first-order condition 
(15.46)
, unemployment under
the optimal policy follows
From 
(15.47)
 and 
(15.48)
, inflation and unemployment gradually converge
to steady state values equal to
where π
*
and u
*
denote steady state inflation and unemployment, respectively.
Equations 
(15.49)
and 
(15.50)
suggest that if the government is using
discretionary aggregate demand policies to pursue an unemployment target
that is lower than the natural rate of unemployment, but it also cares about
inflation (ζ > 0), both inflation and unemployment converge to unique steady
state values. However, steady state inflation turns out to be higher than the
socially desirable inflation target π
A
, and steady state unemployment is also
higher than the socially desirable unemployment target u
A
and is equal to the
natural rate u
0
.
28

---

Note from 
(15.49)
 that the discrepancy between steady state inflation and
the socially desirable inflation target π
A
is larger,
• the larger the discrepancy is between the natural rate of unemployment and
the socially desirable unemployment target u
A
,
• the smaller is the weight ζ of inflation relative to unemployment in the
social loss function 
(15.45)
, and finally,
• the smaller is the impact of unemployment on inflation in the Phillips curve
(15.39)
.
These parameters reflect the incentives of the government to reduce
unemployment and create unanticipated inflation under a discretionary
aggregate demand policy.
In the attempt to reduce unemployment below the natural rate, the
government drives actual inflation and inflationary expectations above the
socially desirable inflation target to a level that balances (1) the marginal
welfare cost of deviations of unemployment from the government target with
(2) the marginal welfare cost of deviations of inflation from the government
target. Note that 
(15.49)
 and 
(15.50)
 satisfy the first-order condition for the
minimization of the social loss function 
(15.45)
. Thus, at the steady state
inflation rate 
(15.49)
, inflation is so high that the government has no further
incentive to use discretionary aggregate demand policies to reduce
unemployment below its natural rate, because this would bring about a
further increase in the already high inflation rate.
We have thus demonstrated that the use of discretionary aggregate demand
policies to pursue an unemployment target that is lower than the natural rate
does not affect steady state unemployment. This ends up at the natural rate u
0
anyway. But it does affect steady state inflation, which turns out to be higher
than the socially desirable government target.