936 Monetary Policy, Beliefs, Unemployment and Inflation
18.5.1 The model
The econometric model of long-run unemployment (equation (18.25)) is taken to
represent the “actual” model of the economy but, when solving the model, this is
amended to conform more closely to equations used so far in this literature. Thus,
the “actual” equation is as shown next:
ut=u∗∗−θ(πt−ˆπt)+d 1 W 1 t+d 2 W 2 t+ν 1 t, (18.26)whereW 1 =OILandW 2 =COM, and the variableu∗∗is the long-run level of
unemployment that obtains in the absence of effects from the external econ-
omy and inflation surprises. The cointegrating equation derived in section 18.4
(equation (18.25)) is thus taken to be (18.26) when inflation surprises are zero.^29
Inflation surprises are introduced when solving the model below, where the param-
eterθin (18.26) is then set at unity, as is standard in the Beliefs literature.
Continuing with the rest of the model, actual inflation is again:
πt=ˆπt+υ 2 t, (18.27)which gives actual inflation as the systematic part of inflation,πˆt, set by the author-
ities by optimizing (18.9) subject to their “approximating” unemployment model
of the economy (18.28):
upt=γ 0 t+γ 1 tπt+δ 1 tW 1 t+δ 2 tW 2 t+ηt. (18.28)As is evident, it is assumed in the “approximating” model that the authorities
know that unemployment is affected by a set of exogenous variables but have a
misspecified form of their effect. As written, (18.28) assumes – as in the Sargent
model – that the authorities also have a mistaken belief in an exploitable trade-off
between inflation and unemployment. As is clear, the time-varying parameters of
(18.28) are (γ 0 t,γ 1 t,δ 1 t,δ 2 t), which are again assumed to be updated using standard
constant-gain recursive least squares formulae,
ξt+ 1 =ξt+gPt−^1 Xt(ut−γ 0 t−γ 1 tπt−δ 1 W 1 t−δ 2 W 2 t) (18.29)=ξt+gPt−^1 Xt(u∗∗−θ(πt−ˆπt)+v 1 t−γ 0 t−γ 1 tπt
+(d 1 −δ 1 )W 1 t+(d 2 −δ 2 )W 2 t) (18.30)Pt+ 1 =Pt+g(XtX′t−Pt), (18.31)whereξt=(γ 0 t,γ 1 t,δ 1 t,δ 2 t)′,Xt= (1,πt,W 1 t,W 2 t), andPtis now( 4 × 4 ).
With this set-up, the solution giving the authorities’ optimal setting of the
control variableπtis:
πˆt=−γ 1 t
1 +γ 12 t(γ 0 t+δ 1 tW 1 t+δ 2 tW 2 t). (18.32)In the remainder of this section, optimal control solutions of the model given by
equations (18.9) and (18.26)–(18.32) are described, and possible interpretations of
these are advanced as we proceed.