To understand how this three-step procedure works, consider the following general linear-
quadratic control problem. Let the economic environment be one in which an n × 1 vector of
endogenous variables, zt, consisting of nι predetermined variables, xt, and n⅛ (n = n — nι)
nonpredetermined variables, yt, evolves over time according to
xt+ι = Axxxt + Axy yt + Bxuut + εt+ι, (12)
Etyt+ι = AyχXt + Ayy yt + Byu ut, (13)
where ut is a p × 1 vector of policy control variables, εt ~ i.i.d. [0, ∑] is an s × 1 (s ≤ nɪ) vector
of white-noise innovations, and Et is the private sector’s mathematical expectations operator
conditional upon period t information. The matrices Axx, Axy, Ayx, Ayy, Bxu, and Byu
contain the structural parameters that govern preferences and technology and are conformable
with xt, yt, and ut as necessary. The matrix Ayy is assumed to have full rank.
Subject to equations (12) and (13) and Xo known, the control problem is for the policymaker
to choose the sequence of control variables {ut}∞ to minimize
∞
Eo ∑ βt [ztwzt + 2ztUut + u^Rut] , (14)
t=o
where zt ≡ [ xt yt ] . Methods to solve this optimal commitment problem are by now well
known (see Oudiz and Sachs (1985) and Backus and Driffill (1986)). For the purposes of this
section, however, what is important is that the equilibrium has the form
|
xt+ι |
= Mxx xt + Mxppt + ≡t+l, |
(15) |
|
Pt+ι |
= Mpxxt + MppPt, |
(16) |
|
yt |
= Hx xt + HpPt, |
(17) |
|
ut |
= Fxxt + FpPt, |
(18) |
where pt is the n^ × 1 vector of shadow prices associated with the nonpredetermined variables
and the system is initialized with xo known and po = 0. These shadow prices are the direct
analog to the Lagrange multipliers employed earlier, and they serve as state variables, keeping
track of the current value of past promises, in the equilibrium (Kydland and Prescott, 1980).
With the solution to the optimal commitment problem in hand, the second step is to
use these equilibrium relationships to derive an expression for the shadow prices. Define
dt ≡
yt
ut
and rewrite equations (17) and (18) as
dt = Gxxt + ɑp Pt,
(19)