where the construction of Gx and Gp is obvious and straightforward. Importantly, since di
contains all of the nonpredetermined variables and Ayy has full rank, Gp is a (n + p) × n
matrix with rank(Gp) ≥ n⅛. Rewriting equation (19) to make pi the subject leads to
Pi — Gp 1 (di - Gxxi) ,
(20)
where Gp 1 represents the generalized (left) inverse of Gp.
In the final step, I substitute equation (20) into equation (15) and the lag of equation (16),
thereby dispensing with the initial condition po — 0. After some reorganization, the timeless
perspective equilibrium can be written as
χi+l
Xi
di
Mxx Mxp (Mpx - MppGplGx)
I 0
Gx Gp (Mpx - MppGplGx)
MxpMppGpl -
0
GpMppGpl
Xi
Xi-1
di-l
I
+ 0 [≡i+l] ∙
0
(21)
To understand why this procedure correctly recovers the timeless perspective equilibrium,
consider the relationship between the optimal commitment problem and the timeless perspec-
tive problem. In both problems the policymaker has access to a mechanism that it uses to
commit to its policy. The value of the central bank’s policy commitments is encapsulated
in shadow prices. Critically, aside from the initial period, the timeless perspective does not
change either the constraints or the objectives in the optimization problem. As a consequence,
the timeless perspective does not change the system’s stability properties, nor does it change
the system’s eigenvalues or whether the shadow prices are predetermined, which is why the
optimal commitment policy and the timeless perspective policy share the same asymptotic
equilibrium. What the timeless perspective does change, however, is the system’s initial con-
ditions, which is why the optimal commitment policy and the timeless perspective policy have
different period-0 transition dynamics and, with discounting, yield different losses.
But, although saddle-point solution methods require the partitioning between stable and
unstable eigenvalues to conform to the partitioning between predetermined and nonpredeter-
mined variables (both unaffected by the timeless perspective), they do not require an explicit
declaration of the initial conditions. Thus, the timeless perspective equilibrium can be found
by first applying standard rational expectations control methods. Then, once the equilibrium
has been obtained for arbitrary initial conditions, the timeless perspective can be introduced
by using the equilibrium relationships to make stationary, and subsequently eliminate, the
shadow prices.
Note the role of the rank condition on Gp. This rank condition ensures that the shadow