The name is absent

rate shock. The parameters {κ, σ} ∈ (0, ∞) describe the price rigidity and the elasticity of
intertemporal substitution, respectively, while {p_u, p_r} ∈ (—1, 1) summarize the persistence
properties of the two shocks.

In addition to equations (2) through (5), the first-order conditions for the optimal com-
mitment policy are

∙"'W∕. + λπt+l  ^{=  0,   t} = ^{0,                          (6)}

pβyt ^{— κ}^πt+l ⁺ βλyt+l  =  ^{0,  t} = ^{0,                     (7)}

β^t ⁺ λπt+l ^{— A}nt ^{— σ}β∣t  =  ^{0,  t}> ^{0,                     (8)}

P^βyt ^{— κ}^πt+l ^{+ β}^yt+l ^{— A}yt  =  ^{0,  t} > ^{0,                     (9)}

vit + σX_yt₊l  =  0,   t > 0,                      (10)

where λ_πt+l and λ^t+l are the Lagrange multipliers associated with equations (2) and (3),
respectively. The time inconsistency of the optimal commitment policy is reflected in the
difference’s between equations (6) through (7) and equations (8) through (9), which imply a
different policy when t = 0 than when t > 0. Notice, however, that these differences disappear
when λ_πo = ∖∣0 = 0. As a consequence, the optimal commitment policy can be obtained by
applying standard saddle-point solution methods to equations (2) through (5) and (8) through
(10), with the initial conditions A_πo = ʌ^o = 0 and Mo, rθ, known.

2.2 Timeless perspective policymaking

To obtain the Woodford (1999a) timeless perspective policy for this model the approach is to
proceed as follows. First, to introduce the timeless perspective, assume that equations (8)
and (9) also apply when t = 0, effectively discarding equations (6) and (7). Then, to obtain
a policy that is implementable, use equation (10) to solve for λ^t+l and equation (9) to solve
for A_πt₊l and substitute these expressions into equation (8) to eliminate the two Lagrange
multipliers. With these substitutions, the timeless perspective policy is

^πt ⁺ K ⁽^t ^{— y}t-l^{) —} _σκβ ^^{β + σκ}^^ ⁽ⁱt ^{— i}t-l^{) — (i}t-l ^{— i}t-2⁾] ⁺ βⁱt = ^{0, t} > ⁰∙     ⁽¹¹⁾

Provided v > 0, equation (11) can be solved for i_t, giving rise to what is known as an explicit
targeting rule. The timeless perspective equilibrium is now obtained by solving for the
rational expectations equilibrium of equations (2) through (5) and (11), with Mo, i-ɪ, and i_-2
known. Notice that the timeless perspective policy depends on the change in the output gap,
a point emphasized by Walsh (2003) in his discussion of “speed limit” policies, and on lags

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