The model contains three stochastic elements: these translate into an aggregate markup
shock, Vt ≡ In
(⅛ ),
an aggregate consumption-preference shock, gt, and an aggregate tech-
nology shock, ut. These three shocks evolve over time according to
|
kt+1 = Pvvt + e7t+1, |
(48) |
|
gt+1 = Pg gt + egt+1, |
(49) |
|
ut+1 = puut + ⅛t+1, |
(50) |
|
where kt is the percent deviation in Vt from steady state, where {p7, pg,pu} ∈ (—1, 1), |
and |
where the innovations {⅞∙t+1 , egt+1 , ⅛t+1 } are i.i.d. with zero mean and finite variance.
5.1.1 The log-linear model
When log-linearized about a zero-inflation nonstochastic steady state, the constraints and
first-order conditions for this model are
|
πt |
/ɔp , (1 — -) (1 — β-β = pEtKt+1 +--------- |
rnct + it, |
(51) |
|
mct |
= aRt + (1 — ɑɔ wt — ut, |
(52) | |
|
.—. |
= -Rt — wt + kt, |
(53) | |
|
.—. Wt |
= χk + σct — gt, |
(54) | |
|
.—. EtRt+1 |
= , P (rt+1 — Et^t+1), P + S |
(55) | |
|
.—. |
= Etkt+1--(rt+1 — Et^t+1 ■ σ |
— gt+ Etgt+1), |
(56) |
|
.—. kt+1 |
= (1 — S) kt + δit, |
(57) | |
|
.—. yt |
= (1 — 7) kt + 7it, |
(58) | |
|
.—. yt |
= ut + okt + (1 — ɑ) it, |
(59) |
where p ≡ 1-- is the discount rate, 7 ≡ 777^71 is the steady-state share of investment in
output, and ε > 1 is the steady-state elasticity of substitution between intermediate goods.
Equation (51) is the New Keynesian Phillips curve linking inflation to movements in real
marginal costs. Equation (52) documents the relationship between real marginal costs and
the costs of the production factors, capital and labor. Equations (53) and (54) describe
labor demand and supply, respectively. Equation (55) summarizes the connection between
the rental rate of capital and the return on the one-period nominal bond that arises from the
household’s portfolio decision. Equation (56) is the standard consumption-Euler equation
19
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