A. Appendix
Now let us return to the case of sticky prices (δ > 0). From the domestic PPI (6) one gets the subsequent
law of motion:
1-θ 1-θ 1-θ
Pt,H = (1 - δ)Pt(h) + δPt-1,H.
Log-linearizing the preceding formula around the zero-inflation steady-state price level PH yields the
following percentage deviations:
Pt,H = (1 - δ)Pt(h) + δp>t-1 ,H.
Now reformulate the price-setting equation as follows:
Et
IXX ( δβ ) s-t [μ PH)θ-1 μ ρH)-1 ( Cw )ι-Λ U
s=t Pt,H Ps
∞ θ -1
—Et Xδβ)∙-t Ks p P-H^ (Cw)1 -ρ ,
s=t ,
where Qt := Pt(h)/Pt,H.
If one log-linearizes this equation around the zero-inflation steady-state, one finally obtains the subsequent
percentage deviations (Q = 1, [θ/(θ — 1)]κfex = 1):
where most of the terms cancel out.
ln
• ( Cw )1 -ρ 1
11 - δβ J
( C w )1 -ρ
1 -δβ
( (∕ww∖ 1 — ρ ∞
l , i qt + X(δβ)s-t (бw)1
1-δβ s=t
ρ[(1 - ρ)cW + (θ - 1)(Et [Ps,H] - pt,H) + (-1)(Et [Ps,H] - Et [Ps])]}
ln
■ //ɔw ∖1 -ρ θ flexx ^
( C ) ^-1K κt
( C w )1 -ρ
1 -δβ
1-δβ
IXt(δβ)s-t(Cw)1 -ρ[(1 - ρ)cw + Et[^s]+ θ(Et[Ps,Η] - pt,H) + (-1)(Et[Ps,H] - Et[ps])]} ,
Solving the remainder for ^t + pt,H, one gets:
^t + Pt,H = (1 - δβ) XX(δβ)s-t{Et [ps,H]+ Et [^s]}
= (1 - δβ)(Pt,H + ^t) + δβ{Et[^t+1] + Et[Pt +1 ,H]}
^ q1 = (1 - δβ)κt + δβ{Et[qt +1] + Et[πt +1 ,H]},
where Et [∏t+1 ,H] := Et [Pt +1 ,H] - Pt,H∙ Due to qt := Pt(h) - P>t,H and P>t (h) = [1 /(1 - δ)]pt,H - [δ/(1 -
δ)]P>t-1 ,H, it follows that qt = [δ/(1 - δ)]πt,H∙ Plugging this result into the above equation one finally
obtains the domestic NKPC (42):
-E∖ lɪ(1 - δ )(1 - δβ )ʌ
πt,H = βEt[ πt+1 ,H] +--δ--------κ t.
Note that the foreign NKPC (43) can be derived analogously∙
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