Macroeconomic Interdependence in a Two-Country DSGE Model under Diverging Interest-Rate Rules



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                    1 1

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-tPH)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 1

11 - δβ J


( C w )1 -ρ

1 -δβ


( (∕ww 1 — ρ      

l , i qt + X(δβ)s-t w)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 θ flexx ^

( C )   ^-1K κt


( C w )1 -ρ

1 -δβ


1-δβ

IXt(δβ)s-t(Cw)1 [(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

πt,H = βEt[ πt+1 ,H] +--δ--------κ t.

Note that the foreign NKPC (43) can be derived analogously∙

34



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