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-θ                    1-θ 1-θ

P_t,H = (1 - δ)Pt(h)    + δP_t-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:

IXX ( δβ ) s^-t [μ PH)^θ-1 μ ^ρH)^-1 ( Cw )^ι-Λ U
_s=t           Pt,H         Ps

∞                     θ         -1

—Et X^δβ)∙^-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)]κ^fex = 1):
where most of the terms cancel out.

Solving the remainder for ^_t + p_t,H, one gets:

^t + Pt,H = (1 - δβ) XX(δβ)^s-t{Et [p_s,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 q_t = [δ/(1 - δ)]π_t,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^{[ π}t+1 ,H^] +--δ--------^κ t.

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

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