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

A. Appendix

Subsequently, Et [πt+1] := Et [pt+1] -pt shall be defined as the expected CPI inflation rate in period t + 1.
In addition, let hatted variables denote the percentage deviations from their zero-inflation steady-state
values (∙^yt := ^yt ^{- y,} Et [^yt+ι] := Et [yt +ι] ^{- y,} it := ⁱt ^- i).

Taking this into account and cancelling the term ρ(n — 1)t on both sides, the last equation rearranges
to:

-ρ(n - 1)tt - ρyt + ρy = it - Et[∏t +ι] - ρ(n - 1)Et[tt +ι] - ρEt[yt+1] + ρy.

Solving this for y_t, one finally obtains the domestic dynamic IS curve (39):

yt = Et[yt +1] + ¹ {Et[∏t+1] - it} - (1 - n)Et[∆tt+1].

^ρ

Note that the foreign dynamic IS curve (40) can be derived analogously.

A.5. New Keynesian Phillips Curves

In period t, a domestic producer willing to reset her price maximizes her expected discounted future
profits with respect to P_t (h):

Et S XX δ^s-tβ^s- μCw) P ∙P^thYs (h) - κ_sY_s (h)] I → max
^t∖=_t ^β Cn Ph ^s( ) ^{s s}( )j → m_w

β ^s-t (C_s^w /C_t^w)^-ρ is a stochastic discount factor, which denotes the marginal rate of substitution of real
(world) consumption between periods s and t. Note that here one has made use of equation (23). In case
of goods market clearing output of an individual producer equals global demand for the differentiated
good (Y (h) = C ^w (h)). Note further that the condition Pt(h) = Ps (h) during the length of the contract
implies for the global demand function (15) for a representative domestic good:

Substituting this into the above equation yields:

Et  IXX (δβ)s-t C -Яf ≡X ¹ -θ PSH^^-- 1 Cw - κs PH)-θ PH^--1 Cwl 1 → max

IS= Cw [∖p_s,hJ ∖pJ ^s Pp_s,hJ Pp_sJ ^s ʃ  p ( h )

^  Et (XX (δβ)s-t ^μCWX -^ρ ɪ "(1 - θ) ^μ≡X -θ ^μPsH^{χ -} 1 + θκs ^μ≡X ^-θ^- 1 ^μPsH^{χ -} 1# cw ) =0

Is=CwrJ Ps,h ∏p_s,_hJ P p_s J Pp_s,_hJ P p_s J S  ^s ∖

Solving this for Pt (h)/Pt,H, one gets after some manipulation the subsequent price-setting equation:

θ                -1

P.(h)= θ Et {p^r-t(^δβ) p∙fe) '      (C?)¹ η}

P^,H ^{θ - 1} Et ½P“ t(δβ)^s-t [(⅛´θ^-1 (^ppH´^-1 (C?)¹ -P]¾

Now consider the case where everybody resets their prices (δ = 0). As each producer charges the same
price (PH = P (h)), the above equation collapses to the following :

Pt (^h ) =   ^θ _κ = ₁.

Pt,H    θ - 1

Again we get the real marginal production cost associated with a flexible-price equilibrium κ^{f lex} :

flex ^{θ - 1}

κi    =       .

^t θ

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