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

A. Appendix

with ₀ⁿ P (h)C (h)dh = PHCH. Now combine the preceding equation with the preliminary demand

function from above. Then one gets for C (h):

C(h) = P (h)^-θ

PH CH

Ro P(h)¹^-θdh

Plugging this into the definition of CH , one gets:

P (h)^-θ

PH CH

Ro P(h)¹^-θdh

Dividing this formula by CH and raising both sides of the resulting equation to the power of (θ — 1)/θ, I
obtain:

1'∙∙ ndι‰R-_rrP^.

θ-1

dh,

which can be solved for P_H to finally obtain the domestic PPI given by equation (6):

1 -θ

PH =

Plugging this formula into the last given equation in C (h), one eventually gets equation (10):

C_H.

A.2. First Order Conditions for a Utility Maximum

The representative household maximizes

, ,.ʃv--‘ C^-'₊ X ^μM∙V^-ε γ r--i∏

U^t = ^; I -^β Pf + 1—ɪШ ^-'L^s J∫

with respect to the decision variables Ct , Mt , Bt , Lt subject to the intertemporal budget constraint (in
real terms)

Wt Bt-1 Mt-1

— Lt ^{+ (}1 ⁺ i^t-¹) ~P~ ⁺ —P—

,Γt(h) _c ,Mtd_t Bt

⁺⁼ C^t⁺ τ∖⁺ π⁺^τ,'

Hence,

Λt

^X β

_C1-ρ

—- Ls + ⁽¹ + is
P

1-ε

_L1-ξ

Γ(h)

])

max

C_t,M_t,B_t,L_t,λ_t