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

A. Appendix

with {λ}_s^∞_=t denoting a sequence of Lagrange multipliers.

From the first partial derivative one obtains C_t^-ρ = -λt and therefore C_t^-₊^ρ₁ = -λt+1. Plugging this into
the fourth one, one gets equation (20):

L-^ξ _W

^γc^-ρ   Pt '

Now, by using C_t^-ρ = -λ_t and C_t^-₊^ρ₁ = -λ_t+1, one obtains from the third partial derivative equation
(18):

c^-ρ

—P— = ^{β (1} + ⁱt ) Et
P_t

Finally, plugging the above expression into the second partial derivative, one obtains equation (19):

A.3. Equilibrium Conditions on World Bond and Goods Markets

The subsequent derivation is based on Obstfeld/Rogoff (2001, pp. 7-9), which itself is based on
reasoning by Corsetti/Pesenti (2001, pp. 430-433).

Start with the market clearing condition for a single good z:

Yt ( : )= nCt ( г ) + (1 - n ) C* ( г ).

Assuming, for instance, that good z is a typical domestic good such that z = h ∈ [0, n] and multiplying
the preceding equation with Pt (h) one obtains:

Pt(h)Yt(h) = nPt(h)Ct(h) + (1 - n)Pt(h)Ct^*(h).

Taking the integral from 0 to n and using equations (6) and (10) yields:

Because of equations (12) and (14) this expression implies:

where the right-hand side of the above equation denotes global demand for domestic goods in domestic
currency. Since Y denotes domestic per-capita output, the left-hand side of the equation can alternatively

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