6. Determinacy of the Rational Expectations Equilibrium
where det(M) = 0.
The impact of the single disturbances contained in v on the structural equations of the model can be
characterized as follows. The positively correlated aggregate productivity shocks Et [∆at +1 ], Et [∆α*+1 ]
affect the respective dynamic IS curves, NKPCs, and interest rate rules. The country-specific cost-push
shocks u,u*, however, only influence the respective NKPCs and interest-rate rules. The country-specific
monetary policy shocks v, v* have a sole impact on the respective interest-rate rules. All macroeconomic
shocks spill over abroad since they explicitly affect the TOT equation (59). In addition, there is an
implicit spill-over effect due to the positive correlation of the aggregate productivity shocks.
6.1. General Case
The system of equations (63) consists of two predetermined variables (it- 1, i*- 1) and five non predeter-
mined ones (xt,x*,πt,H,π↑F, ∆tt). Comparable to the case discussed for the closed economy in Gall
(2008, p. 56), there is a unique stationary solution of (63) if and only if the coefficient matrix M has
five eigenvalues k inside and two eigenvalues k on or outside the complex unit circle (sufficient condition
for equilibrium determinacy). If there were more than five stable eigenvalues, there would be multiple
stationary solutions. If there were more than two instable eigenvalues instead, no stationary solution
would exist at all.
By computing the characteristic determinant det(M - kI7) one obtains one eigenvalue k1 = 0 and a
sixth-degree polynomial in k , which cannot be solved analytically. This polynomial is not displayed here,
but its Matlab code is available on request.
In consequence, we have to assign sensible numerical values to the model parameters in order to determine
the remaining eigenvalues of M.
6.2. Calibration
The numerical exercise is carried out as follows, whereby the length of one period shall correspond to one
quarter of a year. First, the EU and the US can be treated as approximately equal-sized countries such
that n = 1 - n = 0.5. β = 0.97 is assumed to hold for the intertemporal discount factor, which implies
i = i* = r = r* = (1 -β) /β ≈ 0.03 for the zero-inflation steady-state nominal and real interest rates across
countries. Furthermore, ξ = — 1 such that euL,L = eu*,L = 2 holds for the partial elasticity of the utility
function with respect to domestic (foreign) labor. The sensitivity of the Fed to the current foreign output
gap shall be fixed (ι* = 0.5), where this number corresponds to the original value estimated by Tayior
(1993) for the Fed for the time from 1987 to 1992. The Taylor principle, which states that the monetary
authority ought to react to an increase in current PPI inflation by augmenting its policy instrument more
than one for one in order to account for a determinate rational expectations equilibrium (see Woodford
2003, p. 40), shall be fulfilled by both central banks (α = α* = 1.5).25 The degrees of nominal interest-
rate inertia across countries shall also be fixed (ω = ω* = 0.1) implying that both monetary authorities
are supposed to place relatively more weight (1 — ω = 1 — ω* = 0.9) on the adjustment of their short-run
policy instruments to their zero-inflation steady-state value.
Moreover, set ρ = 0.8 such that one gets the following for the slope coefficients 0, 0* of the dynamic IS
25Note that the Taylor principle in its purest form is not a necessary condition for equilibrium determinacy for an interest-
rate rule of type (61). Instead, the condition μ* (α, — 1) + (1 — β)ι* > 0 is a necessary and sufficient condition for
equilibrium determinacy in case of contemporaneous data (see, e.g., Bullard/Mitra 2002, pp. 1125-1126).
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