Investment and Interest Rate Policy in the Open Economy

(14) is satisfied and the transversality conditions hold; (vi) the monetary policy rule is
satisfied, (22) or (23); along with the foreign counterparts for (i)-(vi) and conditions (5),
(6), (20) and (21).

2.6 Local Equilibrium Dynamics

In order to analyze the equilibrium dynamics of the model, a first-order Taylor approxi-
mation is taken around a steady state to replace the non-linear equilibrium system with
an approximation which is linear.⁹ We employ the Aoki (1981) decomposition which
decomposes the two-country model into two decoupled dynamic systems: the aggregate
system that captures the properties of the closed world economy¹⁰ and the difference
system that portrays the open-economy dimension. Consequently for the equilibrium to
be determinate it must be the case that there is a unique solution both for cross-country
differences and world aggregates. The complete linearized system of equations is sum-
marized in Table 1. Note that since money balances are separable, the money demand
equation and its foreign equivalent are irrelevant for equilibrium determinacy and are sub-
sequently ignored. In what follows below it will also be convenient to define x ≡ KL. The
parameters of the model are as follows: σ > 0 measures the intertemporal substitution
elasticity of consumption; φ > 0 measures the inverse of the Frisch labor supply elasticity;
0 < α < 1 measures the production input share of intermediate firms; θ > 0 measures
the elasticity of substitution between aggregate home and foreign goods; λ > 1 is the
degree of monopolistic competition in the intermediate firm sector; Λ₁ ≡ (1^-ψ)ψ1^-βψ) > 0
is the real marginal cost elasticity of inflation, where 0 < β < 1 is the discount factor and
0 < ψ < 1 is the degree of price stickiness; and a ∈ {(0, 0.5) ∪ (0.5, 1)} is the degree of
trade openness measured by the relative share of intermediate imports used in final good
production (1 - a).¹¹

⁹To be precise the model is linearized around a symmetric steady state in which prices in the two countries
are equal and constant (Ph = Pf = P = P = Ph = Pf). Then by definition inflation is zero
(∏ = ∏* = 1), and the steady state terms of trade and nominal and real exchange rates are T = e = Q = 1.
¹⁰The choice of which index of inflation each monetary authority targets is irrelevant for the aggregate
system. This follows since world aggregate inflation (π^W ) is given by

W = π + π* = π^h + π*^f
π ^^ 2 ~     2    '

¹¹The analysis does not consider the case when a = 0.5 since this would imply that purchasing power parity
(PPP) is satisfied and consequently the linearized inflation equation ∏R = (2a — 1)πR(^h-f ) +2(1 — a)∆e_t

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