6. Determinacy of the Rational Expectations Equilibrium
the two central banks are assumed to be non-cooperating. This assumption differs, for instance, from
Clarida et al. (2002), Pappa (2004), or Benigno/Benigno (2006) who consider cooperative equilibria
among other possibilities. A particular reason why cooperative solutions can be ignored is the finding
that there are only quantitatively negligible welfare gains from cooperation between the ECB and the
Fed for empirically plausible parameter constellations (see Pappa 2004, pp. 770-774).
Even though the central banks’ targeting rules fulfill a similar purpose in the EU and the US, it can be
justified to assume that they differ to a certain extent. This is due to the diverging statutes of the ECB
and the Fed, which have also been stated in the introductory Section 1.
Therefore, the interest rate rules shall differ such that the Fed is supposed to conduct its monetary policy
by considering current US PPI inflation π* F and the current US output gap xt, while, for the sake
of simplicity, the ECB is supposed to impose its monetary policy taking into account current EU PPI
inflation πt,H only. This difference is due to the fact that all conceivable policy goals of the ECB besides
price stability can be interpreted as secondary.
Hence, the two interest-rate rules read:
it = α∏t,H + ωit-1 + vt
(60)
(61)
⇔ it = α∏t,H + ωit-1 + (1 - ω)i + vt,
it = α*π*t,F + ι*χt + ω*i*t-1 + v*t
⇔ i*t = αfπ*t,F + ι*x*t + ω*it-1 + (1 - ω*)? + v*t.
The ECB’s interest rate rule (60) can be interpreted as follows: α (α > 0) denotes the sensitivity of the
ECB to current domestic PPI inflation πt,H. Since past decisions cannot be ignored under commitment
(see Pappa 2004, p. 754), the rule incorporates some degree of inertia of the monetary policy instrument
i itself as in Woodford (2003, pp. 95-96), which is measured by the parameter ω (0 < ω < 1). The
parameter 1 - ω , however, measures the degree of adjustment to the zero-inflation steady-state value of
the nominal interest rate i .23
In (60), vt shall denote an exogenously given, stationary AR(1) process of the form vt = ζv vt-1 + ηv,t
(0 < ζv < 1) with the exogenous error term ηv assumed to be i.i.d. ~ N(0, ση2v ). This AR(1) process can
be interpreted as a transitory monetary policy shock, where a positive realization of ηv would denote a
contractionary shock (see Gall 2008, p. 51).
Note that an analogous interpretation for (61) also holds for the US. However, αt may differ from α as
well as ω* from ω. Moreover, v* shall be uncorrelated with v such that domestic and foreign monetary
policy shocks are country-specific. ι* (ι* > 0) denotes the sensitivity of the Fed to the current foreign
output gap x* .24 Since the signs of the elasticities of the central banks’ policy instruments to endogenous
variables are all positive so that they react anti-cyclically to their changes, the policies could alternatively
be characterized to have a ”lean against the wind” property as in Clarida et al. (1999, p. 1672).
Altogether, (49), (50), (56), (57), (59), (60) and (61) form a determined system of six log-linear expecta-
tional difference equations.
6. Determinacy of the Rational Expectations Equilibrium
In order to investigate whether there is a determinate rational expectations equilibrium to the system
of expectational difference equations (49), (50), (56), (57), (59), (60) and (61) it is advantageous to
rearrange it in matrix form.
23Following GAll/MonACeili (2005, p. 723), both rules (60) and (61) could also be denoted PPI (or domestic) inflation-
based Taylor rules (DITR) as opposed to CPI inflation-based Taylor rules (CITR) or a credible peg for the nominal
exchange rate. However, we will not take up these other possibilities of monetary policy design here.
24Note that ι = 0 is assumed to hold for the ECB.
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