5. New Keynesian Framework
Substituting for Et[πt+1] by (54) and for Et[πi*+ι] by (55), the dynamic IS curves (52) and (53) change
to the following:
xt = Et[xt +1] +—{Et[πt+1 ,h] — "it} ÷ $Et δtt + 1] ÷ y----Et[δat+1], (56)
ρ ξ-ρ
1 ξ1
X = Et[X+1]÷-{Et[∏t.F] -et} ÷ $*Et[∆tt+1]÷ ξ---Et[∆⅛], (57)
t t+ ρ t+ , t ξ - ρ t+
where $ := [(n — 1)(ξρ — ξ)]/[(ξ — ρ)ρ] ($ > 0) and $* := [n(ξρ — ξ)]/[(ξ — ρ)ρ] ($* < 0) holds for the
slope coefficients of the dynamic IS curves with respect to the expected movements in the TOT.
Therefore, we need an equation that expresses these movements as a function of the remaining endogenous
variables. In order to do so, let us introduce uncovered interest rate parity as condition for avoiding
currency arbitrage:
E , ■ ʌ _ Et[St+1] ∩ , ■*'
(1÷ it ) = ---St---(1÷ it ),
where its log-linear version reads
it = Et δst+1] ÷ i*∙ (58)
By solving the period t — 1 equivalent of equation (58) for Et-1 [∆st] = ∆st (assuming that past expecta-
tions have been correct) and plugging the result into the log-linear representation of current movements
in the TOT ∆tt = ∆st ÷ π* F — πt,H one obtains:
δtt = it- 1 — it— 1 ÷ π*,F — πt,H ■ (59)
5.3. Monetary Policy Rules
With the derivation of equations (49), (50), (56), (57), and (59) one has obtained a system of five log-
linear expectational difference equations. However, with x, x*, πH, π*F, ∆t, i, i* one has seven endogenous
variables, two more variables than equations. Therefore, we need two more equations which represent
domestic and foreign monetary policy as Taylor (1993) type interest-rate rules in order to obtain a
determined system of equations. Following Woodford (2003, pp. 90-101), these interest-rate rules shall
comprise a feedback from (some of) the endogenous variables. There, those interest-rate rules are first
incorporated into a Neo-Wicksellian cashless economy, but Woodford (2003, pp. 101-106) also shows
that rules of this form produce equivalent results in case of monetary frictions, e.g., in the MIU model
given by equation (1). Even though the Woodford (2003) results have been derived for the closed
economy, they are supposed to hold for the open economy, too, which is due to the isomorphism of the
models.
The feedback is introduced to circumvent price level (and inflation) indeterminacy as shown by Sar-
gent/Wallace (1975), which is typically associated with purely exogenous interest-rate targets (see
Woodford 2003, p. 86). In case the latter type of modeling is avoided, the monetary aggregate is not
a superior policy instrument compared to the short-run nominal interest rate. Moreover, it is assumed
that the central banks are committed to their rules rather than they implement new ”rules” on a period-
by-period basis. This is done in order to overcome time inconsistency of monetary policy. For a debate
on discretion versus commitment in monetary policy and possible welfare gains from the latter see, e.g.,
Ciarida et al. (1999, pp. 1670-1671) or Gall (2008, chapter 5).
Hereinafter, the two monetary authorities shall be called ECB at home and Fed abroad as already fore-
shadowed in the introductory Section 1. In consequence, the home ”country” will be denoted European
Union (EU) and the foreign country United States (US).
These two central banks shall be assumed to conduct their monetary policies autonomously, which means
that they take as given the policy actions of the respective other monetary authority. In other words,
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