5. New Keynesian Framework
with μ := [(1 -δ)(1 — δβ)(p — ξ)]/δ (μ > 0) and μ* := [(1 -δ*)(1 — δ*β)(p — ξ)]/δ* (μ* > 0) representing the
slope coefficients of the NKPCs with respect to the domestic (foreign) output gap. In addition, ut shall
denote an exogenously given, stationary AR(1) process of the form ut = ζuut-1 + ηu,t (0 < ζu < 1) with
the exogenous error term ηu assumed to be i.i.d. ~ N(0, a^u ). This AR(1) process can be interpreted as
a transitory cost-push shock reflecting determinants of real marginal production cost which do not move
proportionally with the output gap (see Clarida et al. 2001, pp. 250-251).
The two NKPCs represent aggregate supply in both countries and are isomorphic to their closed-economy
counterparts, where (49) can be interpreted as follows: the positive short-run ”trade-off” between current
domestic PPI inflation πt,H and the current domestic output gap xt can easily be seen.21 However, this
is not really a trade-off to be exploited since πt,H is also positively related to (discounted) expected
domestic PPI inflation βEt[πt+1,H].22
Note that an analogous interpretation for (50) also holds abroad. However, u* shall be uncorrelated with
u such that domestic and foreign cost-push shocks are country-specific.
It will turn out to be convenient that the following holds for Et [yfleX] — yflex in case one makes use of
the log-linear version of the current domestic flexible-price equilibrium output according to (38) and its
expected counterpart:
f lex f l
Et[ У t +1 ] — У t
f lex f l
Et [yt+1 ] — yt
( n — 1)( P — 1)
ξ — ρ
Et [tt+1 ] +
i—1 Et [ a. + 1] + ^.√ ɪ)
ξ — ρ ξ — ρ .— — 1√
( n — 1)( p — 1)4 ξ — 1
-----1---------tt — 7----at
.—p .—p
. — p'-i . — 1)
7---ln Y
.—p
(n J)(P 1) Et[∆tt +ι] + .-ɪEt[∆at+ι], (51)
.—p .—p
where at is assumed to obey an exogenously given, stationary random process of the form at = ζaat-1 +
φa a↑ + ηa,t (0 < ζa,φa < 1) with the exogenous error term ηa assumed to be i.i.d. ~ N (0 ,σ^ a ).
Note that an analogous equation to (51) also holds abroad.
In consequence, the dynamic IS curves (47) and (48) rearrange to:
χt = Et[χt +1] +—{Et[πt+1] — it} + -----e E^----EEt[δtt+1] + 7----Etδat +1], (52)
p .—p .—p
4 = Et [ x^+1] + 1 {Et [ ∙. —i } + nf—-1 Et [∆ tt+1] + I—1 Et [∆ a^]. (53)
t t+1 p t+1 t . — p . — p t+1
It would also be preferable to express these dynamic IS curves in terms of PPI rather than CPI inflation,
which can be achieved by using the subsequent log-linear representation of the TOT (28): tt := st +
p* F — pt,H. Subtracting this expression from its expected analog one gets: Et [∆tt+1] = Et [∆st +1 ] +
Et[∏*+1 F] — Et[πt+1 ,H] = Et[πt +1 ,F] — Et[πt+1 ,H]. Combining this outcome with the log-linear versions
of the domestic CPI (5) and its foreign equivalent, one obtains the following relations between expected
CPI and expected PPI inflation at home and abroad:
Et[πt+1] ≡ Et[πt+1,H] — (n — 1)Et[∆tt+1],
Et [ πt+1] = Et [ π*+1 ,F ] — nEt δ tt+1] ∙
(54)
(55)
21Note that NKPCs such as (49) and (50) in terms of the output gap sometimes are referred to as aggregate supply (AS)
curves (see Clarida et al. 2001, p. 250).
22If, for instance, some institution had the power to raise domestic output above its flexible-price value (given the deviations
from its zero-inflation steady-state value) by raising πt,H, not only πt,H but also βEt [πt+1,H] would have to rise for
(49) to hold with equality. This means that if output were kept on this artificially high level for an extended period of
time, the respective expected inflation rates would continue to rise at accelerating speed (acceleration theorem).
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