Macroeconomic Interdependence in a Two-Country DSGE Model under Diverging Interest-Rate Rules

5. New Keynesian Framework

Phillips curve it is necessary to assure for price stickiness in addition to monopolistic competition. This
shall be done by introducing Calvo (1983) contracts, which means that each producer is only allowed
to reset her price with probability 1 - δ in any given period, independent of the time since the last
adjustment. Therefore, a measure of 1 - δ of firms reset theirs prices each period, while a measure of δ
of firms keep their prices constant and simply adjust their individual output in order to meet demand.
1 /(1 — δ) then captures the average duration of a price (see Gall 2008, p. 43):

∏tH β βEt [ ∏t+1 ,h ] + ^{(1 δ} )(_δ^{1 δβ} ) ^ t,                                                      (42)

*    - МЧ *    1 . (1 — δ*)(1 — δ*β) ʌ*

^πt,F = ^βEt^[πt+1 ,F^]+        δ*        ^Kt .                                                    ⁽⁴³⁾

In equation (42) π_t,H := p_t,_H — p_t-1 ,_h is defined as current domestic PPI inflation, which typically
differs from domestic CPI inflation. The NKPC (42) states that current domestic PPI inflation πt,H is
an increasing function of both expected domestic PPI inflation Et [πt+1,H] and the deviation of current
domestic real marginal production cost from its zero-inflation steady-state value ^_t := κ_t — κ^flex.

Note that an analogous interpretation for (43) also holds abroad. However, δ^* is assumed to differ from
δ and κ* also from κ, although κ^flex = (κ*)^flex = (θ — 1)/θ. Furthermore, let us assume that ’’setting a
new price at home” and ”setting a new price abroad” are stochastically independent events. As domestic
and foreign firms both set theirs prices in the currency of the countries where they are located, the
present model features producer-currency pricing, which is one of the possible occurrences of pricing to
market.²⁰

Nonetheless, it would be desirable to express equations (39), (40), (42), and (43) in terms of the output
gap, which shall be defined as the difference between the sticky-price and the flexible-price output devi-
ations: x_t := y_t — y^flex and x* := y * — ( y * )^flex. If we want to implement this we have to investigate the
ratio of the sticky-price real marginal production cost κ_t and its flexible-price counterpart κ_t^{f lex} given by
(25):
where Y_t^{f lex} denotes the domestic flexible-price equilibrium output as given by equation (38). Log-
linearizing this expression around the zero-inflation steady-state yields:

^ t = ( ρ — ξ )( y t — y ^ffl^ex ) = ( ρ — ξ ) xt.                                                                  (46)

Hence, by using (46) equations (39), (40), (42), and (43) rearrange to:

²⁰This specification has already been adopted in the theoretical literature (see, e.g., Clarida et al. 2002, p. 885) and can
also be justified by empirical evidence for most of the G7 countries (see Leith/Malley 2007, p. 420).

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