Estimated Open Economy New Keynesian Phillips Curves for the G7

the consumer prices, since the firms are assumed to be owned by domestic
consumers) in period t of the firm producing good z are given by,

Such firms are able to change their price with probability α in a given period,
so that ι-1α measures the length of time a price contract is expected to exist.
This allows us to write the problem facing a firm which is able to change prices
in period t as,

-θ

(^xt)   (cd + gd + m^f * + cf * + gf *) x±

_ptd        t t                        _Pt

_x -θψ

-MCt         (cd + gd + mf * + cf * + gf *)^ψ

( —x^t— ´   ( cd + (7d  + m^f * + c^f * + (7^f * ) —x^t—

Vd+s )   (^ct+s ^{+ g}t+s ^{+ m}t+s ^{+ c}t+s ⁺ gt+s) Pt+s

-θψ

^-M^gt+s dp+s)    (^cd+s ⁺ gt+s ^{+ m}t+s ^{+ c}t+s ^{+ g}t+s)^ψ

j^s=1 rt+j-1

The first order condition for this optimisation is given by,

ψθ(p_t^d)ψ^θMgCt (c_t^d + g_t^d + m_t^f* + c_t^f* + g_t^f*)ψ

P∞ _____w___________

'^t -s'_____________________________Q j = 1 rt+j-1______________________________

(θ ^{- 1})(pd)^θ PT¹ (cd⁺ gd ^{+ mf} *^{+ cf} *⁺ g^f *)

P P∞ ^αp (θ ^- ι)(^pd+s )^{θ p}t+1s (^cd+s +^gd+s +^mf+s +^cf+s +^gf+s )

E^t2-s=l                    Qj = ι rt+j-1

(17)

The first-order condition for the optimal price can be log-linearised to yield,

+(ψ - 1)ye_t+s + P^b_t+_s + θ(ψ - 1)pb_t+s]

where yet = c_t^d +g_t^d +m_t^{f *} +c_t^{f *} +g_t^{f *} is the average firm output supplying domestic
and foreign, private and public demand. This infinite forward summation, can
also be quasi-differenced to give a first order difference equation describing the
evolution of the optimal price set by profit-maximising firms,

„                           r              ----~

α

⁽ г—α ^)Et xt+ι =⁽ r—α ^)xt^- MCt^- (^{ψ - 1})^yt^- Pb^- θ(^{ψ - 1})^pt ■    ⁽¹⁹⁾

The firms which do not perform this optimisation, instead follow a rule of
thumb whereby they set a price equal to the average price set on the previous

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