Estimated Open Economy New Keynesian Phillips Curves for the G7

Appendix I - Operationalising the Open Economy NKPC

To transform our Phillips curve to a form more appropriate for estimation
and to facilitate comparability with existing studies it is necessary to relate the
open economy measure of marginal cost to the typical closed economy proxy,
the ratio of labour income to GDP. To do so recall that average firm output in
the presence of intermediate good inputs is given by,

p_t f

(34)

yt = yt + -d mt
p_t^d

where yet is average firm output and and yt is real GDP. After log-linearisation
this implies,

—

ye_t =

p^dy

p^dy + p^f m^f

^ , Pf^mf f^,f , ^f

^b⁺ pdy + p∣m∣ (^m^f⁺^p^f^-

(35)

Substituting the expression for the optimal level of imported intermediate goods,(12),
and solving for average firm output gives,

_b p^dy ʌ p^f mf

yt p^dy + (1 — ψ)p^f m^f yt p^dy + (1 — ψ)p^f m^f

This allows us to rewrite the Phillips curve as,

1-ρ
w^f am)

^αN (w^f am´ P⁺^α

wb_t^f +pb_t^f —pb_t^d).

(36)

d
πb_t

^βa ,- dd ^ω dd ⁽¹^-^ω)(¹^-^α)(¹^-^aβ) /________w________^

^λ t t⁺1 ^λ t 1 (^{1 +} (^ψ^-¹)θ)^λ (w + ^p^f (_wjf am ´ɔ ^t

∖ ⁺ P ∖ ' .

p^f fwf amʌ^p

P ∖ ^aN )

pf fam)ɔ

P ∖ aN J J

(pb_t^f —

'_b ∖ _l Г I ^p ∖_f Pdy ^
Pt) ⁺ (^ψ - ¹⁾⁽pdy +(1 — ψ)pfmfyt

(37)

1-ρ
^wf θNm)
--------×---------

' ^pd^y⁺ (¹^-^ψ)^pfmf ∖n (wf ^amy^-^ρ + am

+ pb_t

7 7

— pb_t )) + Pt — pb_t ).

Another way of rewriting this is in terms of the labour share variable, e.g.

sbt =Wct —P^bt +N^bt —ybt — (1 — χ)(pbt^d — pbt^f).

(38)

Substituting for the first-order condition for labour supply and using the defin-
ition of consumer prices yields,

sbt = Wct + ψ^byet —

ρ-1
^wf ^amJ

^aN + a_m(^wf am)^ρ^-1

f
wb_t

— p_t

— ybt .

(39)

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