where P is the consumer price index and PH and PF are the respective price indices of
home and foreign intermediate goods, all denominated in home currency:
PH,t =
1 pH,t(i)1-λ
0
1
■ 1-λ
di
PF,t =
1 pF,t(j)
0
1
■ 1-λ
1-λdj
We assume that there are no costs to trade between the two countries and the law of one
price holds, which implies that
PHt = etPH t
P* _ PFt
PFt =
t et
(5)
where e is the nominal exchange rate. Letting Q = e-p- denote the real exchange rate,
under the law of one price, the CPI index (4) and its foreign equivalent imply:
(ɪY-θ = ( Pt ∖1-θ
QQt) p.
aPH-tθ + (1 — a) (etPF,t)1 θ
a (etPF,t)1 θ + (1 — a)PH-tθ
(6)
and hence the purchasing power parity condition is satisfied only in the absence of any
bias between home and foreign intermediate goods (i.e. a = 0.5). The relative price T,
the terms of trade, is defined as T ≡ -PF-.
2.2 Intermediate Goods Producers
Intermediate firms hire labor and rent capital to produce output given a (real) wage rate
wt and capital rental cost rrt . A firm of type i has a production technology:
yt(i) = Kt(i)αLt(i)1-α,
(7)
where K and L represent capital and labor usage respectively, and the input share is
0 < α < 1. Given competitive prices of labor and capital, cost-minimization yields:
w=mct(1—α) (ʌ) (Ka) (8)
rrt=-( ⅛ )( ≡ Γ (9)