Formally, by summing up all the tra

Formally, by summing up all the traded goods in each country, the absolute
version of the PPP hypothesis requires:
(3)
where the weights in the summation satisfy ΣNi =1αi = 1. Alternatively, if the price
indices are constructed using a geometric index, then we must form the weighted
sum after taking logarithms:
(4)
where the geometric weights in the summation satisfy ΣNi =1γi =1 and lower case
letters denote logarithms. The weights αi or γi are based on a national price index
and, according to the seminal Cassellian formulation of PPP, the consumer price
index (CPI). If the national price levels are Pt and P*t or, in logarithms, pt and p*t ,
then (according to whether the arithmetic or geometric index is used) we can use
equation (3) or (4) to derive the (absolute) PPP condition:
(5)
From equation (5) it is easily seen that the real exchange rate, defined here in logarithmic
form:
(6)
may be viewed as a measure of the deviation from PPP.
q s p p t t t t ≡ − + *,
s p p t t t = − *.
i γ i i t γ
N
t i i i t
N p s p , ,, = = Σ = +Σ 1 1
*
α α i i i t
N
t i i i t
N P S P , ,, = = Σ = Σ 1 1