n?u=RePEc:hbu:wpaper:1&r=his

using the market exchange rate to express both in terms of the numeraire (Diewert,

2003).

The cost of the basket in local currency is:

E " =

(

'*+

p ",' ⋅ q '

where g = 1 … G indexes the consumption items in the basket, the q ' ’s are the weights

of each item in the common basket, and the p ",' ’s are the country-specific prices of each

commodity in the local currency. The formula for E . is analogous, with prices p .,' . As

before we convert local currency measures to the numeraire using the market exchange

rate:

NE " = E "

XR .,"

with XR .,. again equal to one by definition. Finally we have

(2) SPI " = NE "

NE .

The SPI gives us the relative price level in the two countries. When SPI exceeds one,

country j’s prices are “too high” relative to country i. Alternatively, its currency is

overvalued relative to purchasing power parity. The hypothetical exchange rate that

would preserve the purchasing power of our money when changing currencies would

be: 14

(3) PPP .," = XR .," ×SPI "

We can use this PPP standard to convert the local nominal expenditures of our two

households to real expenditures expressed in terms of the numeraire.

(4) RE " # = NE " #

PPP .,"

(5) RE . #/ = NE . #/

PPP .,.

= E .

#/

14 An algebraically equivalent formulation is PPP .," = E " /E . .

20