Exchange Rate Economics: Theories and Evidence

90 The economics of the PPP puzzle
The Consumers’ maximisation problem implies that consumers will spend a
share γ and 1 − γ of their income on non-tradable and tradable goods,respectively.
Within tradable goods,the relative expenditure between domestic and
foreign tradables will depend on the relative (domestic versus foreign) prices as
well as on the expenditure bias towards domestic goods. Within each of the three
sub-utility consumption baskets (domestic tradables,foreign tradables and nontradables),expenditure
will be allocated equally among varieties,as the latter are
all symmetric. Formally,the optimal expenditure functions of the representative
consumer of country k on the typical varieties of domestic tradables (cTk k ),foreign
tradables (cTk k ′ ) and domestic non-tradables (cNk k ) are given by:
[
( ) 1/(θ−1)
cTk k = (1/n Tk) (n Tk ) [θ/(θ−1)][(θ Tk−1)/θ Tk ]
pTk
k (δ)
1/(1−θ)]
/[
( ) θ/(θ−1)
(n Tk ) [θ/(θ−1)][(θ Tk−1)/θ Tk ]
pTk
k (δ)
1/(1−θ)
( ) ]
θ/(θ−1)
+ (n Tk ′) [θ/(θ−1)][(θ Tk ′ −1)/θ Tk ′ ] pTk k ′ (1 − δ)
1/(1−θ)
(1 − γ ) E k ,
(3.43)
[
( ) ]
1/(θ−1)
cTk k ′ = (1/n Tk ′) (n Tk ′) [θ/(θ−1)][(θ Tk ′ −1)/θ Tk ′ ] pTk k ′ (1 − δ)
1/(1−θ)
/[
( ) θ/(θ−1)
(n Tk ) [θ/(θ−1)][(θ Tk−1)/θ Tk ]
pTk
k (δ)
1/(1−θ)
( ) ]
θ/(θ−1)
+ (n Tk ′) [θ/(θ−1)][(θ Tk ′ −1)/θ Tk ′ ] pTk k ′ (1 − δ)
1/(1−θ)
(1 − γ ) E k ,
(3.44)
cNk k = 1
n Nk pNk
k γ E k . (3.45)
The firm’s maximisation problem,which is taken from a Dixit–Stiglitz formulation,produces
the following first-order condition:
p k sk =
w k
π sk θ sk
,(3.46)
where π sk represent the marginal productivity of labour in the respective sector of
each country, w is the wage and as is usual in the Dixit–Stiglitz framework, θ sk (a
parameter of the utility function) equals the inverse of the optimal mark-up,and
this is refered to as a measure of product market competition in the respective
sector. 15