Exchange Rate Economics: Theories and Evidence

The flexible price monetary approach 101
The consolidated budget constraint may be obtained by equating (4.18) with (4.19):
c yt + q t c y ∗
t
+ e t ω yt + et ∗ ω yt ∗ = ω yt−1 (y t + e t ) + ω y ∗
t−1
(q t yt ∗ + et ∗ ). (4.20)
Maximising equation (4.15) subject to the consolidated budget constraint for the
home country yields a standard consumption Euler equation of the form:
q t = u∗ c t
u ct
. (4.21)
Equation (4.21) relates the relative price of the two goods to their marginal rate of
substitution (where u ct is the marginal utility of consumption of the y good and u ∗ c t
is the corresponding marginal utility of y ∗ consumption). 4 A standard implication
of this kind of first order condition is that if the agent is behaving optimally then
a reallocation of x units C for C ∗ should not result in any change to total utility.
Similar first order conditions (not reported here) also hold for domestic holdings
of home and foreign equity,and a similar set of first-order conditions are assumed
to hold for the foreign country as well.
Following Mark (2001) the use of an explicit form of the utility function makes
the result in (4.21) more concrete. In particular,if the period utility function is
assumed to be of the constant relative risk aversion (CRRA) form:
u(C, C ∗ ) = X 1−γ
t
1 − γ ,(4.22)
) equa-
where X is a Cobb–Douglas index of the two goods (X t
tion (4.22) can be rewritten as
= C θ
t Ct
∗1−θ
q t = 1 − θ
θ
y θ−1
t
(y ∗ t ) −θ (4.23)
and by dividing the numerator and denominator by (θ − 1) we get
q t = 1 − θ
θ
y t
yt
∗
,(4.23 ′ )
which indicates that in the barter economy the real exchange rate is determined
by relative output levels.
4.3.1.1 Introduction of a cash-in-advance constraint into the barter model
Money can be introduced into this model by assuming a so-called cash-in-advance
constraint; that is,agents are required to use money for the purchase of goods (an
alternative way to get money into this class of model is through the utility function
and this kind of model is considered in some detail in Chapters 10 and 11). In the
Lucas (1982) model the cash-in-advance constraint means that consumers have to