Exercise 3 Consider an economy with flexible prices as presented in this chapter. By hypothesis, assume that there is free mobility of goods with no transaction costs, so that purchasing power parity is true, and that there is free mobility of capital that, together with the hypothesis of absence of exchange risk premium, yields the uncovered interest rate parity. Assume also that the money market equilibrium is given by the following equation, where all variables are expressed in logarithmic terms:

mðtÞ 2 pðtÞ 5 φyðtÞ 2 ηiðtÞ; ð6:26Þ

where the variables have the usual meaning. The foreign economy variables possess analogous annotation, however identified by the symbol . In this model, time is expressed in continuous terms.

a. Show that, by combining the presented hypotheses in the presentation with the Eq. (6.26), we can obtain the following equation sðtÞ 5 f ðtÞ 1 ηEt dsðtÞ

dt n o, where f ðtÞ 5 mðtÞ 2 pðtÞ 2 φyðtÞ 1 ηi

ðtÞ is economic fundamentals, sðtÞ represents the nominal exchange rate, and Etfg represents the hope operator conditional.

b. Assume that the fundamentals of this economy have the following trajectory:

f ðtÞ 5 θt if t , f1

θ

ηβ lnðf2 2 f1Þ 1 βt otherwise 8

<

:

where f2 . f1 and β . θ 1 1. Draw a graph of the fundamentals path. By means of the monetary model, intuitively explain the behavior of the economic fundamentals.

164 PRINCIPLES OF INTERNATIONAL FINANCE AND OPEN ECONOMY MACROECONOMICS

c. Assume that all economic agents know the fundamentals path described in item (b).

What is the exchange rate path? (Tip: the general solution for the differential equation Gðf Þ 5 f 1 ηθ dGðf Þ

df is given by Gðf Þ 5 f 1 ηθ 1 C e f

ηθ, where C is a constant.)

d. Draw a graph of the exchange rate path along with the fundamentals path. Compare between the two paths, analyzing them based on the intuition proposed by the economic theory present throughout this chapter.