Lecture 9: The monetary theory of the

exchange rate

Open Economy Macroeconomics, Fall 2006

Ida Wolden Bache

October 24, 2006

Macroeconomic models of exchange ratedetermination

� Useful reference: Chapter 4 of Sarno&Taylor (2002)

� Some in�uential classes of models

� Portfolio balance models

� The monetary approach

� Flexible-price model

� Sticky-price model

� New Open Economy Macroeconomics

Portfolio balance models

� Useful reference: Branson&Henderson (1985) "The speci�cation and in�uenceof asset markets" in Handbook of International Economics, vol 2.

� Imperfect substitutability between di¤erent assets

� Wealth enters asset demand equations

The monetary approach

� Perfect substitutability between foreign and domestic assets (UIP)

� Exogenous money supply

� Flexible price models:

� Prices respond instantly to shocks

� Output is at its natural level at all times

� Purchasing power parity (PPP) holds continuously (P = SP �)

� Sticky price models (e.g., Dornbush (1976) overshooting model):

� Prices respond sluggishly to shocks

� Output deviates from natural level in the short run

� Short-run deviations from PPP

� Money neutral in the long run

New open economy macroeconomics (NOEM)

� Seminal contribution: Obstfeld&Rogo¤ (1995) "Exchange Rate Economics Re-dux", Journal of Political Economy vol 103

� Useful references: chapter 10 in Obstfeld&Rogo¤ (1996), chapter 5 in Sarno&Taylor(2002)

� Key features:

� Dynamic general equilibrium model

� Explicit microfoundations (intertemporally optimising agents)

� Imperfect competition and nominal rigidities (sticky prices and/or stickywages)

� Changes in the money supply a¤ect real variables in the short run

The monetary theory of the exchange rate

� Required readings: Chapter 4 in Rødseth (2000) and chapter 8.2.7 in Obst-feld&Rogo¤ (1996)

� Preliminaries on �rst-order linear di¤erential equations (see e.g., ch 1 in Syd-sæter et al. (2002) Matematisk Analyse Bind 2).

� De�ne _x(t) = dx=dt

� Consider the equation

_x(t) + ax(t) = b(t)

� For a given initial solution x(t0); the general solution is

x(t) = x(t0) exp(�a(t� t0)) +Z tt0b(�) exp(�a(t� �))d�

� Assumptions

� International goods markets are perfectly integrated �! PPP holds con-tinuously

� Perfect capital mobility �! UIP

� Rational expectations (perfect foresight)

� Flexible prices and wages �! output is always at its natural level

� Money supply is exogenous

� Focus on log-linearised model: lower case letters denote variables in naturallogs (x = lnX and _x = _X=X)

� Model equations

p(t) = s(t) + p�(t)

_s(t) = i(t)� i�(t)m(t)� p(t) = ��i(t) + �y(t);

where p is the domestic price level, s is the nominal exchange rate, p� is theforeign price level, m is the money supply, i is the nominal interest rate, i� isthe foreign nominal interest rate and y is the domestic output level

� Endogenous variables: s; p; i

� Exogenous variables: p�; i�;m; y

� Deriving the di¤erential equation for the exchange rate

1. Solve out for the domestic interest rate from the money market equilibriumcondition

i(t) = �1�(m(t)� p(t)� �y(t))

2. Substitute in for the domestic price level from the PPP condition

i(t) = �1�(m(t)� s(t)� p�(t)� �y(t))

3. Substitute into the UIP condition

_s(t) = �1�(m(t)� s(t)� p�(t)� �y(t))� i�(t)

_s(t) =1

�s(t)� 1

�(m(t)� p�(t)� �y(t))� i�(t)| {z }

z(t)

_s(t) =1

�s(t)� z(t)

� The di¤erential equation for the exchange rate is fundamentally unstable: thehigher is the level of the exchange rate, the higher is the rate of depreciation

� Intuition: s " �! p " (from PPP)�! m � p #�! i " (from the moneymarket equilibrium condition)�! _s " (from UIP)

� The exchange rate is free to jump to any level ! without an initial conditionthere is an in�nite number of solutions for the exchange rate

� The initial exchange rate is determined by the requirement that the exchangerate should tend to a strictly positive value when t goes to in�nity

� Solution in the special case where the exogenous variables are expected toremain constant: exchange rate jumps immediately to the stationary value;i.e, the value corresponding to _s(t) = 0

_s(t) = 0 =) i = i� =) s = m� p� � �y + �i�

� General solution

s(t) = s(t0) exp

1

�(t� t0)

!�Z tt0z(�) exp

1

�(t� �)

!d�

= s(t0) exp

1

�(t� t0)

!

�Z tt0z(�) exp

1

�(t� �)� 1

�(t� t0) +

1

�(t� t0)

!d�

= s(t0) exp

1

�(t� t0)

!

� exp 1

�(t� t0)

! Z tt0z(�) exp

1

�(t� �)� 1

�(t� t0)

!d�

=

"s(t0)�

Z tt0z(�) exp

�1�(� � t0)

!d�

#exp

1

�(t� t0)

!

� For the exchange rate not to explode the term in brackets must go to zero ast goes to in�nity, i.e.,

s(t0) =Z 1t0z(�) exp

�1�(� � t0)

!d�

� Recall that Z baf(x)dx = �

Z abf(x)dxZ b

af(x)dx =

Z caf(x)dx+

Z bcf(x)dx

which impliesZ baf(x)dx�

Z caf(x)dx =

Z acf(x)dx+

Z baf(x)dx =

Z bcf(x)dx

� Insert into the general solution

s(t) =

24 R1t0z(�) exp

��1� (� � t0)

�d�

�R tt0z(�) exp

��1� (� � t0)

�d�

35 exp 1�(t� t0)

!

=

"Z 1tz(�) exp

�1�(� � t0)

!d�

#exp

1

�(t� t0)

!

=Z 1tz(�) exp

�1�(� � t0) +

1

�(t� t0)

!d�

=Z 1tz(�) exp

�1�(� � t)

!d�

=Z 1t

"1

�(m(�)� p�(�)� �y(�)) + i�(�)

#exp

�1�(� � t)

!d�

� The equilibrium exchange rate is the discounted sum of the whole (expected)future path of the exogenous variables where the discount rate is the inverseof the interest elasticity of money demand �.

� Quali�cations

� For the integral to converge z must grow at an absolute value smaller than1=� as t!1

� The above solution is the fundamental solution, denoted s�(t). All solu-tions are related to s�(t) by

s(t) = s�(t) + c exp

1

�t

!;

where c is a constant and there is one such solution for every real numberc. The last term is a rational bubble and is unrelated to the underlyingexogenous variables.

� When the exogenous variables are expected to remain constant the solutionsimpli�es to

s(t) =

"1

�(m� p� � �y) + i�

# Z 1texp

�1�(� � t)

!d�

=

"1

�(m� p� � �y) + i�

# "�� exp

�1�(� � t)

!#1t

=

"1

�(m� p� � �y) + i�

#8>>>>>>>>><>>>>>>>>>:

��exp �1�(1� t)

!| {z }

0

+�exp

�1�(t� t)

!| {z }

1

9>>>>>>>>>=>>>>>>>>>;= m� p� � �y + �i�

� An unanticipated temporary increase in �z

� Assume that

z(t) =

(z0 +�z if t0 < t < t1z0 if t > t1

where z0 and �z are positive constants

� Then in period t0 < t < t1

s(t) =Z 1tz(�) exp

�1�(� � t)

!d�

=Z t1t(z0 +�z) exp

�1�(� � t)

!d� +

Z 1t1z0 exp

�1�(� � t)

!d�

=Z 1tz0 exp

�1�(� � t)

!d| {z } �

s0

+Z t1t�z exp

�1�(� � t)

!d�

� Recall that Zexp(ax)dx =

1

aexp(ax) + C

which implies

s(t) = s0 +�z

"�� exp

�1�(� � t)

!#t1t

= s0 +�z

0BBBB@�� exp �1�(t1 � t)

!+ �exp

�1�(t� t)

!| {z }

1

1CCCCA= s0 + �

1� exp

�1�(t1 � t)

!!�z

� If �z = 1��m

s(t) = s0 +

1� exp

�1�(t1 � t)

!!�m

� E¤ect of a permanent change is found by letting t1 !1

s(t) = s0 +�m

Empirical evidence

� Weak empirical evidence for the monetary model as a model of short-run ex-change rate movements

� Empirical tests tend to reject both

� the joint hypothesis of UIP and rational expectations; and

� the assumption of continuous PPP (see graph!)

� Some evidence supporting the model

� in periods when in�ation is high (e.g., Frenkel�s (1976) study of the Germanhyperin�ation in the 1920s)

� as a model of �long-run�exchange rate determination

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Nominal Broad DollarIndexPrice­adjusted BroadDollar Index

Figure 1: US nominal and real e¤ective (trade-weighted) exchange rate (logdi¤erences)