Lecture 9: The monetary theory of the exchange · PDF fileLecture 9: The monetary theory of...
Transcript of Lecture 9: The monetary theory of the exchange · PDF fileLecture 9: The monetary theory of...
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)