Models with Two Sources of Dynamics1.2 RATEX Dynamics We are going to introduce a second source of...
Transcript of Models with Two Sources of Dynamics1.2 RATEX Dynamics We are going to introduce a second source of...
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Models with Two Sources of Dynamics
Course: Macroeconomics
Professor: Alan G. Isaac
February 23, 2012
Contents
1 Term Structure Model 1
1.1 Disequilibrium Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 RATEX Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.1 Explicit Solution for the Roots . . . . . . . . . . . . . . . . . . . . 6
1.4.2 Solution by Adjoint Matrix Technique . . . . . . . . . . . . . . . . 7
2 Overshooting: An Introduction 10
2.1 Uncovered Interest Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 The Static Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Static Expectations: The Mundell-Fleming Model . . . . . . . . . . 11
2.3 Fixed Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 The Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Floating Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.1 The Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
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2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Regressive Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6.1 The Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7 Rational Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2.7.1 Long-Run Comparative Statics . . . . . . . . . . . . . . . . . . . . 1
2.7.2 Dynamic Adjustment: The Intuition . . . . . . . . . . . . . . . . . 2
2.8 Definite Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.9 Anticipated Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Dornbusch (1976) 6
3.1 Regressive Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Rational Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Blanchard’s Stock Market Model 13
5 The Adjoint Matrix Technique 18
Last modified: 2012 Feb 23
1 Term Structure Model
In this section we will lay out the term structure model more or less at it can be found
in Blanchard and Fischer (1990, ch. 10.4).
We have considered a “Keynesian” IS-LM model with a single interest rate. But often,
a short rate is considered more relevant to the money market and a long rate to investment
decisions. We therefore considered the Tobin 3-asset model, which allowed for imperfect
substitutability between short-term and long-term assets. We are now going to return to
the assumption of perfect substitutability. Term structure considerations are reduced to
the requirement that the total return on short and long bonds be the same. This can still
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illustrate interest term structure considerations in a dynamic setting.
For simplicity, consider a “real console” that pays one unit of output forever. This
will be our long bond. Suppose it costs Q and therefore has an associated capital gain of
Q̂. Then the coupon rate of return is R = 1/Q, while the capital gain is −R̂ = Q̂. So we
find the total rate of return on the long bond is
R− R̂ = 1/Q+ Q̂ (1)
We will simplify slightly from the textbook treatment. Rather than carry around an
exogenous risk premium, we allow the long bond and short bond to be perfect substitutes.
Also, we will ignore expected inflation since we have kept things simple by working with
a fixed price framework.
i = R− R̂e (2)
Our “Keynesian” IS-LM model therefore becomes
Y = A(Y,R, F ) (3a)
M = L(R− R̂e, Y ) (3b)
Given R̂e, there is no fundamental change in structure.
1.1 Disequilibrium Dynamics
One of the first dynamic modifications of the IS-LM model that many students are in-
troduced to allows disequilibrium dynamics in the goods market. The usual motivation
is that the money market clears very quickly and therefore can still characterized by the
LM curve, but we should recognize that the goods market is slower to clear. Output is
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assumed to adjust in response to excess demand in the goods market.
Ẏ = φ̃(A− Y ) φ̃(0) = 0, φ̃′ > 0 (4)
Giving the partial derivatives of A the usual size and sign, we get
Ẏ = φ(R, Y, F ) φR < 0, φY < 0, φF > 0 (5)
These disequilibrium dynamics in the goods market are stable, in the sense that dẎ /dY <
0, and no further dynamics are introduced into this model.
1.2 RATEX Dynamics
We are going to introduce a second source of dynamics: myopic perfect foresight. We
retain our previous descriptions of disequilibrium output adjustments in (5). We add
another source of dynamics with the RATEX assumption:
R̂e = Ṙ/R (6)
Substituting this into our description of money market equilibrium produces
M = L(R− Ṙ/R, Y ) (7)
The equations of motion of the model become
M = L(R− Ṙ/R, Y ) (8)
Ẏ = φ(R, Y, F ) φR < 0, φY < 0, φF > 0 (9)
This is a two-equation first-order system of differential equations (in R and Y ).
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R
YO
@@@@@@@@@@@@@@@@@@@@@@@@@Ẏ=0
�������������������������
Ṙ=0
��
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Figure 1: Term Structure Model
1.3 Phase Diagram
Usually, we approach the explanation of how things change by first describing where they
do not change.
So let us start with an explanatin of why the Ẏ = 0 locus slopes down. The Ẏ = 0
locus is just the combinations of Y and R such that the goods market is in equilibrium.
It is an IS curve. So it slopes down for exactly the same reasons (and in exactly the same
circumstances) as an IS curve.
Next consider what is happening off the IS curve. If we start at a point on the IS
curve and increase R, this reduces investment demand and thereby creates a situation
of excess supply. Our equation of motion for the disequilibrium dynamics of the goods
market tell us that output falls in the face of excess supply, so Ẏ < 0. Similarly, we have
Ẏ > 0 below the Ẏ = 0 locus.
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The Ṙ = 0 locus is a bit trickier to describe, because it is not simply an LM curve.
Generally we think of an LM curve as those special points where the money market is in
equilibrium. But our description of the money market says that it is always in equilibrium.
So the Ṙ = 0 locus is the combinations of Y and R such that we have money market
equilibrium with an unchanging long rate. It is not too misleading to call the a “long-run
LM curve”.
1.4 Algebra
Review adjoint matrix technique.
M = L(R− Ṙ/R, Y ) (10)
Ẏ = φ(R, Y, F ) (11)
implies the linear approximation system
0 = Li
(δR +
Ṙ
R2δR− 1
RδṘ
)+ LY δY (12)
δẎ = φRδR + φY δY (13)
where δR = R−Rss etc.
Recall that all the partial derivatives in this system are evaluated at the steady state.
Noting that Ṙ = 0 at the steady state, we can write this as
Li − LiRD LYφR φY −D
δRδY
= 0 (14)This is in the general form P (D)x = 0. Note that the determinant of P (D) can be written
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as
|P (D)| = LiRD2 − Li
R(R + φY )D + LiφY − LY φR (15)
Now we are going to abuse notation slightly. Up to this point, D has been the differ-
ential operator. Now, in order to avoid introducing extra notation, we will let it represent
a variable. Find the characteristic equation by setting |P (D)| = 0.
LiRD2 − Li
R(R + φY )D + LiφY − LY φR = 0 (16)
or
D2 − (R + φY )D + φYR−LYLiRφR = 0 (17)
Note that the last term is negative, so we know we have saddle-path dynamics. Why?
Recall that the solutions D1, D2 of this quadratic equation can be used to rewrite the
equation as
(D −D1)(D −D2) = 0 (18)
D2 − (D1 +D2)D +D1D2 = 0 (19)
1.4.1 Explicit Solution for the Roots
D1, D2 =φY +R±
√(φY +R)2 − 4(φYR− LY φRR/Li)
2(20)
By convention, we let D1 be the smaller root, which as we have seen must be negative.
Be sure you can explain why
√(φY +R)2 − 4(φYR− LY φRR/Li) > φY +R
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Therefore the smaller (negative) root is
D1 =φY +R−
√(φY +R)2 − 4(φYR− LY φRR/Li
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1.4.2 Solution by Adjoint Matrix Technique
Note that
adjP (D) =
φY −D −LY−φR Li − LiD/R
(21)Therefore δR
δY
= η1φY −D1−φR
eD1t + η2φY −D2−φR
eD2t (22)This is of course unstable (because D2 > 0). But if we could set η2 = 0, the solution
converges. δRδY
= η1φY −D1−φR
eD1t (23)Query: we are used to pinning down the dynamic path by getting the initial conditions
for our two variables. Does this mean η1 is overdetermined?
Answer: No! We turn to the economics of the model to recognize that R is a jump
variable while Y is a predetermined variable. We have an initial condition only for Y .
Setting η2 = 0, given δY0, determines δR0 and thus the initial jump to the convergent
arm.
η1 = −δY0/φR (24)
so δRδY
= −δY0φR
φY −D1−φR
eD1t (25)
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Exercise 1
In the basic term-structure model under rational expectations, show δR/δY > 0 along
the convergent arm.
Exercise 2
In the basic term-structure model under rational expectations, find an expression for δit.
(You can use your exising rational expections model solution.)
Exercise 3
For the basic term-structure model under rational expectations, discuss the effects of a
one-time, permanent increase in F .
Exercise 4
For the basic term-structure model under rational expectations, show how to solve the
model dynamics using the adjoint matrix technique.
For a postively sloped convergent arm, we need φY −D1 > 0. Recall
D1 =φY +R−
√(φY +R)2 − 4(φYR− LY φRR/Li)
2
< 0
(26)
so we have
φY −D1 = φY −φY +R−
√(φY +R)2 − 4(φYR− LY φRR/Li)
2
=φY −R +
√(φY +R)2 − 4(φYR− LY φRR/Li)
2
=φY −R +
√(φY −R)2 + 4LY φRR/Li
2
> 0
(27)
even if φY +R < 0.
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Recall that we have
δR = R−RLR = −(δY0/φR)(φY −D1)eD1t (28)
implying
δṘ = Ṙ− ṘLR = Ṙ = −D1(δY0/φR)(φY −D1)eD1t (29)
Recall (since π = 0 and perfect substitutes)
i = R− Ṙ/R =⇒ δi = δR− δṘRss
+ṘssR2ss
δR (30)
with the last term equal to zero in the steady state. Therefore
δi = −(δY0/φR)(φY −D1)(1−D1Rss
)eD1t (31)
Suppose δY0 > 0. Then δi > δR ∀t. Since iLR = RLR, we know i > R ∀t.
Why? R is falling. Since Ṙ < 0, the short rate must be high enough to offset the
capital gains on the long bond.
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Exchange Rate Overshooting
Although the exchange rate model is discussed in Romer and in Blanchard and Fischer, I
am providing this handout as so that you have an example of the algebraic details of the
adjoint matrix technique. As always, please inform me of any typos in this handout.
2 Overshooting: An Introduction
This handout develops a version of the Dornbusch overshooting model, roughly following
Blanchard and Fischer (1990, p.537). We write the structural equations as
Y = AD(Y, i, F, E) (32)
M = L(i, Y ) (33)
The nominal exchange rate influences aggregate demand by changing the real exchange
rate.1 The nominal interest rate i influences aggregate demand by changing the real
interest rate.
We add perfect capital mobility and perfect capital substitutability in the form of
uncovered interest parity.
i = i∗ +EĖ
E(34)
2.1 Uncovered Interest Parity
Following Romer (1996), we develop uncovered interest parity as follows. Consider two
ways of holding a dollar of your wealth. First, you can invest at rate i so that at the end
of ∆t periods you have new wealth of ei∆t dollars. Alternatively, buy 1/E units of foreign
exchange and invest them at rate i∗, so that at the end of ∆t periods you have new wealth
1We have normalized P ∗ and P to unity, since prices are exogenous. Similarly, we have normalizedexpected inflation to zero.
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of ei∆t/E units of foreign exchange. Then sell them for dollars, with a final outcome of
Et+∆tei∆t/Et dollars. With risk neutral investors, these two approaches should yield the
same expected returns.
ei∆t = ei∗∆tE
Et+∆tEt
(35)
At time t, E, i, and i∗ are known contemporary values, but E(t + ∆t) is realized in
the future. So we can write this uncovered interest parity condition as
ei∆t = ei∗∆tEEt+∆t
Et(36)
Taking the derivative with respect to ∆t, we get
iei∆t = i∗ei∗∆tEEt+∆t
Et+ ei
∗∆tEĖt+∆tEt
(37)
Finally, evaluating this at ∆t = 0, we get
i = i∗ +EĖtEt
(38)
This has a natural interpretation: interest-rate differentials must be offset by expec-
tations of exchange-rate movement.
2.2 The Static Model
2.2.1 Static Expectations: The Mundell-Fleming Model
If the exchange rate is not expected to change, or if it is as likely to rise as to fall,
EĖt = 0 and uncovered interest parity implies simply that i = i∗. This may be an
appropriate assumption in two circumstances: if the exchange rate is truly fixed, or in
some circumstances, if you are doing the long-run analysis of a fixed exchange rate system
(e.g., when both countries involved have zero average inflation).
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2.3 Fixed Rates
The structural equations are
Y = AD(Y, i∗, F, E) (39)
M = L(i∗, Y ) (40)
with M and Y endogenous.
M
YO
IS
�������������LM
Figure 2: Fixed Rates: Mundell-Fleming Model
2.3.1 The Algebra
The comparative statics algebra begins by totally differentiating the structural equations
with respect to the endogenous variables (M and Y ) and the exogenous changes of interest
(in this case, E and F ).
dM = LY dY (41)
dY = ADY dY + ADEdE + ADFdF (42)
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which we can write LY −1−(1− ADY ) 0
dYdM
= 0−ADEdE − ADFdF
(43)Note that this is recursive: we can solve the second equation for dY , and then use this
value of dY to solve the second equation for dM . The solution is
dYdM
= −11− ADY
0 11− ADY LY
0−ADEdE − ADFdF
(44)which simplifies to
dYdM
= 11− ADY
ADEdE + ADFdFLY ADEdE + LY ADFdF
(45)Exercise 5
What is the effect of a one-time, permanent increase in the foreign interest rate: di∗?
2.4 Floating Rates
The structural equations are once again
Y = AD(Y, i∗, F, E) (46)
M = L(i∗, Y ) (47)
but now the endogenous variables are E and Y .
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E
YO
LM
�������������IS
Figure 3: Flexible Rates: Mundell-Fleming Model
2.4.1 The Algebra
The comparative statics algebra begins by totally differentiating the structural equations
with respect to the endogenous variables (E and Y ) and the exogenous changes of interest
(in this case, M and F ).
dM = LY dY
dY = ADY dY + ADEdE + ADFdF
which we can write LY 0−(1− ADY ) ADE
dYdE
= dM−ADFdF
(48)Note that this is recursive: we can solve the first equation for dY , and then use this value
of dY to solve the second equation for dE. The solution is
dYdE
= 1LY ADE
ADE 01− ADY LY
dM−ADFdF
(49)
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which simplifies to dYdE
= dM/LY
1−ADYLY ADE
dM − ADFADE
dF
(50)HW: effect of di∗
2.5 Summary
Summarizing the effects of monetary and fiscal policy under fixed and flexible rates, with
static expectations:
Fixed Rates Flexible Rates
Monetary Impotent Powerful
Fiscal Powerful Impotent
2.6 Regressive Expectations
Suppose Êe = θ(ELR − E)
So we can write,
M
P= L[i∗ + θ(ELR − E), Y ] (LM)
Y = AD[Y, i∗ + θ(ELR − E), F, E] (IS)
We could repeat our previous comparative statics with ELR exogenously given. We
would recover effectiveness of fiscal policy. But what is ELR? We will motivate our un-
derstanding of the long run by introducing slow output adjustment. (In the original
Dornbusch (1976) article, prices adjusted slowly.)
Ẏ = φ̃[AD[i∗ + θ(ELR − E), Y, E, F ]− Y
]= φ[
−Y ,
+
E,+
F ] (51)
At this point, you will notice, we follow Blanchard and Fischer by setting φr = 0 to
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E
YO
@@@@@@@@@@@@@LM�
������������IS
Figure 4: Regressive Expectations
simplify the algebra and, more importantly, to make it easier and more intuitive to draw
the phase diagram.
Take ELR as the value determined by the requirement that Ẏ = 0 and i = i∗. (So that
M/P = L(i∗, Y ) determines YLR.)
@@@@@@@@@@@@@ LM
�������������
IS
Ẏ >0
Ẏ
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The implied dynamics near a steady state are
0 = −LiθδE + LY δY
δẎ = φY δY + φEδE
which we can write as LY −θLiD − φY −φE
δYδE
= 0 (52)The characteristic equation is |P (D)| = 0.
−LY φE + θLiD − LiφY = 0 (53)
This yields a single characteristic root.
D1 =LiφY + LY φE
θLi< 0 (54)
Using the characteristic root, we can apply the adjoint matrix technique to write
δYδE
= η θLiLY
eD1t (55)Note that Y is predetermined, while E is a jump variable. This means that we can
solve for η by noting
δY0 = ηθLi =⇒ η =δY0θLi
(56)
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Using our solution for η, we have
δYδE
= δY0θLi
θLiLY
eD1t (57)=
δY0LY δY0/θLi
eD1t (58)So, δY0 < 0 =⇒ δE0 > 0
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2.7 Rational Expectations
The rational expectations assumption is that EĖ = Ė. Uncovered interest parity can
therefore be expressed as
i = if + Ė/E (59)
As before, use the uncovered interest parity condition to substitute for the domestic
interest rate in the money market. Our description of money market equilibrium becomes
M = L(if + Ė/E, Y ) (60)
Our description of the goods market is unchanged: output adjusts in response to excess
demand.
Ẏ = φ(Y,E, F ) (61)
2.7.1 Long-Run Comparative Statics
To conduct long-run comparative statics experiments, we set Ė = 0 and Ẏ = 0. This
yields the familiar Mundell-Fleming model of flexible exchange rates.
When Ẏ = 0, we are on the IS curve. At a depreciated (higher) exchange rate, demand
shifts toward domestic goods, so the equilibrium level of output is higher. The IS curve
is therefor upward sloping in Y,E-space.
When Ė = 0, Y is determined by the money supply. There is a unique level of output
such that the money market clears. This gives us a vertical long-run LM curve.
The result is familiar: it is just the static Mundell-Fleming model, as represented by
figure 3.
HW: consider the effects of a change in M and of a change in F . Illustrate graphically
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and do the algebra.
2.7.2 Dynamic Adjustment: The Intuition
When we considered the long-run comparative statics, we developed an IS curve and a
long-run LM curve that tell where the dynamic system is not changing. To examine the
dynamics, we need to add to this some information about how the system is changing at
every point in Y,E-space. This information is provided by the equations of motion.
M = L(if + Ė/E, Y )
Ẏ = φ(Y,E, F )
(62)
Recall that any given increase in Y produces a somewhat smaller change in aggregate
demand, and therefore an increase in Y reduces excess demand in the goods market:
φY < 0. Therefore to the right of any point on the IS curve, income is falling. To the left
of the IS curve, income is rising. Note how the dynamics introduced in the goods market
are stabilizing.
The money market provides a contrast. Starting on the long-run LM curve (i.e., the
Ė = 0 curve), and notice that i = if . Now move to the right. We know the interest rate
must be higher, since the higher income raises money demand and money supply is fixed.
So we must be at a point where i > if , or equivalently, Ė > 0.
Figure 6 is the resulting phase diagram. The phase diagram tells us how the system
evolves given any initial position, but how do we determine the initial position of the
economy?
Exercise 6
Consider the basic sticky-income overshooting model under rational expectations. Discuss
the effects of an unanticipated one-time, permanent increase in F . Discuss the effects of an
anticipated one-time, permanent increase in F . No algebra is required for the dynamics:
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�����
���
���
���
���
��HHHHHH
HHHHHHHHH
HHHHHH
6-
?
�
6�
?
-
Ė=0
Ẏ=0
E
YO
Figure 6: Phase Diagram for Overshooting Model
just discuss in detail. But please do the LR comparative statics algebra.
2.8 Definite Solution
If we can determine the initial position of the economy, the equations of motion tell us
how the economy will evolve over time. One approach is to say that the initial position
of the economy, whenever we start our analysis, is determined by the previous history of
the economy. This is not satisfactory, unless we believe the economy is given to explosive
dynamic behavior. Notice that only points along the convergent path will tend to the
steady state; all other points are explosive.
Furthermore, while it is plausible that Y is given historically, it is not plausible for E.
Our model is built around the idea that Y adjusts slowly to economic conditions, so it is
reasonable to treat is as a predetermined variable. However E is a floating exchange rate,
which we think of as moving rapidly in response to any change in expectations. It is a
jump variable. So while we can determine the initial value of Y as historically given, we
need another approach to E. Our approach will be to invoke long-run perfect foresight.
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����
���
���
���HH
HHHHHHHHHHHH
jjjj
Ė′=0Ė=0
Ẏ=0
CA
E
YO
Figure 7: Overshooting: Unanticipated Monetary Expansion II
Recall that under the rational expectations assumption, in this deterministic model,
EĖ = Ė. So the rational expectations assumption is equivalent to myopic perfect fore-
sight. This is embedded in the equations of motion for this system.
Long-run perfect foresight is the assumption that individuals know the long-run eco-
nomic outcome, which we will take to be the steady state of the economy. Rational
expectations are not enough to ensure that we move toward the steady state, but if we
add the assumption that individuals know that the long-run outcome is the steady state
we can ensure that we actually get there.
Consider the historically given level of income, Y0. There is an infinite variety of
exchange rates that are possible with this initial level of income, and we can follow the
equations of motion from any of them. But only one exchange rate gives us an intial
position that evolves by moving toward the steady state. The assumption of long-run
perfect foresight is the assumption that this is the exchange rate determined by the
economy.
Exercise 7
The Algebra: For our simple, “Keynesian” overshooting model, solve for the rational ex-
pectations dynamics using the adjoint matrix technique. Find the slope of the convergent
arm algebraically, and develop it in the phase diagram.
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����
���
���
���
��HH**jj
Ė=0
Ẏ=0
E
YO
Figure 8: Overshooting: Anticipated Monetary Expansion
2.9 Anticipated Changes
Anticipated ↑M to take place at T but announced at t0. Key: not rational to anticipate
asset price jumps (i.e., infinite rates of return). (Say why!)
Therefore E jumps at the time of announcement t0, so that following the rules of
motion, it will land on the anticipated location of the convergent arm at time T .
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3 Overshooting Price Dynamics in Continuous Time
Dornbusch’s explanation shocked and delighted researchers because he showed
how overshooting did not necessarily grow out of myopia or herd behavior in
markets.
rogoff−2002−overshoot25
This section characterizes the sticky price dynamics in continuous time, which is a
common theoretical treatment. The algebraic details in this section may be skipped by
all MA students.
3.1 Regressive Expectations
Recall the basic (partially reduced) structural relationships of the model
H
P= L (i∗ − θ δe, Y ) (63)
AD = AD (i∗ − θ δe, Y,G,EP ∗/P ) (64)
∆p = f
(AD (i∗ − θ δe, Y,G,EP ∗/P )
Y
)(65)
For example, consider the implied movement around the long run equilibrium point
(p̄, ē). Define δp = p−p̄ and δe = e−ē, the deviations of prices and the exchange rate from
their long run level. Then rewrite money market equilibrium and the price adjustment
equation in deviation form. This turns (63) and (65) into (66) and (67).
−δp = λθδe (66)
Dδp = π(ρ+ σθ)δe− πρδp (67)
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You will recognize this as a linear homogeneous first order differential equation system.
One easy way to solve it is to solve (66) for δe, plug this solution into (67), and solve
the resulting differential equation in δp. The dynamics of δe will then be found by time
differentiating (66) after substituting your solution for δp.
Dδp = −π(ρ+ σθ)/λθδp− πρδp
= −π[ρ(1 + 1/λθ) + σ/λ]δp
= Aδp
(68)
Therefore
δp = δp0eAt (69)
Note that it makes sense to solve this in terms of δp0, since p is a predetermined
variable. Since A < 0, we are assured of the stability of the system.
It is also possible to attack the solution directly using the adjoint matrix technique.
First, let’s write (66) and (67) in matrix form using the differential operator.
−1 −λθD + πρ −π(ρ+ σθ)
δeδp
= 0 (70)We can solve the characteristic equation
λθD + π(ρλθ + ρ+ σθ) = 0 (71)
for the unique characteristic root
D1 = −π(ρλθ + ρ+ σθ)/λθ
= −π[ρ(1 + 1/λθ) + σ/λ]
< 0
7
-
Thus stability is assured. Recall that the adjoint matrix technique implies that (72)
is the general solution to (66) and (67).
δpδe
= k exp{D1t} λθ−1
(72)Note that this involves a single arbitrary constant, so that we cannot offer arbitrary
initial conditions for both δp and δe. We supply an initial condition for the predetermined
variable δp0, since prices cannot move instantaneously to clear the goods market. In
contrast, the exchange rate can jump to maintain constant asset market equilibrium.
3.2 Rational Expectations
In this section we replace the regressive expectations hypothesis with the rational expec-
tations hypothesis. Since there is no uncertainty in this model, the rational expectations
hypothesis implies ėe = ė. Given uncovered interest parity (i = i∗ + ė), we can then use
the money market equilibrium condition to express the rate of exchange rate depreciation
in terms of p.
h− p = φy − λ(i∗ + ė) (73)
As we have laid out the model, ė = 0 in the long run. So we know
h− p̄ = φy − λi∗ (74)
Comparing the short-run and long-run we see
p− p̄ = λė (75)
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-
which implies
ė = (p− p̄)/λ (76)
pp̄
e
ė0
ė=0
Figure 9: The ė = 0 Locus
Our basic descriptions of the goods market and of price adjustment are unchanged, so
under the rational expectation hypothesis
ṗ = π[ρ(e− p)− σ(i∗ + ė) + g − y] (77)
Equivalently, in terms of deviations from the equilibrium values (again recalling that
ė = 0 in the long run),
ṗ = π[ρ(e− ē)− ρ(p− p̄)− σė] (78)
Our solution for ė allows us to rewrite this as2
ṗ = −a(p− p̄) + b(e− ē) (79)
making it simple to graph the ṗ = 0 locus.
2Here a = πρ + πσ/λ and b = πρ, and the implied slope along the ṗ = 0 isocline is (de/dp)|ṗ=0 =a/b > 1.
9
-
p
e
ṗ>0 ṗ
-
instability.” Early discussions of this situation suggested that the convergent arm should
be selected from all the possible dynamic paths as the only economically “reasonable”
solution, and a subsequent literature provided some more sophisticated justifications for
this general procedure. Selecting the covergent arm is of course the same as setting η2 = 0
in our solution. In addition to ruling out behavior that many economists consider some-
what perverse–e.g., hyperinflation with a constant money stock–limiting our attention to
the convergent arm allows us to make predictions with the model that would otherwise be
impossible. For although the price level is predetermined, the exchange rate may poten-
tially jump to any level (each corresponding to a different η2). If we assume the exchange
rate jumps to the convergent arm, then we know that the exchange rate overshoots in
response to an unanticipate monetary shock. We also know that inflation and exchange
rate appreciation are correlated. So we get some strong potentially testable predictions.
p
e
-?sssss
�6kk
kk
k
��������������IS (ṗ=0)
ė=0
Figure 11: Rational Expectations and Overshooting
So with rational expectations, we will also see the same overshooting phenomenon.
Comment: We can readily describe the slope of the convergent arm once we note that
for any initial δp0 we can solve η1 = δp0/D1. This implies δe0 = δp0/λD1. But we can
just think of the slope of the convergent arm as δe0/δp0 = 1/λD1. (Why?)
Comment: Recall p̄h = 1. If we are in a steady state at the time of the monetary
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-
shock, p̄h = 1 implies δp0 = −dh. We can then solve for η1 = −dh/D1. This gives us
δe0 = −dh/λD1 > 0. The positive sign indicates overshooting: e has risen above its new
long run value.
Exercise 8
Answer Blanchard and Fisher pg. 557 question #5.
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4 Blanchard’s Stock Market Model
This presentation draws on Blanchard (1981).
Aggregate Demand: d = αq + βy + g with α > 0 and β ∈ [0, 1]
We are allowing q to affect consumption and investment, but we are ignoring changes
in the capital stock over time (typically for such models).
Output dynamics:
ẏ = σ(d− y) = σ(αq + g − by) (83)
where b > 0.
Money market:
i = cy − h(m− p) (84)
where c > 0; h > 0.
We impose the Fisher relation: r = i− ṗ
To keep things simple, assume all profits remitted and characterize the return to
holding equity by
q̇
q+π
q(85)
Finally, we need to characterize the determination of profits. We pick a familiar formula-
tion
π = α0 + α1y (86)
with α1 > 0. This allows us to write the real rate of return on equity as
q̇ + α0 + α1y
q(87)
Financial market arbitrage equates real rates of return:
r =q̇ + α0 + α1y
q(88)
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Model 1: ṗ = 0.
The model can be characterized by two equations of motion:
q̇ + α0 + α1y
q= cy − h(m− p)
ẏ = σ(d− y) = σ(αq + g − by)
The ẏ = 0 locus is clearly upward sloping in (q, y)-space, but the slope of the q̇ = 0
locus is ambiguous, since both r and π depend positively on y.
dq
dy
∣∣∣∣q̇=0
=α1 − cq
cy − h(m− p)(89)
Blanchard calls the downward sloping case the “bad news” case; the upward sloping
case is the “good news” case. The intuition lies entirely in the portfolio balance condition.
Assume cy−h(m−p) > 0 (i.e., that the money market eq takes place at a positve nominal
interest rate). Then we are in the good news or bad news case depending on whether the
response to income (of implied returns) is higher in the equity or the money market. We
will focus on the bad news case.
HW: The good news case.
The goods market clears slowly:
Ẏ = φ(Y, q, F ) (90)
δẎ = φY δY + φqδq (91)
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-
The money market clears at every point in time:
M = L
(q̇ + π(Y )
q, Y
)(92)
0 = Li
(δq̇
q+π′δY
q− πδq
q2
)+ LY δY (93)
Put these together to get
φq (φY −D)−πLi
q2+ LiD
qLY +
Liπ′
q
δqδY
= 0 (94)
dq
dY
∣∣∣∣q̇=0
=Liπ
′ + LyqLiπq
(95)
φq(LY + Liπ′
q) + φY
Liπ
q2− (φY
Liq
+πLiq2
)D +LiqD2 = 0 (96)
φq(qLYLi
+ π′) + φYπ
q︸ ︷︷ ︸C
−(φY +π
q)D +D2 = 0 (97)
C < 0 is necessary and sufficient condition for a saddle.
Condition for a saddle:
dq
dY
∣∣∣∣Ẏ=0
>dq
dY
∣∣∣∣q̇=0
(98)
−φYφq
>Liπ
′ + LY qLiπq
(99)
−φY πq
> +φq(π′ +
LY q
Li) (100)
i.e., π′ dosen’t add too big a positive contribution.
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-
q
Y0
��������������
Ẏ=0@@@@@@@@@@@@@@ q̇=0
HHHH
HHHH
HHHH
HHconv.arm
��rr�
HHjHHj
HHj
Figure 12: Increase in G
q
Y0
����������
@@@@@@@@@@
HHHHHHHHHHHHHH
���
ss
q̇=0Ẏ=0
CA
Figure 13: Stock Market Model: Anticipated Change
16
-
����������
@@@@@@@@@@
HHHHHHHHHHHHHH
���rrSS0
��
q̇=0Ẏ=0
CA
Y0
q
YO
17
-
Appendix
5 The Adjoint Matrix Technique
Consider the first-order linear differential equation
ẋ = Ax (101)
where A is square matrix of real constants. Suppose we can find a scalar λ with associated
vector κ such that Aκ = λκ.4 Then x = exp{λt}κ is a solution, since it implies ẋ =
exp{λt}λκ = exp{λt}Aκ = Ax. The adjoint matrix technique is just an elaboration
on this observation. This technique allows us to determine the general solution to the
homogeneous part of a system of linear differential equations.
Consider a homogeneous system of linear differential equations (not necessarily first-
order) with constant coefficients:
P (D)x(t) = 0 (102)
where P (D) is a matrix of polynomials in D, the differential operator (i.e. Dx(t) =
dx(t)/dt = ẋ(t), D2x(t) = d2x(t)/dt2, etc). We want to find the general solution for this
sytem. The following discussion motivates the ultimate result in some detail, so you may
find it useful on first reading to skip immediately to equation (107).
Let λ be a constant and let v be any nonzero n × n matrix independent of t. Note
that
P (D)eλtv ≡ eλtP (λ)v, (103)
Since P (λ) is a square matrix of constants, so is its adjoint P †(λ). Recall, P (λ)P †(λ) =
4If Aκ = λκ, we call λ an eigenvalue of A and κ is an associated eigenvector. The eigenvalues andeigenvectors characterize many of the important properties of the matrix A. We find the set of eigenvaluesby noticing (A− λI)κ = 0 only if det(A− λI) = 0, which is called the characteristic equation of A.
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|P (λ)| I.5 Then our last result implies
P (D)eλtP †(λ) = eλtP (λ)P †(λ)
= eλt|P (λ)|I(104)
Now consider the characteristic equation of our differential equation system:
| P (λ) |= 0, (105)
where | P (λ) | is an nth order polynomial in the variable, λ.
The roots of this equation are called the characteristic roots of the differential equation
system. Choosing any root λi of the characteristic equation will give us
P (D) exp{λit}P †(λi) = 0 (106)
since |P (λi)| = 0. Thus, denoting by vi an arbitrary column of P †(λi), we know that
exp{λit}vi is a solution of the homogeneous system
P (D)x = 0 (107)
We will assume that all of the characteristic roots (the λis) of our differential equation
5This is just expressing the determinant through expansion by cofactors. (Remember that an expan-sion by alien cofactors is null.) Of course when the inverse exists P †(λ) = P (λ)−1|P (λ)|. However, wewill care most about the case when the inverse does not exist.
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-
system are distinct.6 In this case, the general solution to P (D)x(t) = 0 is:
xc(t) =n∑i=1
ηieλitP †j (λi) (108)
for arbitrary constants η, where λi is the ith root of |P (λ)| and P †j (λi) is the jth column
of P †(D), the adjoint matrix of P (D), with λi in the place of D.7 If the original system
was non-homogeneous, then the general solution can now be written as
xg(t) = xp(t) +n∑i=1
ηieλitP †j (λi) (109)
and the unique definite solution can be found by solving for the arbitrary constants ηi
using appropriate boundary conditions.
xd(t) = xp(t) +n∑i=1
civi exp{λit} (110)
where vi satisfies P (λi) vi = 0.8
6In general, since repeated roots are not robust in the sense that they disappear with small changesin model parameters, we are not very interested in the case of repeated roots. However, suppose thereare k distinct roots λi(i = 1, . . . , k), each with multiplicity ωi. In a manner similar to a single equation,we have the general solution of nonhomogeneous system (??) as follows:
xg(t) = xp(t) +
k∑i=1
ωi−1∑si=0
cisivsitsi exp{λit}
where xp(t) is a particular solution of (??), cisi stands for an arbitrary scalar and vsi(si = 0, 1, . . . , ωi−1)are linearly independent column vectors of P †(λi). If λi is a multiple root with multiplicity wi, then thereexist wi linearly independent columns in P
†(λi)(at least in the cases that we will consider; see Murata3.2 for specific restrictions, esp. Th.6 for some details).
7You will generally be able to choose the columns of the adjoint matrix P †(λi) arbitrarily: due totheir linear dependence this will only change the constants η in an offsetting manner and will have noeffect on the definite solution. However, it is possible to get a zero vector in P †(λi), and this (althoughstill linearly dependent with the other columns) should obviously not be used.
8Equivalently, if we have a first order system ẋ = Ax + u, then (λiI − A)vi = 0. I.e., the vi arecharacteristic vectors of A.
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References
Blanchard, Olivier and Stanley Fischer (1989). Lectures on Macroeconomics. Cambridge,
MA: MIT Press.
Blanchard, Olivier Jean (1981, March). “Output, the Stock Market, and Interest Rates.”
American Economic Review 71(1), 132–43.
Dornbusch, Rudiger (1976, December). “Expectations and Exchange Rate Dynamics.”
Journal of Political Economy 84(6), 1161–76.
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