Models with Two Sources of Dynamics1.2 RATEX Dynamics We are going to introduce a second source of dynamics: myopic perfect foresight. We retain our previous descriptions of disequilibrium

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

1

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

2

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

3

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 ).

4

R

YO

@@@@@@@@@@@@@@@@@@@@@@@@@Ẏ=0

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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.

5

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

6

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

7

Therefore the smaller (negative) root is

D1 =φY +R−

√(φY +R)2 − 4(φYR− LY φRR/Li

2

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)

8

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.

9

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.

10

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.

11

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).

12

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)

13

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 .

14

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)

15

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

16

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

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)

18

Using our solution for η, we have

δYδE

= δY0θLi

θLiLY

eD1t (57)=

δY0LY δY0/θLi

eD1t (58)So, δY0 < 0 =⇒ δE0 > 0

19

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

1

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:

2

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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.

3

����

���

���

���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.

4

����

���

���

���

��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 .

5

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)

6

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)

8

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

11

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.

12

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)

13

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)

14

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.

15

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.

18

|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.

19

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.

20

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.

21