DEPARTAMENTODE ECONOMÍA

DEPARTAMENTO DE ECONOMÍAPONTIFICIA DE?L PERÚUNIVERSIDAD CATÓLICA

DEPARTAMENTO DE ECONOMÍAPONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA

DEPARTAMENTO DE ECONOMÍAPONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA

DEPARTAMENTO DE ECONOMÍAPONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA

DEPARTAMENTO DE ECONOMÍAPONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA

DEPARTAMENTO DE ECONOMÍAPONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA

DEPARTAMENTO DE ECONOMÍAPONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA

DEPARTAMENTO DE ECONOMÍAPONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA

DEPARTAMENTO DE ECONOMÍA

DEPARTAMENTO DE ECONOMÍAPONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA

DEPARTAMENTO DE ECONOMÍAPONTIFICIA DEL PERÚUNIVERSIDAD CATÓLICA

DOCUMENTO DE TRABAJO N° 344

IS-LM STABILITY REVISITED: SAMUELSON WAS RIGHT, MODIGLIANI WAS WRONG Waldo Mendoza Bellido

DOCUMENTO DE TRABAJO N° 344 IS-LM STABILITY REVISITED: SAMUELSON WAS RIGHT, MODIGLIANI WAS WRONG

Waldo Mendoza Bellido

Noviembre, 2012

DEPARTAMENTO DE ECONOMÍA

DOCUMENTO DE TRABAJO 344 http://www.pucp.edu.pe/departamento/economia/images/documentos/DDD344.pdf

© Departamento de Economía – Pontificia Universidad Católica del Perú, © Waldo Mendoza Bellido

Av. Universitaria 1801, Lima 32 – Perú. Teléfono: (51-1) 626-2000 anexos 4950 - 4951 Fax: (51-1) 626-2874 econo@pucp.edu.pe www.pucp.edu.pe/departamento/economia/

Encargado de la Serie: Luis García Núñez Departamento de Economía – Pontificia Universidad Católica del Perú, lgarcia@pucp.edu.pe

Waldo Mendoza Bellido IS-LM Stability Revisited: Samuelson was Right, Modigliani was Wrong Lima, Departamento de Economía, 2012 (Documento de Trabajo 344) PALABRAS CLAVE: Historia del pensamiento económico, teoría macroeconómica, modelo IS-LM.

Las opiniones y recomendaciones vertidas en estos documentos son responsabilidad de sus autores y no representan necesariamente los puntos de vista del Departamento Economía.

Hecho el Depósito Legal en la Biblioteca Nacional del Perú Nº 2012-14847. ISSN 2079-8466 (Impresa) ISSN 2079-8474 (En línea) Impreso en Cartolán Editora y Comercializadora E.I.R.L. Pasaje Atlántida 113, Lima 1, Perú. Tiraje: 100 ejemplares

IS-LM STABILITY REVISITED: SAMUELSON WAS RIGHT, MODIGLIANI WAS WRONG

Waldo Mendoza Bellido

RESUMEN

En el modelo IS-LM de Hicks, cuando la propensión marginal a gastar es mayor que uno, el multiplicador keynesiano es negativo y la IS tiene pendiente negativa. Keynes declaró este caso como completamente inestable. La literatura posterior, desde Modigliani (1944) hasta nuestros días, incluyendo a Varian (1977) y Sargent (1987), determinó que en este caso especial el modelo IS-LM es estable cuando la pendiente de la LM es mayor que la de la IS. En línea con Samuelson (1941), este artículo muestra que en este caso especial el modelo es estable cuando la pendiente de la IS es mayor que la de la LM. Sin embargo, en este caso estable, el modelo no tiene un significado económico útil. Primero, los valores de equilibrio de la producción y la tasa de interés son negativos. Segundo, dado que el multiplicador keynesiano es negativo, una expansión de la demanda reduce la producción. Palabras clave: Historia del pensamiento económico, teoría macroeconómica, modelo IS-LM. Códigos JEL: E11, B22.

ABSTRACT In Hicks’s IS-LM model, when the marginal propensity to spend is greater than one, the Keynesian multiplier is negative and the IS has a positive slope. Keynes deemed this case as totally unstable. Further literature, from Modigliani (1944) through our days, including Varian (1977) and Sargent (1987), determined that in this special case the IS-LM model is stable when the LM slope is greater than the IS. In line with Samuelson (1941), this article shows that in this case the model is stable when the IS slope is greater than the LM slope. However, in this stable case, the model does not carry a useful economic meaning. Firstly, the equilibrium values of production and interest rates are negative. Secondly, since the Keynesian multiplier is negative, an expansion of demand reduces output. Keywords: Economic thinking, macroeconomic theory, IS-LM model. JEL codes: E11, B22.

1

IS-LM STABILITY REVISITED: SAMUELSON WAS RIGHT, MODIGLIANI WAS WRONG

Waldo Mendoza Bellido1 1. INTRODUCTION

In the traditional IS-LM model created by Hicks (1937), when the marginal

propensity to spend is greater than one, the Keynesian multiplier is negative

and the IS has a positive slope. Hicks (1937) warned that in this special

case, only under certain conditions the IS-LM model could be stable, and

Keynes, in a letter published in Gilboy (1939), pronounced this occurrence

“completely unstable”.

Later developments -starting with Modigliani (1944) and including articles in

specialized journals, as well as in macroeconomics and mathematics for

economics textbooks- agree that, in this atypical case, the IS-LM model is

stable when the LM slopes more steeply than IS.

This article shows that:

i) The case the literature pronounces as stable is unstable. The model is

stable when the IS is steeper than the LM. This result is consistent with

the views expressed by Samuelson (1941).

ii) However, in this special stable case, the model’s linear version provides

results without of no useful economic significance. Firstly, the

equilibrium values of production and the interest rate are negative.

Secondly, since the Keynesian multiplier is negative, expanding demand

results in shrinking output.

1 Professor and researcher, Department of Economics, Pontificia Universidad

Católica del Perú (PUCP), 1801 Universitaria Ave., Lima 32, Lima. Telephone: +511-6262000 (4950). E-mail adress: waldo.mendoza@pucp.pe I wish to thank the comments by Juan Antonio Morales and Oscar Dancourt, which helped me to improve a former version. I am sole responsible for any remaining errors.

2

The following section reviews the literature on the special case. Section 3

discusses the stability conditions of the standard IS-LM model. Section 4

repeats the exercise for the special case and argues why the traditional

treatment is incorrect. Section 5 proposes a complementary argument to

show the instability of the case the literature pronounces as stable, by

simulating the dynamic effects of an expansive monetary policy. Section 6

shows that in the linear version of the stable model, the equilibrium values of

the interest rate and output are negative, and that the model forecasts a

reduction in output as a consequence of expanding demand. Finally, the

conclusions appear in section 7.

2. BACKGROUND

In the IS-LM model, when the marginal propensity to spend (propensity to

consume plus propensity to invest) is greater than one, the Keynesian

multiplier is negative and the IS curve has a positive slope.

In Hicks’s presentation (1937), equilibrium in a goods market requires ex

ante equality between saving )(S and investment )(I 2. Thus,

),(),(+++−

= YiSYiI

The slope of the IS curve, graphed on the ),( iY space, is given by:

ii

YY

ii

YY

IS IS

IC

IS

IS

dY

di

−−−

−=−−

−=1

Where xYYx ∂∂= / is the generic form representing the partial derivative of Y

with respect to X.

The denominator is undoubtedly positive, while the numerator can be

positive, in the typical case where the propensity to spend is lesser than one

)1( <+ YY IC , or negative, when the marginal propensity to spend is greater

2 Hicks’s text has been adapted to current terminology.

3

than one ).1( >+ YY IC Both values are possible, according to Hicks (1937),

because the propensity to invest is pro cyclical. The model can still be stable

under certain conditions:

“If there is a great deal of unemployment, it is very likely that IC ∂∂ /

will be quite small; in that case IS can be relied upon to slope

downwards. This is the sort of Slump Economics with which Mr.

Keynes is largely concerned. But one cannot escape the impression

that there may be other conditions when expectations are tinder, when

a slight inflationary tendency lights them up very easily. Then IC ∂∂ /

may be large and an increase in income tends to raise the investment

rate of interest. In these circumstances, the situation is unstable at

any given money rate; it is only an imperfectly elastic monetary

system-a rising LL curve- that can prevent the situation getting out of

hand altogether” (Hicks, 1937, p. 158) (Hicks’s LL is the current LM

and IC ∂∂ / is the marginal propensity to invest.)

One of the innovations in Keynes’s General Theory was the introduction of

the concepts of marginal propensity to consume and the multiplier. In a

letter published in Gilboy (1939), Keynes warned that if the marginal

propensity to consume is greater than one, the model would be unstable:

“My theory itself does not require my so-called psychological law as a

premise. What the theory shows is that if the psychological law is not

fulfilled, then we have a condition of complete instability. If, when

incomes increase, expenditure increases by more than the whole of

the increase in income, there is no point of equilibrium.” (Gilboy,

1939, p. 634).

All the subsequent literature dealing explicitly with this special case (negative

Keynesian multiplier and positively sloping IS) —that first appeared in

specialized journals and then was adopted by macroeconomics and

math

to be

Modi

and L

That

slope

This

point

wher

the L

unsta

hematics f

e stable, th

gliani (194

LM slopes

“Stabilit

of the e

concave

is, for the

e must be

argument

ts A and B

re the IS

LM slope is

able.

for econom

he LM curv

44) was th

. For the c

y is also p

equilibrium

e toward it

e model to

greater th

t appears

.B For Mod

cuts the L

s greater

mists textb

ve must b

he first in

case in wh

possible w

m points

ts convex s

o be stable

han that o

in Figure

digliani, th

LM from it

than that

4

books— ha

e steeper

treating t

ich IS slop

when the I

as long a

side” (Mod

e in the ne

of the IS.

1. The L

he equilibr

ts concave

of the IS.

as establis

than the I

this specia

pes positiv

S curve ri

as it cuts

digliani 19

eighborhoo

LM and IS

rium point

e to its co

. At point

shed that

IS curve.

al case in t

vely, he st

ises in the

s the LL c

944, p. 64)

od of equi

intercept

t would b

nvex side

,A equilib

for the m

terms of t

tates that:

e neighbor

curve from

).

librium, th

t each oth

be stable a

. At this p

brium wou

model

he IS

rhood

m its

he LM

her in

at ,B

point,

uld be

Later

spec

Since

stret

mult

Acco

great

Later

reach

(194

r, Hudson

ial case a

e Hudson

tch and a

iple equilib

“Stability

upwards

position o

1957, p.

rding to H

ter than th

r, Dernbu

hed the s

44) or to H

(1957) p

and also e

’s IS is n

nother wi

brium, like

of equil

less stee

of unstabl

382).

Hudson the

hat of the

rg and D

same inde

udson’s w

resented t

erroneousl

non-linear

ith positiv

e the one

ibrium th

eply than

e equilibr

en, the at

IS.

Dernburg (

pendent c

work (1957

5

the IS-LM

ly conside

r and sinc

ve slope,

shown in

en requir

the LM

rium, while

ypical mo

(1969), S

conclusion

7).

model’s s

ered the u

ce it show

Hudson’s

Figure 2. I

res that t

schedule.

e C and

del is stab

Smith (197

n without

stability co

unstable c

ws a neg

IS-LM m

In this mo

the IS sc

Consequ

´C are sta

ble when t

70), and

referring

ondition fo

case as st

atively slo

odel is on

odel:

chedule s

ently, B

able.” (Hud

the LM slo

Varian (1

to Modigl

or the

table.

oping

ne of

lopes

is a

dson,

ope is

1977)

iani’s

Varia

arriv

mult

IS.

Figur

Huds

inter

LM,

slope

point

show

Subs

main

show

Burro

(200

The

teach

an (1977)

es at the

iple equili

re 3 —wh

son, that

rsection po

as in poin

e is great

t A, or wh

wn by the d

sequent pu

ntain witho

wn in work

ows (197

4).

special ca

hing of ma

) builds, a

e same re

bria, the s

hich reprod

the region

oints betw

nt B of Fi

er than th

en the IS

direction a

ublications

out alterat

ks by Silve

4), Cebul

ase and i

acroecono

as Hudso

esult: whe

stable regi

duces Var

n is unsta

ween the I

gure 3. T

hat of the

slope is p

arrows of t

s in specia

tion these

er (1971),

a (1980),

its stabilit

mics in bo

6

n, a mod

en the IS

ion is that

rian’s Figu

able (i.e.

S and LM

The interse

e IS (whe

ositive, bu

the phase

alized jour

authors’

Chang an

, Wang (

ty conditio

oth old (Sm

del with m

S slopes p

t where th

ure 3 (197

configures

, when th

ection poi

n the IS

ut less tha

diagram.

rnals deali

proclaime

nd Smyth

1980), Pa

ons are a

mith, 197

multiple e

positively

e LM is st

77, p. 268

s a saddle

e IS is ste

nts are st

slopes ne

an the LM,

ng with th

d stability

(1972), P

atinkin (1

also incorp

0, p. 313;

quilibrium

and there

teeper tha

8)— show

e point) a

eeper tha

table if th

egatively,

at point C

his special

y condition

Puckett (19

1987) and

porated to

; Dernburg

m and

e are

n the

ws, as

at the

n the

he LM

as in

C), as

case

ns, as

973),

d Ros

o the

g and

7

McDougall, 1976, pp. 244-247; Branson, 1972, pp. 222-223), and recent

text books (Sargent, 1987, p. 59; McCafferty, 1990, p. 149.; Jiménez, 2006,

pp. 432-433).

When referring to the special case, Sargent states that:

“This condition is automatically satisfied when the LM curve is upward

sloping and the IS curve is downward sloping. It can still be satisfied

if the IS curve is upward sloping, provided that the LM curve is more

steeply sloped” (Sargent, 1987, p. 59).

The same argument appears in text books on mathematics for economics,

including Ferguson and Lim (1998, p. 129), Shone (2002, p. 449), and

Zhang (2005, p. 243).

In short, all the literature starting with Modigliani (1944) states that in the

case in which the IS has a positive slope, for the IS-LM to be stable, the LM

curve must be steeper than the IS curve.

The following sections show that the IS-LM model is stable in the special case

when the IS slope is greater than the LM slope.

3. IS-LM STABILITY: THE STANDARD CASE

This section presents Hicks’s model (1937), in its dynamic version3. In the

goods market, it is assumed that adjustment is by quantities and that output

increases )0(0

>Y when there is an excess of demand in that market )(EDB . In

the money market, the adjustment variable is the interest rate, which rises

)0(0

>i when there is excess demand )(EDM .

0);(0

>= εε EDBY (1)

3 See Gandolfo (1996), Ferguson and Lim (1998) and Zhang (2005).

8

0);(0

>= ηη EDMi (2)

Where ε and η are, respectively, the production adjustment velocities and

the interest rate vis-à-vis excess demand in the goods market and the money

market.

If Y represents output and dY stands for the demand for goods, in a closed

economy without government, the demand for goods comes from

consumption and private investment. Consumption and investment are

positive functions of income and negative functions of the interest rate.

Consequently, demand for goods is given by:

),(),(+−+−

+= YiIYiCY d ; 10 << YC (3)

Excess demand in the goods market therefore equals:

),(−−

=−= iYEDBYYEDB d ; 0;01 <+=<−+= iiiYYY ICEDBICEDB (4)

In the money market, if s

M is the money supply and P is the price level, the

real money supply equals:

P

Mm

ss = (5)

If dm is the real demand for money, which is directly related to output and

inversely related to the interest rate:

),(−+

= iYmm dd (6)

9

Consequently, the excess demand in a money market equals:

),(−+

=−= iYEDMmmEDM sd ; .0;0 <=>= dii

dYY mEDMmEDM (7)

Replacing equations (7) and (4) in equations (1) and (2), respectively, we

obtain a differential equation system to discuss the stability conditions of the

traditional IS-LM model.

0;),(0

>

=

−−εε iYEDBY (8)

0;),(0

>

=

−+ηη iYEDMi

(9)

Where [ ].ε and [ ].η are rising monotone functions which are differentiable and

satisfy the [ ] [ ] 000 == ηε condition.

To evaluate if a model is stable or not, it is worthwhile taking into account

Gandolfo’s warning (1996) about the appropriate formulation of adjustment

dynamics in markets:

“The dynamic formalization of the Walrasian assumption is the following

[ ])()(0

pSpDfp −=

Where [ ] [ ],...sgn...sgn =f [ ] 00 =f , 0)0(' >f

The notation [ ] [ ]...sgn...sgn =f means that f is a sign-preserving function, i.e.,

the dependent variable has the same sign as the independent variable (which

in this case is excess demand): therefore, if excess demand is positive

(negative) the time derivative of p is positive (negative), i.e. p is increasing

(decreasing)” (Gandolfo, 1996, p.172).

10

Which, when applied to the IS, implies:

“ [ ]),(),(1

0iYSiYIY −= ϕ , [ ] [ ]...sgn...sgn 1 =ϕ ” (Gandolfo, 1996, p. 328)4.

Taking the case shown by Gandolfo as an example, since 01 >ϕ , the influence

of the investment-saving )( SI − gap on production, which equals the effect of

the excess demand, also must be positive. In our presentation, since 0>ε , the

influence of the good excess demand on the product must be positive.

The system made up by equations (8) and (9) is non-linear and its study

implies analytical difficulties, which we will avoid. To this purpose, we will

discuss some valid properties in a local equilibrium context, through Taylor’s

expansion.

If eY and ei represent stationary (local) equilibrium values of production and

the interest rate in the IS-LM model, equations (8) and (9) can be presented

in a matrix as follows:

−−

=

e

e

iY

iY

ii

YY

EDMEDM

EDBEDB

i

Y

ηηεε

0

0

Or its equivalent,

−−

+−+=

e

e

di

dY

iiYY

ii

YY

mm

ICIC

i

Yηη

εε )()1(0

0

(I)

Or in its compact form:

e

ΥΩ=Υ0

4 We have adapted Gandolfo’s terminology to ours.

11

Where:

+−+=Ω d

idY

iiYY

mm

ICIC

ηηεε )()1(

The necessary and sufficient conditions for this system to be asymptotically

stable, that is a system where all the movements flow cyclically or non-

cyclically towards a stationary equilibrium point, are that the Ω Jacobian

matrix trace be negative and its determinant positive.

i) 0<ΩTr

ii) 0>ΩDet

In this version of the IS-LM, the assumption that the propensity to spend is

less than one )1( <+ YY IC ensures that the two stability conditions are met

without restrictions.

i) ;0)1( <+−+=Ω diYY mICTr ηε

ii) [ ] 0)()1( >++−−−=Ω iidY

diYY ICmmICDet εη

Stability can also be discussed from the qualitative point of view, by relating

the described conditions to the graphic representation of the IS-LM model.

To this purpose, we use a phase diagram to represent the goods market and

the money market in stationary equilibrium, that is, when output and the

interest rate reach equilibrium and stay constant ).0(00

== iY

Stationary equilibrium in the goods market is reached when output equals

demand (excess demand is null), and output remains stable

)0(0

==−= EDBYYY d . From equation (8), the IS equilibrium equation is

obtained in the goods market when 00

=Y .

),(),( YiIYiCYY d +== (10)

In th

prop

Keyn

Whe

Let u

in Fig

unde

inves

exce

incre

point

supp

point

he ),( iY s

ensity to

nesian mul

=IS kdY

di

re 1−

=C

k

us assume

gure 4, w

er the IS

stment, w

ss deman

eases outp

t M . The

ply reduces

t.

pace, the

spend is a

ltiplier.

)(1 <+ ii ICk

01 >− YY IC

e that the

here outp

S curve,

which gene

d in this m

put. This is

same rea

s output, a

IS curve

assumed

0<

0 is the Ke

interest r

put equals

a lower

erates an

model, wh

s the mean

asoning ap

as shown

12

has the u

to be less

eynesian m

ate slides

demand.

interest r

excess de

hose dynam

ning of the

pplies to a

by the left

usual nega

s than one

multiplier.

starting f

At this po

rate incre

emand in

mics is ex

e arrows p

a point su

t-pointing

ative slope

e, resultin

from any p

oint, let u

eases con

the good

xpressed in

pointing to

uch as N ,

arrow tha

e, becaus

ng in a po

point on t

s say poin

nsumption

ds market

n equation

o the right

, where ex

at starts at

e the

sitive

he IS

nt M

and

. The

n (8),

from

xcess

t that

The

seen

5. Th

prod

equa

smal

Curv

Figur

upwa

exce

and p

Stati

the d

rate

mark

mechanis

more clea

he 45º cu

uction )(Y

ation (10)

ler than o

ve D has th

re 5 show

ards to 1D

ss deman

production

onary equ

demand fo

(0

−= mmi d

ket, the LM

= mP

M s

m allowin

arly in the

urve show

) equals d

with a

on, slope.

he interest

ws that if

.1 Excess

d increase

n raises to

uilibrium i

or money,

= EDMms

M, when 0i

),(+−Yid

ng for an

e classic 45

s the equ

demand (

positive i

This refle

t rate as p

the intere

demand a

es output.

o .1Y

s reached

i.e. exces

)0= . We r

0= in equ

13

excess d

5º diagram

uilibrium p

).( dY Curv

ndepende

ects a pro

parameter

est rate fa

appears at

The equil

in the m

ss demand

reach the

uation (9).

demand to

m such as

points in t

ve D is a

ent compo

opensity to

.

alls from

t the initia

librium po

money mar

d is null lea

equilibrium

.

o increase

the one s

the goods

linear rep

onent and

o spend lo

oi to ,1i

al output

int moves

rket when

ading to a

m equatio

e producti

shown in F

market,

presentatio

d positive

ower than

curve D

level 0Y .

s from A t

supply e

stable int

n in the m

(1

on is

Figure

when

on of

, but

one.

shifts

Such

to ,B

quals

terest

money

1)

Let u

Figur

point

large

of th

the i

from

there

show

In a

stabl

asym

statio

The LM

=LMdY

di

us now ass

re 6, whe

t such as

er, creatin

he money

nterest ra

point M

e is an ex

wn by the d

general e

le, and d

mptotically

onary equ

curve slop

0>−di

dY

m

m

sume that

re money

M , to the

g excess

market re

ate upward

. Symme

xcess of m

downward

equilibrium

directional

y stable

ilibrium.

pes positiv

output in

demand

e right of t

of demand

epresented

ds. This ex

etrically, to

money sup

ds pointing

m conditio

l arrows

equilibrium

14

vely.

creases st

equals su

the LM wh

d in that

d by equa

xplains th

o the left

pply, henc

g arrow sta

n we can

show th

m. At po

tarting fro

pply. Larg

here the de

market. If

tion (9), e

e upward

of the LM

e the inte

arting at t

see the s

at we ar

oint ,( 00 iY

m any poi

ger output

emand for

f we know

excess de

pointing a

M at a po

erest rate

hat point.

standard I

re in the

)0 the m

int of the

t takes us

r money is

w the dyna

mand will

arrows sta

oint such a

has to fa

IS-LM mod

e presenc

model rea

LM in

s to a

s also

amics

shift

arting

as N ,

all, as

del is

ce of

aches

Final

posit

Ther

more

Start

these

secti

4.

When

Keyn

ly, since

tive determ

di

dY

km

m>−

efore, the

e steeply t

ting with M

e same sta

on shows

IS-LM S

n the pr

nesian mul

=IS kdY

di

the secon

minant, is

)(1

ii ICk +

e standard

than the IS

Modigliani

ability con

that this a

STABILIT

ropensity

ltiplier is n

)(1 >+ ii ICk

nd conditi

equivalen

d IS-LM m

S.

(1944), th

nditions tha

approach is

TY: THE SP

to spend

negative (k

0>

15

on for sta

t to

model is s

he literatur

at apply to

s incorrect

PECIAL C

d is grea

)0<k and t

ability, tha

stable whe

re has exte

o the stan

t.

ASE

ater than

the IS has

at corresp

en the LM

ended to t

ndard case

one ( YC

s positive s

ponding to

M curve s

the special

e. The follo

)1>+ YY I ,

slope.

o the

lopes

l case

owing

, the

16

If the IS and the LM have positive slopes, we have to determine which slope

is greater to reach stability.

Our review of the literature found that stability conditions in the special case

are reached starting from the same equation system (8) and (9). As a result,

the same necessary and sufficient conditions for stability are reached as in

the standard case. So, even if 1>+ YY IC , the requirement is the same as in

the traditional model: the LM must slope more steeply than the IS:

)(1

iidi

dY

ICkm

m

+>− .

This procedure is not correct. The error stems from considering that the

dynamics of the goods market, as represented in equation (8), still holds for

the special case5.

0;),(0

>

=

−−εε iYEDBY (8)

What does equation (8) tell us? That excess demand in the goods market

increases output. Therefore, parameterε is positive.

If the IS has a positive slope, that is when 1>+ YY IC , is it still true that

excess demand in the goods market will increase output?

In the traditional case, excess demand in the goods market disappears when

output rises, as represented in equation (8).

Let us assume a fall in the interest rate. A falling interest rate increases

investment and consumption, and generates excess demand in the goods

market. Excess demand increases output, which, in turn, through its effect 5 Besides, if condition i) holds in the special case, additional restrictions for

stability are needed because the sign for ΩTr is undetermined. In our review of the literature we found no trace that this issue has been addressed.

17

on private expenditure, increases demand again. Since demand increases

less than output― because the marginal propensity to spend is less than

one―the excess demand shrinks. In the movement towards a stationary

equilibrium, the excess demand continues to fall until reaching zero, thus

reestablishing equilibrium in the goods market.

For a positively sloped IS, the dynamics of adjustment in the goods market is

different. Therefore, the differential equation reflecting the adjustment

dynamics in this market must be reformulated to apply the usual stability

conditions. Omission of this aspect has led to the incorrect treatment of

stability conditions in the atypical IS-LM model.

Let us assume, as previously, the interest rate falls, investment and

consumption rise, and excess demand is created in the goods market. What

would be the adjustment mechanism required to restore equilibrium? What

must happen to output so that such excess demand will be canceled?

If output increases as a response to excess demand, given that the

propensity to spend is greater than one, the demand would increase more

than production, thus the excess demand in the goods market, rather than

reduced, would rise. The excess demand would continue to grow indefinitely,

preventing the system to reach steady state. This dynamic is clearly

explosive.

For the process to be convergent output must contract rather than increase

as it is erroneously presumed. As output shrinks, demand for goods falls

more than output because the propensity to spend is greater than one, thus

reducing excess demand. That is, in this special case, excess demand in the

goods market reduces output. Or, likewise, if the Keynesian multiplier is

negative, an increase in demand reduces, and does not increase, output.

The adjustment dynamics equation in the goods market must reproduce the

mechanism we just described. In the goods market’s differential equation,

output increases should be related to excess supply in the goods market

18

)( dYYEOB −= , not to excess demand. The equation that appropriately

reflects the new adjustment dynamics of the goods market can be written

as:

0);(0

>= εε EOBY (12)

Excess supply in the goods market is defined by:

),(+−

=−= iYEOBYYEOB d ;

0)(;01 >+−=<−−= iiiYYY ICEOBICEOB (13)

In this way, the new adjustment dynamics in the goods market, which

replaces equation (8), is now written as:

;0;),(0

>

=

+−εε iYEOBY (14)

This formulation is consistent with Gandolfo’s mathematical requirement

(1996, p. 328) noted above.

Since there is no change in the money market, the new differential equation

system in the atypical case is comprised of:

;0;),(0

>

=

+−εε iYEOBY (14)

0;),(0

>

=

−+ηη iYEDMi (9)

As before, linearizing the equation system (14) and (9) using Taylor’s

expansion yields the following:

−−

+−−−=

e

e

di

dY

iiYY

ii

YY

mm

ICIC

i

Yηη

εε )()1(0

0

(II)

19

Or in its compact form:

e

ΥΩ=Υ 1

0

Where:

+−−−=Ω d

idY

iiYY

mm

ICIC

ηηεε )()1(

1

The necessary and sufficient conditions for this differential equation system

to be stable are, as before, for the matrix 1Ω trace to be negative and its

determinant positive.

iii) ;0)1(1 <+−−=Ω diYY mICTr ηε

iv) [ ] 0)()1(1 >++−−=Ω iidY

diYY ICmmICDet εη

The first condition is met with no restrictions. To meet the second condition,

we must have

di

dY

ii m

m

ICk−>

+ )(1

,

Otherwise said, when the propensity to spend is greater than one and the

Keynesian multiplier is negative, for IS-LM to be stable, the IS must be

steeper than the LM:

LMIS dY

di

dY

di >

We reach the same conclusion using the phase diagram system. Since no

change has occurred in the money market, let us focus our attention on the

positively sloping IS.

Let u

Figur

the

exce

mark

expre

erron

point

supp

Figur

Figur

is th

slope

one.

the v

below

Let u

the i

us assume

re 8, wher

lower int

ss deman

ket, in th

essed in e

neously as

t M . By

ply that in

re 8.

re 9 replic

e linear re

e is greate

In these

value of o

w.

us simulat

nterest ra

e a fall in

re output

erest rate

nd in the

he framew

equation (1

ssumed. T

the same

creases o

cates the 4

epresenta

er than o

condition

output is

e, as befo

ate falls, c

n interest

equals an

e increase

goods m

work of

14)―contr

This expla

e reasonin

output, as

45º diagra

tion of eq

ne becaus

s, equilibr

negative.

ore, the int

curve D mo

20

rates sta

nd demand

es investm

arket. Su

this mod

racts, and

ains the a

ng, at a p

symboliz

am for the

quation (1

se the pro

rium can o

We will

terest rate

oves upwa

arting at s

d. At point

ment and

uch excess

del―the d

does not

arrow to t

point such

ed by the

e atypical c

0). The d

opensity t

only occur

return to

e falls from

ards to .1D

some poin

t ,M unde

consump

s demand

dynamics

increase,

the left th

h as N t

e right-poi

case. As b

ifference

to spend i

r in the q

this poin

m 0i to .1i

At the in

nt of the

r the IS c

ption, cre

d in the g

of which

output, as

hat starts

here is ex

inting arro

before, cur

is now th

s greater

uadrant w

nt in sect

In Figure

itial produ

IS in

curve,

eating

goods

h are

s it is

from

xcess

ow in

rve D

at its

than

where

ion 5

e 9, if

uction

level

dema

from

In g

diagr

than

6

,0Y ther

and reduc

A to ,B

eneral eq

ram, reve

the LM6.

In section equilibrium

re is an

ces output

and outpu

quilibrium,

ealing that

5 we will

m values fo

excess de

t, it does

ut falls toY

Figure 1

t the IS-L

show that r output an

21

emand in

not increa

.1Y

10 combin

LM model

in the specnd the inter

the good

ase it. The

nes figure

is stable

cial case ofrest rate ar

ds marke

e equilibriu

es 1 and

when the

f a linear ISre negative

t. This ex

um point

2 in a s

e IS is ste

S-LM mode.

xcess

shifts

single

eeper

el, the

In Fi

IS, h

gure 11 th

he model is

he phase

s unstable

diagram s

e, of the sa

22

shows that

addle poin

t when th

nt type.

e LM is steeper tha

n the

23

5. STABILITY, INSTABILITY AND EXPANSIVE MONETARY POLICY

An additional argument in discussing instability conditions is provided by

changing an exogenous variable and evaluating how the endogenous

variables will adjust. In stable models, after changing an exogenous variable,

the endogenous variables temporarily depart from their initial equilibrium

values, but then they change until they reach a new stationary equilibrium.

When models are unstable, endogenous variables move away from the initial

equilibrium and never reach a new stationary equilibrium.

In Dernburg and McDougall (1976) the effects of an expansive monetary

policy are illustrated within the context of a positively sloped IS. Their

presentation -which assumes stability when the LM is steeper than the IS - is

useful to evidence the error of considering that, in this model, excess

demand in the goods market leads to increased output.

Dernburg and McDougall (1976) repeat the exercise with a diagram as the

one showed in Figure 12, assuming that the money market is always in

equilibrium and the goods market may be temporarily in disequilibrium.

In Figure 12, an expansive monetary policy (an increase in sM ), shifts the LM

to the right. According to Dernburg and McDougall:

“The shift in the LM curve causes the interest rate to fall immediately

to 10i . This again means that intented investment exceeds saving and

that income must therefore rise. But the rise in income stimulates

further investment because of our assumption that investment is a

function of the level of profits and income. Consequently, the original

monetary disturbance causes income to rise; this cause additional

investment to be induced, and this, in turn, causes income to rise still

further.

Figur

arrow

What

falls

atypi

Ther

outp

Will inco

be foun

rate to

income

interest

than the

Consequ

1i . The

LM curv

re 12 show

ws highligh

t is the m

to point

ical IS-LM

efore, sin

ut should

ome contin

d? In the

rise and t

stimulate

that keep

e rate of in

uently, a n

path of ad

e” (Dernb

ws the abo

ht their pr

mistake in

B , the

model ex

ce in poin

fall and th

nue to rise

present c

to dampen

es further

ps the mon

nterest th

new stable

djustment

urg and M

ove mentio

roposed ad

this reaso

resulting

xcess dem

nt B ther

he arrows,

24

e indefinite

case the r

n investm

investme

ney marke

at keeps t

e equilibriu

again foll

McDougall

oned auth

djustment

oning? Po

excess de

mand reduc

re is exce

, the blue

ely, or will

rise in inc

ent more

ent. In ot

et in equili

the produc

um point w

lows the a

1976, p. 2

hors’ graph

dynamics

siting that

emand in

ces output

ss deman

ones, sho

l a new eq

ome caus

rapidly th

ther word

brium rise

ct market

will be rea

arrows upw

244)

h and argu

s.

t when th

creases o

t, it does n

nd in the

ould point

quilibrium

es the int

han the ri

ds the ra

es more ra

in equilib

ached at Y

ward alon

ument. Th

he interest

output. In

not increa

goods ma

to the left

point

terest

ise in

te of

apidly

rium.

1Y and

g the

e red

t rate

n this

ase it.

arket,

t, not

to th

a sta

Figur

befor

term

same

dema

mark

rate

along

statio

The

incre

cons

effec

prop

he right. T

ationary eq

re 13 sho

re, the ex

m equilibriu

e producti

and in the

ket. As ou

slips as w

g the new

onary equ

fact that

easing the

istent wit

cts of an

ensity to s

herefore,

quilibrium.

ows the st

xpansive m

um is reac

ion level.

e goods m

utput falls

well. Dow

w LM cu

ilibrium is

t an exp

e interest

h Samuel

expansive

spend grea

the mode

.

table case

monetary

ched at p

As in B t

market, ou

, the dem

wnwards a

rve until

s reached.

ansive m

rate as

son’s rem

e monetar

ater than

25

l is unstab

e, when th

policy shif

oint B , w

the lower

tput must

mand for m

djustment

it reache

monetary p

in the un

mark seven

ry policy i

one, he es

ble and the

he IS is s

fts the LM

with a low

interest r

t fall to re

money als

t continue

es ( 1Y , 1i ).

policy sho

nstable ca

n decades

n differen

stablished

e system

steeper th

M to the ri

er interes

rate has c

store equ

so falls, an

es (followin

. At this

ould redu

se shown

s ago. By

nt scenario

that:

does not r

an the LM

ight, and

st rate and

created ex

ilibrium in

nd the int

ng the arr

point, a

uce instea

Figure 1

simulating

os, includ

reach

M. As

short

d the

xcess

n that

terest

rows)

new

ad of

11, is

g the

ing a

26

“..the only theorem which remains true under all circumstances is that

an increase in the amount of money must lower interest rates if

equilibrium is stable” (Samuelson, 1941, p. 120).

If the LM were steeper than the IS, as in Figure 12―which reproduces

Dernburg’s and MacDougall’s arguments (1976)―an expansive monetary

policy would increase the interest rate.

6. STABILITY, NEGATIVE KEYNESIAN MULTIPLIER AND THE RELEVANCE OF THE ATYPICAL CASE

We have shown that in the atypical case, IS-LM model is stable when the IS

is steeper than the LM. In this section we will show that in this special stable

case, the model yields results of no useful economic significance.

Firstly, in the model’s linear version, output and interest rate equilibrium

values are negative. Secondly, since the Keynesian multiplier is negative,

expanding demand results in falling output.

We have adopted the linear version of the IS-LM model to illustrate how to

obtain analytical equilibrium values for output and the interest rate.

Equilibrium in the goods and money markets is given by:

iaYaIiaYaCYY d320100 −++−+== (15)

ibYbmmPM dss10 −===− (16)

Where ooAIC =+ 00 is autonomous private expenditure, sM is the nominal

money supply, P is the price level and all the parameters are positive.

The IS and LM equations are drawn from these formulae, respectively:

27

Yaakaa

Ai o

)(1

3131 +−

+= (17)

Yb

b

b

PMi o

s

11

+−−= (18)

Where 01

1

20

<−−

=aa

k is the Keynesian multiplier.

Equilibrium in the goods market can also be expressed in such a way that

the consequences of having a negative Keynesian multiplier will be more

clearly apparent. In this case, as shown in equation (19), greater demand―

prompted by rising autonomous expenditure or a sliding interest rate―leads

to smaller output.

[ ]iaaAkY o )( 31 +−= (19)

The IS and LM slopes are given by:

0)(

1

31

>+

−=aakdY

di

IS

; 01

>=b

b

dY

di o

LM

To meet the second stability7 condition, the IS must be steeper than the LM:

1

0

31 )(1

b

b

aak>

+−

That is,

0)( 0311 >++ baakb

7 The trace condition is met without restriction.

Furth

rate

Since

equil

whic

Figur

large

reach

This

shoc

hermore,

are obtain

1bY e

+=

1bie

+=

e due to

librium va

h makes n

re 14, whe

er than th

hed in the

IS-LM m

ks produc

the statio

ned from e

)( 31

1

aak

kb

++

)( 31

0

baak

kb

+

o the sta

alues of o

no econom

ere equati

he LM slop

quadrant

odel conf

e analytica

onary equi

equations

10

0 bA

b ++

10

0 bA

b +−

ability con

output and

mic sense.

ons (17) a

pe, reprod

t where int

figuration

ally strang

28

ilibrium va

(17) and (

)()(

31

31

aak

aak

+++

)(1

31 baak +

ndition 1b

d the inte

and (18) a

duces this

terest rate

leads us

ge outcom

alues for

(18).

(0

PMb

s −

)(0

PMb

s −

( 311 ++ aak

erest rate

are repres

s result. S

e and prod

to a situ

mes.

output an

)P

)

0) 03 >b ,

are evide

sented, wi

Stationary

duct values

uation whe

nd the int

(20)

(21)

the statio

ently nega

th the IS

equilibriu

s are nega

ere exoge

terest

onary

ative,

slope

um is

ative.

enous

For e

inves

gove

This

mark

mone

In Fi

upwa

shifte

7.

This

the I

chall

stabi

example,

stment o

ernment e

is becaus

ket, reduc

ey market

gure 15, i

ards. In t

ed, both o

CONCLU

paper sho

IS-LM mo

enges the

ility is reac

in this m

r, if we

expenditur

se greater

cing outpu

t, bringing

n view of

the new p

output and

USIONS

ows that i

odel to be

e literature

ched when

model, an

introduce

es, would

expenditu

ut. Falling

the intere

a larger a

point of i

d the intere

in the spe

e stable, t

e that star

n the LM is

29

autonomo

e govern

d lead to

ure genera

output re

est rate do

autonomou

ntersectio

est rate ar

ecial case

the IS mu

rting with

s steeper t

ous expan

ment in

falling ou

ates exce

educes de

own.

us expend

n with cu

re smaller

where IS

ust be ste

Modigliani

than the I

nsion of c

the mod

tput and

ss deman

emand for

iture, the

urve LM,

.

is positiv

eeper than

i (1944) e

S.

consumptio

del, expan

interest r

d in the g

money in

IS curve

which has

vely sloped

n the LM.

established

on or

nding

rates.

goods

n the

shifts

s not

d, for

This

d that

30

Only Samuelson (1941), by simulating the effects of an expansive monetary

policy in different scenarios, including the case of a propensity to spend

greater than one, proposes a result that is consistent with ours.

However, in this special stable case, the model yields results of no useful

economic significance. Firstly, in the model’s linear version, the equilibrium

values for output and the interest rate are negative. Secondly, since the

Keynesian multiplier is negative, expanding demand results in falling output.

31

BIBLIOGRAPHY Branson, W. 1972. Macroeconomic Theory and Policy, New York, Harper & Row, Publishers. Burrows, P. 1974. The Upward Sloping IS Curve and the Control of Income and the Balance of Payments, The Journal of Finance, vol. 29, no 3, 955-961. Cebula, R. 1980. IS-LM Stability and Economic Policy Effectiveness: Further Observation, Journal of Macroeconomics, vol. 2, no. 2, 181-183. Chang, W. and Smyth, D. 1972. Stability and Instability of IS-LM Equilibrium, Oxford Economic Papers, New Series, vol. 24, no. 3, 372-384. Dernburg, T. and Dernburg, J. 1969. Macroeconomic Analysis. An Introduction to Comparative Statics and Dynamics. Massachusetts, Addison-Wesley Publishing Company, Inc. Dernburg, T. and McDougall, D. 1976. The Measurement, Analysis, and Control of Aggregate Economic Activity (fifth edition), U.S.A., McGraw-Hill Book Company. Ferguson, B. and Lim, G. 1998. Introduction to Dynamic Economic Models. UK, Manchester University Press. Gandolfo, G. 1996. Economic Dynamics (third Edition), Germany, Springer. Gilboy, E.1939. The Propensity to Consume: Reply, The Quarterly Journal of Economics, vol. 53, no. 4, 633-638 Hicks, J. 1937. Mr. Keynes and the “Classics”: A Suggested Interpretation, Econometrica, vol. 5, no.2, 147-159. Hudson, H. R. 1957. A Model of the Trade Cycle, The Economic Record, vol. XXXIII, no. 66, 378-389. Jiménez, F. 2006. Macroeconomía: enfoques y modelos. Perú, Fondo Editorial de la Pontificia Universidad Católica del Perú. Kaldor, N. 1940. A Model of theTrade Cycle, Economic Journal, vol. 50, no. 197, 78-92. McCafferty, S. 1990. Macroeconomic Theory, New York, Harper & Row, Publisher. Modigliani, F. 1944. Liquidity Preference and the Theory of Interest and Money, Econometrica, vol. 12, no. 1, 45-88. Patinkin, D. 1987. In Defense of IS-LM, UCLA Working Paper no. 537.

32

Puckett, R. 1973. Monetary Policy Effectiveness: The Case of a Positively Sloped I-S Curve: Comment, The Journal of Finance, vol. 28, no. 5, 1362-1364. Ros, J. 2004. Una nota sobre expectativas y equilibrio múltiples en un modelo LMIS − , in Ruiz, P. and Serrano, P. (eds), Enseñanza y Reflexión Económica: Homenaje a Carlos Roces, México, Plaza y Valdés. Samuelson, P. 1941. The Stability of Equilibrium: Comparative Statics and Dynamics, Econometrica, vol. 9, no. 2, 97-120. Samuelson, P. 1965. Foundations of Economic Analysis, New York, Atheneum. Sargent, Th. 1987. Macroeconomic Theory (second edition), Florida, Academic Press, INC. Shone, R. 2002. Economic Dynamics. Phase Diagram and their Economic Application (second edition), New York, Cambridge University Press. Silver, W. 1971. Monetary Policy Effectiveness: The Case of a Positively Sloped IS Curve, The Journal of Finance, vol. 26, no.5, 1077-1082. Smith, Warren. L. 1973. Macroeconomía, Amorrortu editores, Buenos Aires. Varian, H. 1977. The Stability of a Disequilibrium IS-LM Model, Scandinavian Journal of Economics, vol. 79, no. 2, 261-270. Zhang, Wei-Bin 2005. Differential Equations, Bifurcations, and Chaos in Economics, USA, World Scientific Publishing Co. Wang, L. 1980. IS-LM Stability and Economic Policy Effectiveness: a Note, Journal of Macroeconomics, vol. 2, no. 2, 175-179.

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Dancourt. Octubre, 2012. No. 341 “Reducción de costos de transporte por medio de la innovación campesina: una

ruta por recorrer”. Javier M. Iguiñiz. Octubre, 2012. No. 340 “Explaninig the Determinants of the Frequency of Exchange Rate Interventions

in Peru using Count Models”. Edgar Ventura y Gabriel Rodríguez. Octubre, 2012.

No. 339 “Inflation Expectations Formation in the Presence of Policy Shifts and Structural

Breaks: An Experimental Analysis”. Luis Ricardo Maertens y Gabriel Rodríguez. Octubre, 2012.

No. 338 “Microeconomía: Teoría de la empresa”. Cecilia Garavito. Octubre, 2012. No. 337 “El efecto del orden de nacimiento sobre el atraso escolar en el Perú”. Luis

García Núñez. Setiembre, 2012. No. 336 “Modelos de oligopolios de productos homogéneos y viabilidad de acuerdos

horizontales”. Raúl García Carpio y Raúl Pérez-Reyes Espejo. Setiembre, 2012. No. 335 “Políticas de tecnologías de información y comunicación en el Perú, 1990-

2010”. Mario D. Tello. Setiembre, 2012. No. 334 “Explaining the Transition Probabilities in the Peruvian Labor Market”. José

Rodríguez y Gabriel Rodríguez. Agosto, 2012. No. 333 “Los programas de garantía de rentas en España: la renta mínima de inserción

catalana y sus componentes de inserción laboral”. Ramón Ballester. Agosto, 2012.

Departamento de Economía - Pontificia Universidad Católica del Perú Av. Universitaria 1801, Lima 32 – Perú.

Telf. 626-2000 anexos 4950 - 4951 http://www.pucp.edu.pe/economia