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A PATH OF OPTIMAL ECONOMIC DEVELOPMENT BENHABIB, Jess NISHIMURA, Kazuo Keio Economic Society, Keio University Keio economic studies Vol.20, No.1 (1983. ) ,p.1- 22 Journal Article http://koara.lib.keio.ac.jp/xoonips/modules/xoonips/detail.php?koara_id=AA00260492-198300010001

A PATH

OF OPTIMAL

ECONOMIC

Jess BENHABIB and

1.

Kazuo

DEVELOPMENT

NISHIMURA

INTRODUCTION

Consider a two-sector economy consisting of an industrial and an agricultural sector.' The agricultural sector produces a consumption good and the industria l sector produces a composite good which can be consumed or used in production. Is it possible, under factor-intensity assumptions on technology and by classifying the demand for goods as normal and inferior, to characterize the optimal dynamic behavior of relative prices and outputs? The standard two-sector models in the literature assume that only one of the goods can be consumed. Such assumptions are quite restrictive and limit substantially the possible interactions of technology with the demand side of the economy. Relaxing these assumptions leads to a much richer characterization of the movement of relative commodity and factor prices along an optimal path, and could be especially useful for applications to planning models of developing economies.' In the present model we assume that utility is a concave function of both quantity consumed of a second good which can also be used in production. We then (in Section 3) derive stability conditions of stationary states under joint or separate assumptions on factor intensities and demand conditions, i.e. the normality or inferiority in consumption of the respective goods. We also clarify the relation between stability and uniqueness of stationary states (see Section 5). In Section 4, we assume that the pure consumption (agricultural) good is labor intensive' and non-inferior. We then show that if the economy is initially endowed with a low capital-labor ratio, the price of the pure consumption good relative to the consumable capital good and the wage rate relative to the rental of the capital good, both increase monotonically. It is interesting to note that either one of those results may not hold if the pure consumption (agricultural) good is inferior relative to the (consumable) capital good. Such demand-side considerations may be highly relevant for planning purposes in developing countries, where the strong income ' See Srinivasan (1964) and Uzawa (1964) in the linear utility case and Ryder (1969) and Hadley and Kemp (1971) for a concave utility case. z For the two -sector models of economic development see Jorgenson (1961), (1967), Marglin (1966), Dixit (1970), (1973), Harris and Todaro (1970), Marino (1975), McIntosh (1975) & (1978), Todaro (1969) and Zarembka (1970). For a comprehensive treatment, see Taylor (1979), Blitzer, Clark and Tagor (1975). 3 This is contrary to the stability condition the two-sector descriptive models used in such as Shinkai (1960), Uzawa (1961), (1963). See So low (1961), Inada (1963), Stiglitz (1967) and Burmeister and Dobell (1970), too. 1

2

JESS

elasticity

of demand

BENHABIB

for

and

KAZUO

NISHIMURA

goods

is sometimes

manufactured

underestimated

by

planners.

2.

THE MODELANDASSUMPTIONS

We consider a two-sector model with linear homogeneous production functions . Y3=fi(Kj, Li)

j= 1, 2

The first good Yr is a pure consumption good and the second good Y2 is a capital good as in Srinivasan (1964) and Uzawa (1964). Here we shall assume that the capital good is also consumable, i.e. vi= Cl, Y2= K+ C2. Capital K and Labour L are allocated between the first sector and the second sector Kl+K2
j=1,2

kl+k2
where r is a rate of time preference. Shrinivasan and Uzawa model is to be a special case of the above where u(cl, c2) = cl. To simplify the formulation of the problem we introduce the social production function y2= T(yr, k), with labour normalized to be unity. T is a maximand of Max f2(k2, 12) s.t.

J llkl, ll) ~yr kl+12
c

A PATH

we formally

state

OF

OPTIMAL

the assumptions

ECONOMIC

3

DEVELOPMENT

we use.'

(U.1)

u(cl, c2) is twice continuously for j=1,2.

differentiable in (cl, c2) > 0 and

(U.2)

u(cl, c2) is strictly concave in (cl, c2) > 0 so that the usual Hessian conditions are satisfied. rim u(cl,c2)=rim

(U.3)

u(cl,c2)= —co

C,-oc2-oo

(U.1) and (U.2) do not need justification. (U.3) is used to avoid specialization (for treatment of specialization see Hague (1970)), since we, at least in the present paper, are not concerned with it. (F.1)

f'(Kl, L) is linearly homogeneous, continuous in (K', L') >0 and continuously twice continuously differentiable in (K', L')> 0 and satisfies the usual bordered Hessian condition is satisfied for j= 1, 2. fl(0, Li) =f'(Kl, 0)=0

(F.2)

OffOf aK>o'Li JJ

j= 1, 2 .

for (Kl,Li)>0

limak---= lim---00 Li= +

j=1,2.

JJ

(F.3)

KioJ

for

Li-co

rim Of' = rim ----=0 KJLi-. 04

for

(KJ, Li) > 0 j=1, 2 .

(Kl, Li)> 0 j=1, 2 .

essentiallyinivasan The assumptions Uzawa. We can prove the following proposition.

and

PROPOSITION 1. Social production function y2= T(yr, k) is twice differentiable in (yr, k) when yr, y2, k> O. Proof

See the appendix in Benhabib and Nishimura (1979). 3.

A two production

sector

optimal

LOCAL STABILITY CONDITIONS

growth

problem

we solve

is restated

by using

a sociial

function. Max

4 (xi

continuously

fco

u(cl, c2)e-indt

, x2) > 0 means that x, >_0 a.no x2 >_0, (x, , x2) > 0 that xi > 0 and x2 > 0 and (xi, x2) > 0 that (x,, x2) >_0 and (x, , x2) 0 O.

0

4

JESS

(1)

BENHABIB

and

KAZUO

NISHIMURA

S.t.k=T(yr,k)—gk—c2 y2=c2+gk+k Yr=Cl

where p=r —g. In frequently used notation, (2)aT(c1'

k)=—P(cl, k) ;aT(cl,

k)= w(cl, k) aclak

where p is the price of the first good in terms of the second good, or the marginal rate of substitution between the two outputs, w is the rental of the first good in terms of the price of the second good, or the marginal productivity of the capital stock in the output of y2. To solve the problem we set up the Hamiltonian: H=e-fr_g)t[U(cl, c2)+q(T(cl, k)—gk—c2)] The Maximum Principle yields the first order conditions as follows: (3)k

= T(cl, k) —gk —c2 4=q(—w(cl,k)+r)

(4)U2(cl,

c2)=q Ut(cl, c2) = p(cl, k)q

where Ut=aU/ad, U2=aU/ac2. Under (U.1)—(U.3) and (F.1)—(F.3) one can obtain "saddle-point stability" conditions that are economically meaningful. Let us name the first good (whose output is denoted by cl) the pure consumption good. THEOREM1.5 Let (k, q) be one of the steady state of the optimal problem (1). Under (U.1)—(U.3)and (F1.)—(F.3)the steady state is locally saddle point stable if any of the following hold: (i) Both goods are non-inferior in the neighborhood of the steady state. (il) The capital intensive good is inferior or neutral. (iii) The pure consumption good is neutral. It was shown by Brock (1973) if the utility function exhibits normality for all values of feasible consumption, the steady state is unique. Then by virtue of corollary to Theorem 1 we have the saddle point stability. Our (i)—(iii)in Theorem 2 shall be local conditions. We will establish some structural relations before solving the stability problem. From Eq. (2) the Hessian of T(cl, k) is given by 5 A good is neutral if its consumption does not change with income at constant prices inferior if it is normal or neutral.

. A good is non-

A PATH

OF

OPTIMAL

ECONOMIC

—

ap

[A] =

5

ap —a k

ac,

(5)

DEVELOPMENT

aw

aw ad

ak

It is well known that, under non joint production, factor prices uniquely determine commodity prices. This relationship can be written as P =P(w) The associated Jacobian is ap/aw. We can use this information in the Hessian matrix above. A change in the capital stock changes factor prices which in turn change prices. Thus

apHaPaw

(6)ak(cconstant)=

ak

Similarly,

(7)Op]

Lacl

Opiocl irOwl aw

From the symmetry of the Hessian of T(cl, k), denoted by [A] in (5), we have

awl

ap

ad

ak

Using this and also substituting (6) and (7) into [A] we obtain

(8)

[A] =

k

[aPirwrPlaka-~awTwi _~ak~~awl[Owl

We proceed to analyze further the elements of the matrix [A]. Under non joint production prices, which have been normalized by the price of yr, are given by (9)EP,

1]=[we,w]

ao1 a02 all a12

where we is the wage rate and aoi and au are the amounts of labour and capital used to produce a unit of the j-th good. The total differential of (9) is

(10)

1 [op, 0]=[dwo, ow]aolao2[we, w]daoldaao2 112112]

Since we dao,,+ wda;,,= 0, j= 1, 2 hold by the cost minimization in each industry, the second term of the left hand side of the above equation vanishes. Using this result and eliminating dwo by solving (10) we obtain

6

JESS

BENHABIB

and

KAZUO

NISHIMURA

op _1 (11) dwa~l— aai2aol 02 The

same

relationship

set the amount

can be obtained

of labor

equal

(i2)[aoi

to unity.

=b

for the output We have,

under

side of the economy.

We

full employment,

ao2Yi=1)

allal2y2

k

If we take the differential for given prices, we obtain, after eliminating dy2, dk1 (13) =all--a~2aoi=b dYiao2 This follows because the cost minimizing input coefficients a1j(we,w) do not change with a small change in capital stocks when prices are fixed and if both goods are being produced.' Note that substituting (13) into (8) the Hessian of T(di k) can now be written as aw2aw ak bb

(14)

[A] _

ak

awaw akbak

where b = ap/aw = aklay, . We will make use of (14) in analyzing the stability of the dynamic system given by Eqs. (3) and (4) a little further along. Consider the two equations given by (4). Taking the differential we obtain Ull-q--

Op

U 12

(15) U21

U22

de, (pdq+qdk dc2dp

where U ` Let D be the determinant

auo2U a ct,U`i— ac.ac;

of the matrix D

.

on the left hand

=U llU22—Ul2U2l—q

side.

Then

Op acU22•

1

The transformation

function T(cl, k) is known to be a concave function of cl, k

(see Samuelson (1966)) and its Hessian must be at least negative semi-definite. Thus —ap/ad <_0. Note that ap/ad tells us how prices change when outputs are varied and the capital stock is fixed. It reflects the curvature of the production possibility frontier. Also note that (Ult U22—U12U21) is the Hessian of the utility 6 From (11) and (13) we have dk/dy1=op/ow. This is the well-knownduality between the Rycbizinskiand Stapler—Samuelson theoremsin trade theory.

A PATH

OF

OPTIMAL

ECONOMIC

7

DEVELOPMENT

function U(cl, c2). Assumption (U.2) assures that D> O. From (15) we obtain ad _1 ap akD gU22 ak lac2

(16) ad

ac2_1 ak

ap D U2lq ak 1 a Ult app aq =DUll—aclq—U2lP acq—U2lP

aq=D(PU22—U12)

Consider the Jacobian of the differential equation (3): (17)

ak ak

ak aq

aq ak

aq aq

aClac2&(ac2—

Pak

ak

(w

9),Pa

q

J=

aq

awlaCl q—ac -—aw,(—w+r)+q— clakakac,

aw

aq

We will prove Theoremm 1 by the following order. LEMMA1. Let J be a Jacobian of the differential equation (3) at a steady state state. . Then the following are true: true: (i)

d

etJ=

low [(Pu22-ul2)((w-g)b-P)+(Pul2—ult)] z2ul2)((w-g)b—P)+(Pui2—ult)]

aw

ak

akq

q

2u (il)detJ=~[-p2u22+Pu2l+Pul2—uii+(Pu22—u12 akqq 22+Pu2l+Pul2—uii+(Pu22—u12)(w—g)b] ak aw

The proof of Lemmaa 1 requires requires lengthy lengthy calculation. calculation. So it is given given in Appendix Appendix (AI). LEMMA2.

aw/ak <0 at the steady states.

Proof It is well known that aw/ak is non-positive. To see that it must be in fact. negative, we fix outputs, (F.2) implies the both inputs are used, for each 'sector at the steady states. Hence by the non joint production assumption and the bordered Hessian condition in (F.1), the input coefficients kl, k2 are differentiable functions of w. Hence the total required capital k = kl + k2 is a differentiable function k(w) in w. On the other hand as w = aT/3k is a differentiable function in k, the differentiation of the equation w= w(k(w)) with respect to w gives =

aw ak I ak • —Ow

Hence aw/ak is non-singular and it is negative.Q LEMMA3. Proof

(w —g)b —p < 0.

If b< 0, w > g implies that (w—g)b—p <0

If b> 0, consider

1

.E.D.

8

JESS

BENHABIB

and

p—wb=p—w(all

KAZUO

NISHIMURA

—al2aol)>p—wait

>0

Hence wb—p—gb <0Q.E.D. The roots of J determine the local stability of the steady state. For "saddle-point stability," we require that the two roots be of opposite sign. Since the product of the roots is the determinant of J, saddle-point stability would require def J< O. Therefore we shall prove that the conditions (i)—(iii)imply def J< O. Proof of Theorem 1. (i) We use the expression of def J in (i) of Lemma 1. The non-inferiority of both goods in the neighborhood of steady state implies (18)pu22

—u12 G 0 ,

ul l —pu2l G 0

Also by Lemmas 2 and 3 —Ow ak< 0

and

(w —g)b —p < 0

As the argument after Eq. (15) shows that D is positive and one of two inequalities in (26) is strict by (U.1) def J<0 This implies that the roots are real and of opposite sign and the saddle point stability follows. (il) We use the expression of def J in (il) of Lemma 1. Assumption implies (19)—p2ull

(U.2)

+Pu2 +Pul2 —ul 1 > 0

We will show that the condition (il) of Theorem 1 makes (pu22 —u12)(w—g)b > 0. Suppose that the pure consumption good cl as capital intensive and inferior. Then b=aol

all_12 aim

>0,pu22—u12>0 a02

This is the case where (pu22—u12)(w—g)b is positive and detJ<0 follows . On the other hand suppose that the second good y2 is capital intensive and inferior or neutral. Then by assumption (U.1), the first good is normal and pu22 —U12<0 . But the capital intensiveness of y2 implies b < 0. Accordingly def J < 0 and saddle point stability follows in this case, too. (iii) If the pure consumption obviously makes def J<0.Q

good is neutral, we have pu22 —u12 = 0 which .E.D.

Remark 1. Brock (1973) proved that if all consumption goods are normal, then a steady state is unique. The result was later improved by Brock and

A PATH

OF

OPTIMAL

ECONOMIC

DEVELOPMENT

9

Burmeister (1976) and Benhabib and Nishimura (lgiga). Theorem 1 here proves that it implies not only uniqueness but also the saddle point stability of a steady state in our two sector model. We also note that local stability of the steady state in Srinivasan—Uzawa model is a special case of Theorem 1 as their utility function u(cl, c2) = cl satisfies condition (i). 4. In

this

section

we

consumption

goods

assumptions,

then

optimal

follows.

path

goods changes as the interest intensity

THE PATH OF THE RELATIVE PRICE

make

and

factor

the steady

specific

global

intensities state

becomes

We first analyze

assumptions

of both

of

products.

unique

how the relative

and

the

normality

If we impose

the global

stability

of those of an

price p of two consumption

along optimal paths, and second, how p at the steady state changes rate varies. The assumption we make will be called the global labour

condition.

Global labour intensity condition: cl is non-inferior and labour intensive. Labour intensity condition implies that pu22—u12<— 0 and b < O. If cl is considered to be a product of agriculture sector, then the condition cl is noninferior and labour intensive, is a natural assumption to make. We shall proceed to show several results under Global labour intensity condition. LEMMA4.

All steady states are locally saddle point stable.

Proof If c2 is also normal, then both goods are normal and this implies the saddle point stability by Theorem 1. If c2 is inferior, then this is the case where the capital intensive good is inferior, and the saddle point stability again follows by Theorem 1.Q .E.I. Let qt(k) and q2(k) be solutions g of k(k, q) = 0 and q(k, g)= 0 given k. qt(k) and q2(k) are shown to be unique (see Appendix (AII)). Then we can show the following sign properties of the derivatives. LEMMA5. (i) q2'(k) < 0 for any (k, q) > 0 (il) qt '(k) < 0 when (k, q) > 0 and w >_r (iii) qt '(k) —q2'(k) > 0 when (k, q) > 0 and w>_r. Proof

We use the expression of Jacobian (17) and let

(20)j=dlldl2 Then

2122

ioJESS

BENHABIB

and KAZUO

NISHIMURA

q1'(k)= –dl-,q2'(k)=–d2 2 1222

(21) q1'(k)—q2'(k)=—dlld22–dl2d2l d i2d22 (i) First we shall show d22<0. d22=(r–w)+Owq_j.8c1 a q since Ow Op ad=ak d

Op ac, 22=qak•aq+r–w Op Ow 1 =q— Ow ak D [pu22–u12]+r–w =q

_

Ow • 1 ak D [pu22–u12]b+r–w

We already know that awlak < 0 and D> O. Labour intensity condition implies (p U22—U12)b? 0 Hence Ow 1 (22) q ak • D [pU22– U12]b<0 If d22is evaluated at (q2(k),k), then w=r and d22<0. Let us check the signs of d21. _

(23)

d21= q–a

qF =D— _ _

_

awl aclaw c,•akak

awl

Op Ow

oct qu22ak–akD

q[owOw

Dak(ul

lu22 —ul2u2l)+qu22–

Opaw,ap

ak • ad + ac, • ak

By the strict concavity of utility function and the concavity of social production function (24)ult

(25)ak

u22—U12/121 >0 Ow OpOwlOp

ad–ad

_2 ak–TkkTci–Tk~,>_0.

A PATH

OF

OPTIMAL

ECONOMIC

DEVELOPMENT

11

Hence d21> 0 and q2'(k) < 0 follows. (il) First we shall check the sign of d12. 1 21 12=D(Pu22-Pul2)— =— D

D

ap (26)d ult—aCq—u2lP

—P2u22+2Pul2—ult+

-- q

By ap/ad= —a2T/acl2 > 0 and the strict concavity follows. On the other hand =

P

ad ak

of utility function, d12 >

&C2 (29)dlr ak + w—9

lap _—Dq(Pu22—u21)a

k+w-g 1 aw =—Dq(Pu22—u21)b ak+w—g —Dq(Pu 22—u21)bak +r—g which follows as long as w>_r, since dl 1 is positive by (22). Hence q1'(k) <0 follows. (iii)

q1'(k)—q2'(k)=—------J dlld22 def

We know that def J is negative at w= r. It is easily shown that it is still negative as long as w? r (see (i) of Lemma 1). Hence qt '(k) —q2'(k) > 0 holds . Q.E.D. In the proof of Lemma 5 we observed that d21depends on the Hessians of the utility and social production functions. In fact the Hessian of the social production function becomes singular if no joint production exists (see Samuelson (1966)). So it is the strict concavity of utility function which makes d21non-zero (on this point see Benhabit and Nishimura (1981) for extensive discussion on the heterogeneous capital goods model) which in turn make 4=0 curve downward sloping. Let kl (q) and k2(q) be solutions of k(k, q)=0 and q(k, q)=0 given q. w(cl(k, q), k) given q is shown to be decreasing in k and k2(q) is uniquely determined (see Appendix (AIII)). As q2'(k) < 0, we know that {(k, q) : 4(k, q) = 0} is a connected curve. On the other hand kl (q) is unique on the region S= {(k,q) : 4(k, q)_ 0} (Appendix (AIII)). Hence {(k,q) : k(k, q) = 0} is also a connected curve on S. Since any steady state must exist on S, a steady state must be unique. These consideration justify the phase-diagram in Figure 1.

1

12

JESS

BENHABIB

and . KAZUO

NISHIMURA

qk

--------------------------------------------------------------------------k 0 Fig. 1. LEMMA 6. We shall rental

The steady

now study

price of capital

state

is unique.

changes good

in the relative

price

of two consumption

and the wage rate of labour.

Below

goods,

we crucially

the

use the

global labour intensivecondition and the fact that gk <0 if k OE, along optimal paths. THEOREM 2. p > 0 if kc < E and p <0 if kc > k. Proof

We differentiate p=p(cl,k) with respect to t.

(28)p=8pk+8p[ac,k+a~lq ak

ap

ad

ak

Op

aclOpac,.

aq

akacakk+acaq

using (16)

liq

Op ap ak + ac,

•

ac, ak

Op Op qu22 Op ak ac, D ak DakD+qu22p

(29)

A PATH

OF

OPTIMAL

ECONOMIC

DEVELOPMENTls

_1ap

ap ap)

D ak ullu22—ui2u2i—qacu22+acqu22

1 ap 1 D ak (ullu22 —ul2u2l)

We note that >0andapp> 0 andapapawD akawakbaw>o. 1k

Hence

(30)ak+ap•--->0 1

On the other hand Op aclap _ ad aq ac,

1 (31) D(Pu22-ul2)~0

Since qk <0 for k 0 0 along optimal path, (32)pk

>0

if

k00

It implies the statement of Theorem 2.Q.E.O. THEOREM3. Proof

If k < k, then w < 0 and 14)0>0. If k> k-, then 14)>0 and we < 0.

We differentiate w = w(cl, k) with respect to t.

(33)w_Owk+Ow[Ocik+a~lq akac, ak

aq

[Ow awaclaw ac, . ak+ad akk+ad aq----q From (23)—(25), aw

aw oct (34)

ak +ad

ak <0

and aw oct

ap aclaw

ocaaka—bak•D[Pu22 1qq

1 (35) —u 12] > 0

Hence wk <0 if k o 0. Thus the sign of vi, is determined Theorem. In order to obtain ivo we can use the relation

as in the statement of

14

JESS BENHABIBand KAZUO NISHIMURA

(36)dwoal2<0 02

implied by Eq. (10).Q.E.D. Next we consider the change of the steady state value when the discount rate p varies. By differentiating the left hand side of Eq. (3). (37) L_-9dP dgJL

It is easy to show d —>

dk op <0

(38)

0 . op

THEOREM4. op

<0,

P

ow>0 anddoo<0 PP

where p, w and we are evaluated at a steady state. Proof

We differentiate p = p(cl, k) with respect to r.

(39)

op Loci ap oct dk + Loci op ad dq op ak +apak op aqidp

In the proof of Theorem 2 we have shown dp8clapolsoc, dclak+ak>o'ac,

P

P

<0 (40) aq

Sincedk P<0anddq>0 (41)op<0 follows. Similarly

(42)op

ow

awocl

awdkawacldq

Lac, ak + ak dp+ acl aq dp> 0

and dw0/op<0 follows from (36).Q.E.D. If we use expression (20) of the Jacobian J, then (41)dk_gdl2dqll

go opdef J

op

def J

By substituting (26) and (22) we see that d12 >0 follows from the concavity of

A PATH

OF

OPTIMAL

ECONOMIC

DEVELOPMENT

15

utility and production functions. Hence dk/op is negative or positive according to the sign of def J. However dq/d p does not depend on the sign of def J (i .e. the stability or instability of the steady state) alone . (pu22 —u21)b >_0 is crucially used to show that dq/op is positive. On determining the sign of op/op we used the assumption of the normality and labour intensiveness of the pure consumption good. Remark 2. It is interesting to compare our global labour intensity condition with the sufficient condition used by Uzawa (1961) to assure the stability of a steady state in the two sector descriptive model . He assumed that a pure consumption good is capital intensive , contrary to our assumption, and he ((1963) p. 109) refer to Gordon (1961) for empirical evidence in the case of the United State economy. However at least for developing countries , it seems more reasonable to us to assume that a pure consumption good is more labour intensive than a consumable capital good. 5.

THE GENERAL ONE CAPITAL MODEL

An interesting question is how much one can relax the technological restriction we imposed. There are a few papers dealing with more general production functions. For example see Liviatan—Samuelson (1969) for a model involving joint production (see also Mag ill (1979)), Kurz (1968) for a model with wealth effects and Ryder and Heal (1973) for a model of intertemporally dependent preferences . On the other hand Kurz (1968) reported a one-capital good example due to Arrow which has a unique and totally unstable steady state . This example however requires not only utility saturation but also that the discount rate has to exceed the marginal productivity of capital (which was a constant for the example) at the steady state. Elsewhere we also have shown that , with two or more capital goods, the loss of "saddle-point stability ," is possible when uniqueness is preserved, and it can result in optimal paths that form closed orbits around the steady state (see Benhabib and Nishimura (1979)) . It is not clear from the literature what the general relationship between the uniqueness of the steady state and the saddle point stability is in models with one-capital models. One also could ask if such relation could depend on the size of discount rate or not , as Cass (1965) in one sector model and Uzawa in two sector model proved that the optimal path converges to a unique steady state independently of the size of the discount rate . In this section we study the relation between the multiplicity of steady states and saddle point stability. We provide a rigorous analysis and establish some general results which support the phase-diagramatic conclusions of Liviatan —Samuelson (1969). We shall further show that the uniqueness of a steady state implies the local saddle point stability. Following Brock and Scheinkman (1976) , we start with a general modified Hamiltonian system: H: Q—*R is continuous and of C2-class on Q where S2=

16

JESS

BENHABIB

and

KAZUO

NISHIMURA

{xe R"]x > 0} and Q is the interior of Q. We obtain the dynamical system (1')k

= 011(k, q q) 4

aH = Pq—ak(k, q)

Definition 1. A steady state is a point in

{(kq)e~x SI/Pq — ak(k, q)=aH (q, k)=0 We assume

the following.

(A.1)

The set of steady state M(p) lies in the interior of a compact set D which is in the interior of Q for all p E [0, p] for some positive value p.

(A.2)

When p= 0, the dynamical system (1') has a unique non-trivial steady state which is locally saddle point stable.

The Jacobian matirx J can be written as J(P) = —Hu Hqk

pl Hqq Hkq

where H

o2Ha2Ho2H, qq=-----aq2 ,Hkk=

ak2 '

Hqk=agakand

Hkq=Hqk.

We will also assume that all the steady states of an economy for a given discount rate have non-singular Jacobian J: (A.3)

Jacobians J(p) are non-singular on M(15)and Jacobians J(0) are non-singular on M(0).

Note that (A.3) does not rule out singular Jacobians for p= p and p=0. We now derive results for an economy with discount rate p. LEMMA7 (Bendixon's theorem). Let f: R2 R2 be of o-class on R2 and denote its Jacobian J. If trace J has constant non-zero sign, then no limit cycle appears as a solution of .z=f(x). Proof.

See Hsu and Meyer (1968).

LEMMA8 (Poincare—Bendix on). Every bounded path in R2 converges to a equilibrium or has an accumulation point on a limit cycle. Proof.

See Hirsch and Smale (1974).

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Since Lemma 7 follows from Green's theorem on the plane, it can be applied to the system (1) defined on 6. This result was used by Brock and Scheinkman (1977) for a dynamical system defined on a two-dimensional capital domain. Definition 2. A solution path of the dynamical system (1) with an initial point in S is an interior path if it is bounded above and no subsequence along it has an accumulation point on the boundary of Q. THEOREM 5. Let p= p > 0. (i) The number of steady states is odd and the number of saddle point stable steady states exceeds the number of totally unstable steady states by one. (il) An interior path starting from any point other than the totally unstable steady in S converges to one of the saddle point stable steady states. Proof (i) 6 is a smooth manifold. Then the dynamical system defined by (1) is a vector field on 6 as the right hand terms of (1) is a continuous function. By the Hopf's lemma (see Milnor 1965, p. 36), the boundary degree of the vector field on D defined by (1) is equal to the sum of def J(0) evaluated at steady states. But by assumption (A.2) and (A.3), def J(0) = —1. Hence the boundary degree of (1) on D is also equal to —1 when p = O.As p increases from 0 to p, the right hand term of the vector field (1) changes continuously in p, and it never vanishes on the boundary of D by (AI). Then the boundary degree remains constant through this deformation (see Benhabib and Nishimura (1979, Theorem 1)). Hence again by Hopf's lemma

E defJo= —1

M(p)

must be true. As def J(p) 0 on M(p) by (A.3), def J(p) <0 if and only if a steady state is locally saddle point stable and def J(p) >0 if and only if a steady state is totally unstable. The second part follows since the trace of J(p) is equal to p, but this is also equal to the sum of eigenvalues of J(p); hence the steady state can never be totally stable. Accordingly the number of steady states is odd and the number of saddle-point stable steady states must exceed the number of totally unstable steady state by one. (il) If a path (k(t), q(t)) is of interior, there is a uniform m > 0 and an M> 0 such that M> k(t) > m, M> q(t) > m by Definition 2. Lemma 2 implies that this path converges to an equilibrium or has an accumulation point on a limit cycle. But Lemma 1 says that a limit cycle does not exist, therefore an interior path must converge to a steady state.Q.E.D. Theorem 1 implies that if a steady state is unique, then the result of Cass (1965) follows: COROLLARY 2. If a steady state (k(p), q(p)) is unique, then it is locally saddle point stable and every interior solution path of (1) converges to the steady state. In the following section we study the multiplicity and "saddle-point stability" of

18

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BENHABIB

and

KAZUO

NISHIMURA

steady states in a two-sector model with both goods consumable. Remark 3. In this section we did not use the underlying concavity properties of the technology or the utility function. Proposition 1 implies that under assumptions (A.1)—(A.3) alone the general dynamical system generates (for given p = p) phase diagrams of the Liviatan—Samuelson (1969) type. The above Corollary implies that under (A.1)—(A.3) any attempt to construct a totally unstable steady state cannot succeed. Remark 4. A few words on the example due to Arrow (Kurz (1968)), are in order. His dynamical equations are (1)'k=q+ak—c q=(p-a)q in our framework. As his steady state is on the boundary of the domain, we might include the set of negative current prices as the subset of the domain as well, to see the entire picture of the dynamical system. Denote this extended domain A. In his example, for the discount rate p less than the marginal productivity a, a unique steady state (k(p), 0) is saddle point stable. However once the discount rate becomes equal to the marginal productivity, then the whole line q + ak —c = 0 in A becomes the set of steady states. Hence the set of optimal paths coincides with the set of steady state. Therefore assumption (A.2) does not hold in his example. APPENDICIES

(AI)

Proof of Lemma 1. From (17) the determinant of J is: etJ=

ac,ac2aW —Pak ak+w—g(—w+r)+q

aCi (39) aca1q

d

act ac2aw aclaw Pa q aqqacl akqak Evaluating the determinant at the steady state we have w= r from the second equation of (3). We use this to eliminate (—w+ r) from the right hand side of (39). Substituting for act/8k, act/aq i= 1, 2 from (16) we get

def J=—ls2-lq2(ZVI:a(PU22— U12)2+Dl(x'-g)qa ci_ aw(pU22 — U12) —Dq22ac awapl iU22ak—

aw D qak

X —p2U22+PUl2+PU2l—Ult+q

1

a aP

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a aw 1 1

=—g2[p2Ui2+Ui2U2l-2U2lU22p]

ak ad D2

_q2_p2Ui2+pUi2U22+pU2iU22—UiiU22aaPg xi--a apawlawpU22—U kaclDZ+(w9)g

12

ad

D

)

+Dqak—p2U22+pUi2+pU2i—Ult+ap ----g Simplifying furtherandcollecting terms,

(40)

rawD +q[-a---

def J=—q2

Dac ad+(w

—9)q—awpU22—

D Ui2

aw2apl

kpU22+pU2l+pUl2—U1~+qacD

We now use the relationships derived in the analysis of the Hessian of T(c,, k), given by the matrix [A] in (14). Comparing (5) and (14), we have

-ad=L [awlaw aid • Since

[g
I[°-1w_2aw2 (41)'PPaclak

Comparing (5) and (14) we also see that

(42)[jib

[4aaw2

Substituting (41) and (42) into (40), we obtain

def J=Dqakl[(Pu22—U12)((w—g)b—p) Ult) +bakb2ak q2awgaw =akq[(pU22 —U12)((w—g)b—p)+(pUl2 —Ult)]

i

2oJESS

BENHABIB

Rearranging,

we can also

write

and KAZUO NISHIMURA

def J as follows:

(41) def J=D—p2U [22+PU2l+PUl2—Ult+(PU22—U12)(w—g)b]

akq Q.E.D.

(AII) Uniqueness of qt(k) and q2(k). Given k, we differentiate w(cl, k) with respect to q. Then Ow aw oct —>o aqoc 1 aq holds by (35). This implies that a solution q2(k) of w(cl(k, q), k) = r is unique. On the other hand if we differentiate T(cl, k) —gk —c2 with respect to q, we get the following by the use of (16) a(y2—gk—c2) ac, 0c2 aq— —Paq—aq

— —PD(PU22U12)---(Ult— aClq-U2l P 1

aP

=D—P2U22+2Ul2p-Ull+~pq >0 Hence T(cl(k, q), k) —gk —c2(k, q) = 0 has a unique solution qt(k). (AIII) Uniqueness of kl(q) and k2(q). Given q we differentiate w(cl, k) with respect to k. Then Ow ad Ow ac, ak+ak`0 holds by (34). Hence a solution k2(q) of w(cl(k, q), k) = r is unique. Next we differentiate T(cl, k) —gk —c2 with respect to k. Then as long as w>=r, ac, 0c2 —p-----+w—g>0 must hold by (27). Hence a solution kl(q) of y2 —gk —c2= 0 is also unique on the region defined by w(cl(k, q), k) New York UniversityUniversity andand University of Southern California

of Southern California Tokyo Metropolitan University

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REFERENCES

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