INTERNATIONAL ECONOMIC REVIEW

Vol. 20, No. 1, February, 1979

ON THE UNIQUENESS OF STEADY STATES IN AN ECONOMY

WITH HETEROGENEOUS CAPITAL GOODS

BY JESS BENHABIB AND KAZUO NISHIMURA'

1. INTRODUCTION

It is well known that in multisector models of optimal growth, optimal paths

converge to a unique steady state when future utilities are not discounted.

Sutherland [1970], Kurz [1968], and Liviatan and Samuelson [1969] gave ex-

amples of multiple steady states when future utilities are discounted. Beals and

Koopmans [1969] and Iwai [1972] used intertemporal utility functions, which

also yield multiple steady states.

The uniqueness of steady states with multiple consumption goods when future

utility is discounted has been studied by Brock [1973]. He showed tlhat if we

assume that none of the goods are inferior in consumption (he calls this the

normality condition for the utility function) the uniqueness of the steady state is

assured. Brock did not allow for pure consumption goods. Later Brock and

Burmeister [1976] generalized Brock's result to allow pure consumption goods

as well (Morishima [1974] type). Brock also formulated an alternative approach

where uniqueness is assured under the assumption of a non-vanishing Jacobian

for every non-negative discount rate. He writes however that the non-singularity

of the Jacobian "is an obscure assumption" and that it would be worthwhile to

relate it to the normality condition of the utility function.

We first propose to weaken Brock's assumption of non-vanishing Jacobian for

every discount rate. Then in Section 3, Theorem 2, we show that the normality

condition on the utility function implies a non-vanishing Jacobian. In Theorem

3 we weaken the normality condition for the uniqueness of the steady state by

investigating the conditions for a non-vanishing Jacobian. We then clarify the

economic content of our weaker conditions. In the final section we observe that

the normality theorem can be proved for the joint production case (Mirrlees

[1969] type) using a technique due to McKenzie [1963, 1973].

2. JACOBIAN CONDITIONS FOR THE UNIQUENESS OF THE STEADY STATE

Our problem is the following:

* Manuscript received May 21, 1976; revised September 13, 1977.

I The research was done while Jess Benhabib and -Kazuo Nishimura were students of

Columbia University and the University of Rochester, respectively. We are grateful to Pro-

fessors L. McKenzie and K. Lancaster for their guidance and encouragement. We are heavily

indebted to Professor L. McKenzie for his help and comments on various aspects of this paper.

We benefited greatly from correspondence with Professor W. Brock. For any remaining

errors we are entirely to blame.

59

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60 J. BENHABIB AND K. NISHIMURA

oc

Max U(co, Ic)e-P'dt

(1)

s.t. co = T(y, k)

where Y=(Yi, Y2.-- Yn), k=(kl, k2,..., kn), 1c=(cl, c2-. Cn ),2 v1 =ki+gki+cj,

i= 1, 2,..., n, g is the rate of population growth, p is the rate of discount, co is the

level of the consumption of the pure consumption good, the ci's are the other

consumption levels, the ki's are capital stocks and the J'i's are the outputs of

goods. The utility function and the transformation function are assumed to be

concave. Both functions are of the Cl-class on the interior of a domain and

of the C2-class on the set of steady states. We rule out the possibility of corner

solutions.

Set up the Hamiltonian

H = ePtU(co, 1c) + qo[- co + T(y, k)] + Eqj[y; - j- gkj].

By the maximum principle

0 = ? j = O.

aT

4i -gqj + qo a j- j-l =. n

aT

qoeI + qj- j-l . n

kj =, Y- c.-j -l g kj n.

Put pj - U j = 0 .., Il.

Then

Pi = (P + 9)P - PO akj j=I

(2)

kj = J=; - cj-gkj n.

Let the consumption good be a numeraire. Now let pj, wj j =1.. n be the

prices and the rentals of the goods in terms of the price of the pure consumption

good. Let r=p+g, p=(pP... p), w=(w,... w,). The following are the

steady state equations:

wj(y, k) - rpj(y, k) = 0

(3) Pi = &C

yj= gkj + cj

2 Variables (y, k, c,, 1c) are represented by per capita levels.

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UNIQUENESS OF STEADY STATES 61

where

-T pj(y, k)

aT

-k-- =w(y,k) n.

The steady state equations will then be functions of the vectors k, Ic and the

rate of interest, r. We can write the above equations as

Gi(lc, k; r) w,(gk + 1c, k) - rpi(gk + lc, k) i = 1,..., n

Fi(lc, k; r) _pi(gk + Ic, k) 8u(T(gk + 1c, k), 1c) Au = n.

ac0 c

Let G=(G1,..., G) and F=(F1,...,F).

(Al) For any r _ g, (k, 1 c) belongs to Q = {x E R2,Ix _ O}, and the domain of the

functions G and F can be restricted to a compact and convex 2n-dimensional

manifold D in Q, containing all steady state solutions in its interior.

For large amounts of capital k, the capital stock cannot be sustained. So the

set of steady state solutions is bounded above (see McKenzie [1968], p. 357).

If (3) has a solution (k, 1 c) on the upper boundary, the vector field (4) can be con-

tinuously extended to a neighborhood of (Ik, Ic). By extending the domain of G

and F (if necessary), the steady state solutions can be prevented from lying on the

upper boundary of the domain. It is through the positivity of steady states that

assumption (Al) imposes any restriction on our problem. The positivity of

steady states can be assured by assuming suitable conditions on the partial deriva-

tives of the transformation function and the utility function for the zero level of

each variable. Consider a vector field on D:

kj = Gj(1c, k; r)

ei = Fj(,c, k; r) j ,.,n.

Let Mr={(ic, k)eR2 IG(,c, k; r)=O, F(1c, k; r)=O}.

Brock proved that if Mg is a singleton and the Jacobian of the above system with

respect to Ic and k (say J,) over Mr is non-singular for every r_g, then Mr is a

singleton for every r?g (Brock [1973], p. 555). But we are now interested in

the uniqueness of the steady state for a specific r > g. We want to show that even

if det Jr vanishes for some r E (g, F), we still have uniqueness for F.

We will depend heavily on Hopf's lemma in the discussion below.

LEMMA (Hopf).3 Let B be an m-dimensional ball (up to diffeomorphism).

If v: B-*Rm is a vector field with isolated zeros, and if v points inward on the

boundary aB of B, then the degree of a mapping v restricted to aB is (- 1)m.

I For the exposition and proof of this lemma, see Milnor [1965].

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62 J. BENHABIB AND K. NISHIMURA

Moreover if 0 is a regular value of v, then v-l(O) consists of finitely many

points and (- 1)m= , sign detJxv wlhere J v is a Jacobian ojf v with

xe v I (0)

respect to x E B.

THEOREM 1. Suppose that Mg is a singleton and J. is non-singular over Mg.

If detJr has a uniform sign over M,, then M, is a singleton.

PROOF:

Consider H: D x [g, r]-*R2" defined by

Hj(lc, k; t) = Gj(1c, k; t) j = 1. n

Hn+j(lc, k; t) = Fj(lc, k; t) jl.. n.

This gives a homotopy between G(1c, k; g), F(jc, k; g) and G(,c, k; r), F(1c, k;

r). Since for r=g the vector field has a unique zero point in the interior of D,

and Jr is non-singular, the vector field has degree (+1) on the boundary of D.

This may be seen from the proof of Hopf's lemma. The inward pointing property

of the vector field is not needed here. The degree of a vector field restricted to the

boundary is equal to the sum of the signs of the determinants at zero points of

the vector field.

As t changes from g to r, the vector field changes continuously and never has

a boundary equilibrium by (Al). Then the vector field restricted to the boundary

for r = - has the same degree as the vector field for r = g since the degree is

homotopy invariant. On the other hand M. consists of finitely many points (say

(1ci, ki) j= 1,..., s), since det Jr# 0 and M, lies in a compact set by (A 1).

Applying Hopf's lemma, (-I) or (? l)= E sign detJF evaluated at (1ci, ki).

Therefore s 1. Q.E.D.

Theorem I is a weakening of Brock's theorem ([1973], p. 555). For some

re-(g, r) Jacobian J, may vanish on Mr, or even uniqueness may fail to hold

(see Figure 1). This theorem may conveniently be applicable to other models

in growth theory.4

In Theorem I, a direct generalization of Brock's result, the non-singularity of

the Jacobian and a unique steady state for r=g are crucial assumptions. Since

Mr, for every r e [g, r], lies in the interior of the domain, we always know the

boundary degree of the vector field for r = r as well. On the other hand, we

cannot, in general expect a vector field to point inward. Hence we could not

apply Hopf's lemma directly to a vector field as Dierker [1972] or Varian [1975]

did in general equilibrium theory (compare with Nishimura [1976] and [1978]).

In our model, if the following condition holds when r=g, we can circumvent

I For example, for a modified Hamiltonian dynamical system in the price-capital space, we

cannot expect the inward pointing property to hold. Then Theorem 1 may be directly appli-

cable.

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UNIQUENESS OF STEADY STATES 63

C

I~~~

______~I I____

g7

1 1~FIUR

this difficulty. For simplicity assume that D is rectangular. Given the values

of k at any level (not just at steady state solutions), let F(lc, k, g) = 0 have a

solution but not on the lower or upper boundary of D with respect to 1c.

Similarly, given 1c at any level, let G(1c, k, g)=0 have a solution but not on the

lower or upper boundary of D with respect to k. Then, if r =g, we have

G(,C, k; g) = OT(c + gk, k)

ak

F(1c, k; g) = au(T(c + gk, k), 1c)

__T au

T and u are concave in k and 1c. Hence k = _Mk and 1c= -@ c point inward at

the upper and lower boundaries with respect to k and 1c for r = g. Thus we

obtain the boundary degree of the vector field as (- 1)2n -1. Since, by (Al),

this degree is preserved as r increases, even though the inward pointing property

may be lost, the sign uniformity of the Jacobian on M. yields the uniqueness of

the steady state at r = r.

In Section 3 we show that if the pure consumption good is normal and the

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64 J. BENHABIB AND K. NISHIMURA

utility function has strictly concave contours, the Jacobian is non-singular and

the steady state is unique for r=g. In such a case we can rely on Theorem 1.

But if the normality of the pure consumption good or thie strict quasi-concavity

of the utility function fail to hold for the solution at r=g, the above result, which

dispenses with these requirements, may become very useful.

It should be noted that the above approach also gives a very simple proof of

the existence of steady states. The non-zero degree of the vector field on the

boundary of the domain implies the existence of zero points on the interior of the

domain. This follows without using Jacobian conditions nor differentiability.

(Compare with the existence proofs in Sutherland [1970], Peleg and Ryder [1974],

and Brock [1973]).

3. THE RELAXATION OF THE NORMALITY CONDITION AND ITS RELATION

TO THE JACOBIAN CONDITION

In this section we relate the non-vanishing Jacobian condition to the normality

condition of the utility function. Then, in Theorem 2, we weaken the normality

condition and replace it with a weaker condition that yields a non-vanishing

Jacobian of uniform sign for a given discount rate. In conjunction with Theorem

I this yields a unique steady state.

We make the following standard assumptions on the technology:

(A2) All goods are produced by production functions homogeneous of degree

one, twice differentiable, and which satisfy the usual second order con-

ditions.'

(A3) Labor is required directly in the production of the consumption good

and either directly or indirectly, in the production of at least one capital

good, i.e. aoo0 0 and a.0 #0.

Given the rate of interest we can use the non-substitution theorem to generate

a unique technology matrix which is independent of the pattern of consumption.

Let the technology matrix be given by the usual Leontief matrix A

Let A= 0

Full employment of capital goods requires

L ? 0 o C ,O 0 k

In a steady state

LCO] = Lgk' ] +

I Specifically, the i principal minors of the bordered Hessian alternate in sign with the princi-

pal minors of order i having sign (- 1)i.

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UNIQUENESS OF STEADY STATES 65

Combining the above we get

[I? = [I - gA]1l CC,O.

Full employment of a unit of labor will require

[aoo, ao.][ y] = [aoo, ao.] [I - gA1-1 [ IC] = 1.

Using this we can express co in terms of 1c and the rate of interest since the interest

rate determines the coefficients of the technology matrix.

Consider now the equation system

wi(y, k) - rp1(y, k) = 0 1= 1,..., n

pi(y, k) - Vj(1c, r) = 0 where V1(1c, r)- au/ac0 = 1,..., n.

Here yj=gk +cj. We have eliminated co from the Vi's and replaced it with r.

Let

OG Ow ow DP -~~~~~~~~~~~~~~~[r rg[

[ Ak ] [Ak 2 0ay 2 k0 [ay]

where bracketed terms are (n x n) matrices. We obtain the Jacobian of the above

equations as

[ ap -[] F&L. K g ]2

ay ] Lak ay

Law] L ] aL

- ayl ay~j F k

where [V][-a8i] i,j=1,...,n.

We will now simplify the above Jacobian to derive our results. Consider first

the Hessian of the function co= T(y, k),

H - a.. ..[.... ............

raw]aw]

r ay r akw ]

From its symmetry we have L ak ] =- Lay] Since, from profit maximization,

we know that given factor rentals, relative prices are determined uniquely, we have

n functions

pi = p(w) i=1,...,n.

Their Jacobian [A can be used to simplify the expressions in the matrix [J].

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66 J. BENHABIB AND K. NISHIMURA

We have

__ O _w -w Fw k 8w

0 k- ak ]; i L w]k[ k l and

oP 1 rP Orwl _ p _rP aw rap1

LWS22-L@WJ } -w [Y]aw ]L OIYawl

Using the above equations we obtain

OW Ok Iw

t Ak ! I r0 Au ]0a-l0- 8P-0

and the Jacobian [J] can be written as

[J]=F - L~i~,]L~]Ll'-VI [ k j ] g[wP]] 1

_ w a]0 k ] w ak

We can now use the theorem for partitioned matrices stating

det = det [D] det [A - BD-'C]

provided det [D] #0. Let us identify the matrices from the partitioned Jacobian

matrix J as follows:

[p = -aw F 81) 1w

[C=-LIk L= 8-w w 8 k i 8w]

For the moment let us also assume that [8G]is non-singular. Then, by simple

k- .rr raD1w akt r nwt

substitution of these matrices from the partitioned Jacobian [J] into the above

theorem, we obtain

det [J] =-det [-] det [VL] =

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UNIQUENESS OF STEADY STATES 67

Let us now explore the non-singularity of [v-] We must start by investigat-

ing the matrix The equation for prices, normalized by the price of the

consumption good is given by

[E1. P] = [wo, w] a,, ao.

Taking the differential we get

[0, dp] = [dwo, dw]L a0 a0 ]

a0 A

Let us emphasize at this point that we do not assume fixed coefficients. In view

of (A2), the aij's are differentiable functions of wi's, the input prices. However,

from the homogeneity of degree one of the cost functions, we have

[wO, w] dL -_ = o

Thus the above result follows when we take the differential (see Samuelson [1967],

pp. 61-69). Since dwoaoo + E dwiaiO = 0, we can solve the above matrix equation

for dwo in terms of the dwi's. Substituting for dwo we then obtain the reduced

matrix equations

dw [A- a.oao.] = dp

or [P ]'[A -a-oao.

Let [B'] =LA--'a - ao. We can then write I - rL &]]=[I-rB] and

I- g8Pw ]=[I- gB']. Since we want to prove the non-singularity of [Ak 1

=[I -rB] 8wk[Ij- gB'] we must show that I- rAjB# 0 and I- g;.iBO where

Aj8 is the i-th root of the matrix B.

Consider first the matrix [A]. It is non-negative, square and has a dominant

root is. We would like to argue that r).A < 1.

PROPOSITION 1. Let r be the rate of interest and [A] be the associated

input coefficient matrix when the economy is at a steady state. If AA iS the

dominant root of [A], then r).A < 1 if the wage rate is positive, at least one capital

good requ ires some labor (i.e., a.0 # 0) and [A] is indecomposable. Below x _ 0

means that xj>Ofor all j, x>O that x_O and xj>O for some j, and x>O that

xj > 0 for all j,6

6 Note that (A3) guarantees positive prices in terms of the wage rate. See Burmeister and

Dobell [1970], page 237.

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68 J. BENHABIB AND K. NISHIMURA

PROOF. Steady state prices of goods used in production are given by

p = a.0wo + rpA.

Since wo >0 and aO# 0,

p ? rpA; --p ? pA.

Debreu and Herstein ([1953], section 4) have proved that if the above inequality

holds for [A] indecomposable, then it follows that

-r > iA; I > riA

Q.E.D.

Let us now consider whether [I - rB] is a non-singular matrix. We need

1- rAjiB #0, all i. If AiB is a complex root we have no problem. We also have,

from Proposition 1, 1 - rLA >0. If we can show that AA > jB where )jB is a real

root of the matrix [B] we will have proven the non-singularity of [l- rB].

PROPOSITION 2. LetL aoo ao 1 be a non-negative squar-e matrix partitioned

a. I A

such that aoo is a scalar. Let [B]-=A-i-j-a.o ao]. Let )A be the dominant

root of [A]. Then tjB?AA where XjB is any real root of [B].

PROOF. Any root of [B] must satisfy det A - a1 a0a0 - iJBj=I 0. Assume

then that tjB> ;A* From a theorem on partitioned matrices we have

det LA -a LaoaO. -AjI] =

det [A - AjBI] - al -ao. (A - jBI)a. oJ

(See Graybill [1969], page 165). If ijB> 'A' from standard theorems in linear

algebra we get

det(A - AjBl) # 0; (A - ,jBI)- < 0.

Thus (I -al ao.(A-A{JBII a.o)# 0 since a'o, ao >0 aoo > 0. But then det LA-

al a,0a-AjBI] #0 and we have a contradiction. Thus if )LjB is a real root of

B, then )L.B_
We have shown [I - rB] to be non-singular. Since we also have r > g, [I - gB]

is also non-singular.

Consider now [ aw Since wi = wi(y, k), i =1,..., n, for given output vector

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UNIQUENESS OF STEADY STATES 69

y, the capital stock vector k uniquely determines wi's. On the other hand, in

view of (A2), unit input coefficients are differentiable functions of factor rentals.

Thus given w,'s we can determine input coefficients. If outputs of capital goods

are fixed, the labor constraint immediately yields the output level of the con-

sumption good. With all outputs and input coefficients given we immediately

obtain the required capital inputs. Thus we have the differentiable functions

k,= k(w), i =,..., n for fixed outputs of capital goods. Since these are the

differentiable global inverse functions of the functions wi = wi(k), where the out-

puts y are fixed, the Jacobian aw- cannot vanish.

L8ki

Thus the following must hold:

det LG ]=det LI-rI 8p i det 8k j det I - L ]

The non-singularity of the Jacobian [J] depends solely on the non-singularity

of the matrix [V] since xaG] is shown to be non-singular under standard

assumptions. Consider then the matrix [V], given by

[ O(ul/u0) j n

aU.~~~c

where ui= au i=0 1,..., n. From the labor constraint we have (ao0, ao).

[I - gA] ( Cc )= 1. Let (ao, a) =(ao0, ao0) [I-gA]-1. In view of (A3), (ao, a)

>0 (see Burmeister and Dobell [1970], page 237). Using the labor constraint

we can solve for c0. We obtain

n

1- aici

Co =- i=l

aO

Define the row vector d=(d1,..., d)=-I(a,,..., an)=aa>0. Let uj= 2U

ao ao aci Ojcj

and let the matrix [uij] have elements uij. Evaluating matrix [V] we find

[Vij]=-[uo(uij-djuo)--ui(uoj-u oodj)].

Since uiuo =pi, rearranging, we have

[V] 1 [[uiu - p'uo.] - (u0 - p'uoo)d]

U0

092U

where u0 =(u01,..., u0"), u 0=(ujo,..., uno)' and u0=--. We will proceed

to show sufficient conditions for [V] to be non-singular. We require the follow-

ing assumption:

(A4) The utility function is twice differentiable, has positive first partials and

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70 J. BENHABIB AND K. NISHIMURA

satisfies second order conditions to assure strict quasi-concavity.7

Define the following matrices:

[M] [uij - p'U0o]

O 1 PI .. Pn

H UOO U01 P . ' UOn

[H] = PI 11,I u,I X .. 111',

Pn Un0 Un I , nn

[M] is one of the matrices defining [V] and [H] is the bordered Hessian of the

utility function. The price of the pure consumption good, co, is set equal to

one. The pure consumption good will be a normal good if the income effect is

positive. From the well known Slutsky equation this requires

- det

- 2=_ Pn Unl... Unn < 0

where IHI denotes det [H] and HI 2 is the minor corresponding to the element of

the first row and second column. (We use minors of [H] with explicit signs,

rather than cofactors.) In general, ci will be a normal good if

.(1-_1)i fH ,i+2 < O 1.. n

where H1,i,2 is the minor corresponding to the element of the first row and

i-th column of [H].

DEFINITION: A good is defined as weakly normal if I)i+OHl,i+2
A good is defined as neutral if the equality holds.

Consider now the matrix [M]. From a theorem on the determinants of par-

titioned matrices, we have

Uo ...X** Uo,

det [M] = det [uij- 'uo.] = detK u0 , j

(see Graybill [1969], page 165). But note that the matrix on the right hand side

I Second order conditions require i-th order principal minors of the bordered Hessian of the

utility function to have sign (-- l)i.

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UNIQUENESS OF STEADY STATES 71

is the same as the matrix defining H1 2. Thus det[M]=H1 2. If the pure

consumption good is normal, H1,2 cannot vanish and has the same sign as IHI.

Therefore [M] must be non-singular.

Let us now turn to the matrix [V]. Let the column vector e=(u.0-p'uOO).

We can write [V] as follows:

[V] = [[M] - ed]( )

Using once again the theorem oni the determinants of partitioned matrices, we

obtain

det [V] = det K ) = (det [M]W)( - dM-'e)(--).

In the following theorem we relate the normality condition on the utility function

to the non-vanishing Jacobian condition.

THEOREM 2.

i) Let the pure consumption good be normnal and all othier goods be weaklv

normal at the steady state. Thien, under (AI)-(A4), tiie Jacobian [J] is non-

singular at the steady state.

ii) (Brock) Let tile utility function be suchI that the above normality con-

dition holds for all income levels (the income level is defined as Xpici). Then

under (A I)-(A4) the steadY state is unique.

Proof of part i).

The non-singularity of the Jacobian [J] was shown to depend on the non-sin-

gularity of [V]. But we have

det [V] = (det [M]) (1 -dM-Ie) J_

where [M] was shown to be non-singular under the assumption of normality of

the pure consumption good. We must show, then, that (1 -dM-1e)#O if all

capital goods are weakly normal. We will show, in fact, that (I -dM-1e)>O.

We refer to all goods other than the pure consumption good as capital goods.

First note that the vector d is non-negative. It is obvious from the inspection

of Figure II that this fact is crucial, since it makes the consumption possibility

frontier have a non-positive slope. Thus, to show that (I - dM Ie) is positive,

it is sufficient to show that M-'e is non-positive.

LEMMA. If the pure consumption good is normal and capital goods are

weakly normal, then

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72 J. BENHABIB AND K. NISHIMURA

H1,3

H1 ,2

HI,2

M-le= 1-)1;+lH,i+2 < O i -,. n.

HI,2

H1,2

PROOF. The non-positivity of the right hand vector follows because H1H,3

- H1N41. (-_ l)I1H 1,+29 are all non-negative and represent the income

IHI ' I'HI H2

effects on weakly normal goods. is strictly negative since the pure con-

sumption good is normal rather than just weakly normal. The ratio of each

of the above to I-H I is the right hand vector in the lemma and therefore non-

positive.8 We must show, then, that the equality in the lemma holds. We

have

U11 - PU01 U12 P-U02 Uln PlUOn

_Un1 PnUOU U1n2O2- . Unn PnUOn

M-1 det M] (adjoint [M])

NI ...(- 1)n+1Nl -

Adjoint [M]=

- I)n"+Nn, ..(1)2nNnn.

The element (-1)i+iNij is obtained in the following way: delete the j-th row

and the i-th column of the matrix [M], take the determinant of the resulting

matrix and multiply by (-l)i+j. To make the proof easier to follow we write

the matrices appearing in adjoint [M] explicitly:

U22 P2Uo2 U2n - P2UOn

Un2 - PnUO2 Unn U PnUOn

U12 PlUO2 Un - PIUO,

N1"n

un- 1,.2 -Pn-1U02 "' Un- l,, PnlPn- lUOn

[ U21 -p2U01 U2,n- 1 - P2Uo,n- I

Unl - PnUOl ... Un,n-I1 - PnUO,n- I

8 We conjecture that the normality, rather than the weak normality of the pure consumption

good is not a restriction. Since at least one good must be normal, that good, rather than the

pure consumption good, could be chosen as the numeraire. A proof, however, would probably

be cumbersome.

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UNIQUENESS OF STEADY STATES 73

U11 -UpPlUol . Ul,n- -PlU0,n-

Nnn =

L Un-11 Pn- U01U" Un-l,n-1 Pn- IUO,n-1 .

We note the following:

M-e det tM [adjoint [M e =e [adjoint

dt[M] dt[M] U20 P2UOO

Uno -PnUO .

The first element of M-le, then, is a weighted sum of determinants: M

Idet 1 Y(-t)+iNljej. The k-th element of M-le is given by Mk=

det[M] (_l)k+iNkiei. The weighted sum of the determinants, however,

is equivalent to the determinant of a matrix larger by one row and one column.

Consider Mk. Observation shows that

Mk =det [M] l)k+iNkiei;

X(-1)k+iNkiei =

U0O PlUo0 U1 - P1U01 U1,k-1 PIUo,k-1

Ul,k+l PIUo,k+l ... Uln PlUOn

(_ 1)k+Idet I

UnO- PnUoo Unl - PnUO . Un,k-l PnUO,k-I

Un,k+I - PnUO,k+ 1 ' Unn PnUOn j

Denote the matrix above, on the right, as [Qk]. Note that the first column of the

matrix [Qk] is e. We obtained [Qk] by deleting the k-th column of [M] and

adding e as the first column. Careful inspection will show that evaluating its

determinant by expansion, using elements of its first column e, and multiplying

by (-1)k+1 yields exactly (-1)k+Nkiei. Multiplying by (- 1)k+ is necessary

because we place e as the first column and always expand by the first column

whereas minors of the transpose of [M], the Nij's, acquire signs depending on

column as well as row positions. For example, we can write M1 as follows:

)2 UIO -PUOO U12 -PlU02 .. Uln -PlUOn

1 det M L UnO PIUOO Un2 - PnUO2 a Unn- Pnuon J

The above matrix, [Q1], is obtained by replacing the first column of [M] by e.

We need one more operation on Mk before we can conclude the proof of the

Lemma. Using the theorem on determinants of partitioned matrices that we

used earlier (Graybill [1969], page 165), we can express the determinant of [Qk]

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74 J. BENHABIB AND K. NISHIMURA

as follows:9

1 UOO tIO,k- IUO,k+ 1-0O

det [Qk] - det Pi u I 0 It 1,k- U 1,k+ 1 U I n

Pn UnO. Un,k- 1 Un,k+ I Urnn

The determinanit on the right in the above equation is exactly Hl k+2. Fur-

thermore, as we have shown earlier, det[M]=Hl 2#0. Thus we can express

Mk as

Mfk = --('' H ) H1,k+2

H1,2

where M-'e M,

M2

Mn

This concludes the proof of the lemma. The first part of the theorem trivially

follows since with M-'e?O, (I -dM-e)>O and det[V]=(- 1)(det[M])(l -

dM-'e)#O. Since the non-singularity of [J] depends on the non-singularity of

[V] the first part of the theorem is proved.

Proof of part (ii). If the normality conidition holds for all income levels,

(1-dM-1e) is always positive. Similarly, det [M] = H 1 is non-zero since the

pure consumption good is normal. Finally the sign of

Ldet G = det [I -rB] det r a v det [I - gB']; B -- j

must be uniform. B consists of the elements of the input coefficient matrix

which are uniquely determined by r, irrespective of whether the steady state solu-

tion is unique or not. [I - rB] and [I-gB] were shown to be non-singular

earlier, and given that r uniquely determines B, the signs of their determinants

must also be determined by r. | , as shown before, is negative-definite and

the sign of its determinant depends on its dimension, i.e., on the number of capital

goods. Thus

det [J] = det L? jG det [1V]

must have a determinate uniform sign for given r.

Partitioning the matrix on the right and labelling as [ .a where [c] is (n x n), a is (I x n),

his (n x 1), we hiave det_[ _ajdet [c--ba]. Btit [c--ba] is exactly [Qk]-

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UNIQUENESS OF STEADY STATES 75

The steady state solution for r = g can easily be shown to be unique, irrespective

of whether the goods are normal or not. Using the steady state condition, w = rp,

we can write steady state prices as

(pO, p)= r(po, p)A4 + ao.;

(Po, P)= a .[I - rAL.

Here A, as before, is the matrix r ?0 ] and the wage rate is set equal to one.

La.ol IA wg aei e qaooe

The bar over p indicates prices are normalized by the wage rate. When r=g

we can express the labor constraint as follows:

ao. [I - gA1[ coH (Po, P)C co 1

This implies that the consumption possibility frontier, expressed by the labor

constraint, is tangent to the contours of the utility function. To see this consider

Max U(co, 1c) subject to (po,P)Lco ]= ]C

Co, IC

The solution to the above problem satisfies all the steady state conditions, wi= rpi,

yi= gki, pi - -Y- and the labor constraint. If the utility function satisfies second

U0

order conditions, i.e., has strictly concave contours, the steady state solution is

unique.

We have shown, then, that det [J] is non-singular, that it is of uniform sign for

given r, and that the steady state solution is unique for r = g. All requirements

of Theorem 1 are satisfied. Then the steady state must be unique for given r.

Q.E.D.

Part (ii) of the above Theorem is intended to establish the relation between the

non-vanishing Jacobian condition and the normality condition which leads to

the uniqueness of the steady state. We can exploit this relation to establish a

condition weaker than the normality condition.

THEOREM 3.

i) Let the pure consumption good be non-neutral at a steady state solution

for given r. Then the Jacobian evaluated at that steady state solution is non-

singular if and only if (1-dM-1e) #O.

ii) Let (1-dM-1e) be of uniform sign at steady state conditions corre-

sponding to given r and let the pure consumption good be non-neutral for all

income levels. Then the steady state, for given r, is unique.10

Proof of part (i). This follows trivially from Theorem 2. The non-neutrality

10 We would like to emphasize again that the normality condition on the utility function

implies (1 -dM-le)>O, but not vice versa, as we have shown in Theorem 2. Thus (I -dM-Ie)

*0 is weaker than the normality condition.

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76 J. BENHABIB AND K. NISHIMURA

of the pure consumption good at the steady state solution assures that det [M]

is non-zero. Since L-OG_ was shown to be non-singular and (1 -dM-'e) is

non-zero by assumption, det [J], defined as

det [J] = det L 'ldet [ [M] ( - dM- I e)]( Jl)

must be non-singular.

Proof of part (ii). If the consumption good is non-neutral for all income levels

det [M] (=H1,2) cannot change sign. It remains either positive or negative and

must have uniform sign. It was showni in the proof of part (ii) of Theorem 2

that the steady state solution is unique for r=g, irrespective of whether all goods

are normal or not. We now show, without using normality assumptions, that for

r=g the Jacobian J is non-singular. Note that

(PO, p) = ao.[I - gA] = (ao, a).

Hence we have

(d) (P) = (p)=

Consider now dM-'e:

From the lemma in the proof of Theorem 2 we have, since (d) = (p),

H1 _3 L )i+1pI i+2

dM-1e = p Hi

H1,)i+lIli+2

H, ,2

(_)n+l 11, sn+2

H1,2

i- I,..., n.

Now consider the determinant of the bordered Hessian of the utility function:

IHI=detl 0 1 P, Pn

1 U00 U0' U0n

P, 2inO ... . U,n

Evaluating by expansion, using elements of the first row, we get

- H1,2 + (-l)i+IpjHji+2 = HI.

Dividing by H1 2 and rearranging,

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UNIQUENESS OF STEADY STATES 77

(-l)1+'piHlHi+2 __ IH 1

H1,2 H1,2

Thus (1-dM-le)= 1- -1___ Hi #0 The last inequality follows

from the assumption that the consumption good is non-neutral.

We have det[J]=det[_0] det[M](1-dM-1e)-L-. Det[ _-] is non-zero

and of uniform sign as shown in the proof part (ii) of Theorem 2. Det [M] is

non-zero by the assumption of non-neutrality of the consumption good for all

income levels and therefore cannot change sign. (1-dM-le) is non-zero for

r = g as shown above and non-zero and of uniform sign for given r by assumption.

Furthermore, the steady state is unique for r=g, irrespective of whether goods

are normal or not as shown in the proof of Theorem 2. Hence we can apply

Theorem 1 to establish that the steady state is unique for given r. Q. E. D.

It remains to discuss the economic content of the crucial assumption (1-

dMIe) #0 in the theorem above. From the Lemma in the proof of Theorem 2

we observe that M-le yields the negative of the ratios of the income effects for

capital goods. The elements of the vector d correspond to the ratios of direct

C2

a

0 a so CI

FiGURE IL

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78 J. BENHABIB AND K. NISHIMURA

plus indirect labor inputs used to produce capital goods, to the direct plus indirect

labor inputs used to produce the consumption good. The scalar term AM-'e,

then, is a weighted sum of inacome effects for capital goods in terms of the income

effect for the consumption good. The weights are direct plus indirect or total

labor inputs for capital goods in terms of the total labor inpuLt for the consump-

tion good. Consider, for example, the case where the i-th good is strongly

inferior. The i-th element of M-'e will be positive and large. If, however, the

i-th good is produced with a total labor input that is small relative to that of the

consumption good, the i-th elemenit of 1I will be small. Thus, the term (I -

dM-le) may remain positive in spite of the presence of strong inferiority.

We showed that for r=g the steady state is unique and that the ratios of total

labor inputs given by the vector d become equal to the price vector p, where prices

lhave been normalized by the price of the consumption good. Consider in-

creasing r above g. Input coefficients will change as r changes. This will cause

changes in the consumption possibility surface and, since prices will also change,

in the shape of the Engel curve. In terms of Figure 1I, a change in r twists both

aa and OB. Depending on relative factor intensities and the shape of the

isoquant map we may move away from or closer to the possibility of multiple

solutions.

4. THE JOINT PRODUCTION CASE

In the case of joint production, the normality theorem depends only on the

availability of a suitable non-substitution theorem. A non-substitution theorem

when joint production exists was conjectured by Samuelson [1967], and proved

by Mirrlees [1969]. Brock [1973] conjectured that a normality condition is

sufficient for uniqueniess even wlhen joinlt productioni exists. We will see how this

case can be managed following McKenzie's unpublished proof of a nlon-sub-

stitution theorem [1973]. 1

We cannot consider the activities yielding one unit of output under joint pro-

duction. Accordingly, we will use activities combined with one unit of labor.

Suppose that we have n strictly concave production functions represented by

I =fi(v1. y'n, kl,..., kn) J = 1,..., ii.

The left hand side represents the unit labor input. We may draw a production

possibility locus for each j. Let k = 0. Then

I =fi(c I + gk.cn + gkn, k. kn)

where c = (c, ..., cn) is determined, given k = (k1,..., k,) through the produc-

tion function. So each production function fi gives a set of feasible variables

(k, c). Call this vector, (k, c), an activity. Assume free disposal for outputs

and inputs. k is bounded above through y and fi. So activities (c, k) can be

11 See also McKenzie [1963].

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UNIQUENESS OF STEADY STATES 79

i~~ \

FI2R IV

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80 J. BENHABIB AND K. NISHIMURA

considered to be chosen from a convex, compact, non-empty set Di in R2 .

Then Bi = {c - (r - g)k I(c, k) e DJ} is also compact, convex and non-empty.

Suppose that the convex combination of all Bi's has non-empty intersection

with the strictly positive orthant of RI, and the convex combination of any

n - 1 Bi's has an empty intersection with the non-negative orthant. Then we

can apply the usual argument of the non-substitution theorem to subsets Bl,...,

nn

Bn of Rn, i.e., if E is a set of efficiet points of { E2ac bil , j C 1, ocj 2O, bi E B1

j=1 j=1

j .=. ., n}, and E=En {xER"Ix O}, then E is represented by a linear com-

bination of columns of a matrix B=(b,. .., bn) where bi= J-(r - g)ki.

n

where 1 Xj= 1, c is a final demand.

j=1

-3 B1 Z

(1,..., l)x' = 1.

This gives the production possibility frontier. The rest of the argument is

the same as that used by Brock [1973]. Figures IIl and IV borrowed from

McKenzie [1973] show the idea of the above argument.

5. FINAL REMARKS

a. Uniqueness theorems in Section 2 also hold for the joint-production model.

However these theorems are true for the economy with the very general pro-

duction function

Co = T(kl ,... kmi k1,..., km, C1,.., cn).

This case is reduced to Liviatan-Samuelson's example [1969] when m = 1, n= 1.

We can easily check that in their example of multiple equilibria, the uniform

sign condition is violated.

b. It is possible to modify our model in Section 3 to allow for pure capital

goods or pure consumption goods. If, for instance, we have n - m pure con-

sumption goods our equations for the steady state become

pi(c1 + gk,, C2 + gk2,..., Cm + gkm, Cm+l ,- Cnj1 kl,..., ki)

- Vi(Ci,..., Cm, Cm+l,-- cCn)O =

wi(c1 + gk1,..., Cm + gkm, Cm+t X... Cnj, kI,.X, km) - pi(c1 + gk,.

Cm + gkm, Cm+l, , Cnj, kl,..., km) = ?

yi= c + gki ,m

Yi=c i-m +,.. ,n.

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UNIQUENESS OF STEADY STATES 81

There are n pi's, m wi's, m k,'s, and n Yi'S.

Let us introduce the following notation:

[OplOy] is an (n x n) matrix; [Optm/y] will be an (m x n) matrix where only the

first m prices, those that are the prices of goods used in production as well, are

included. Similarly [Op/Oym] will be an (n x m) matrix where only the first m

outputs, those used in production also, are included. We also use [OwlOym]

etc., in the same fashion. Our Jacobian will be (n + m) by (n + m):

.. P ] - [ ][ y k............ ..... .. . .... --------------------- ........... . ..... .. . .. . _ ._ . ... .......... ...... t........ r " [V + g I - r y[Opiny II

[y] [OG;7] Ly k ym ] k y

The following equations will hold:

[Op/Ok] = [Op/Ow] [Ow/Ok] = - [OwlOy]'

(n x m) (n x m) (m x m) (n x m)

[Op/Oy] = [op/ow] [lw/ay] = - [Op/w] [Ow/Ok] [Op/Ow]'

(nxn) (nxm) (mxn) (nxm) (mxm) (mxn)

[Opr/Oy] - [Opm/Ow] [OwlOy] = - [0pm/Ow] [3w/Ok] [Op/Ow]'

(mxn) (mxm) (mxn) (mxm) (mxm) (mxn)

[op/Oym] = [Op/Ow] [Ow/oym] = - [Op/Ow] [Ow/Ok] [0pm/Ow]'

(n xm) (n Xm) (mxm) (mxm)

[apm/Oym] = -[Opm/Ow] [Ow/Ok] [Opm/Ow]'.

These matrix equations are not different from those used in our one pure con-

sumption good model; they are adjusted for the changes in dimension which only

complicate notation. We can still re-write the Jacobian as

J _V r-[ap/Ow] [Ow/Ok][Op/Ow]' - [V]

L -[Im - r[Opml/w]] [Ow/ak] [Op/Ow]' [

[Op/Ow] [Ow/Ok] [Im - g[pm/Ow]']

[I-r[Opm/Ow]][Ow/lk] [I-g [Opm/Ow]]|

As in the original case det [J] simplifies:

det [J] = - det [V] det L0%I where

i Ek rL 0w ][Ak g Ow

To simplify, we use the same theorem on partitioned matrices as we used in our

one pure consumption good model above. Note, however, that [V] and [OG/ek]

are of different dimensions.

University of Southern California, U.S.A.

Tokyo Metropolitan University, Japan

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82 J. BENHABIB AND K. NISHIMURA

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