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INTERNATIONAL ECONOMIC REVIEW

Vol. 22, No. 2, June, 1981

STABILITY OF EQUILIBRIUM IN DYNAMIC

MODELS OF CAPITAL THEORY*

BY JESS BENHABIB AND KAZUO NISHIMURA 1,2

1. INTRODUCTION

In this paper we study the stability of equilibrium in models of capital theory.

We first give alternative proofs of several local stability results obtained by Magill

[1977] without making use of the reduced form for the equations of motion.

This allows us to dispense with assumptions requiring the existence of the optimal

path and the value function or the differentiability of the value function. (See

Section 3). We then clarify the restrictions on technology and preferences

required for the principal stability condition (the Ka condition) to hold. This

stability condition has been widely used in the literature. (See for example

Brock and Scheinkman [1976], Magill [1977], Brock [1978], [1979], Brock and

Magill [1979]). Our results serve to clarify the restrictions that must be imposed

at the individual firm level in order for the K6 condition to be satisfied. Particular

restrictions discussed in Section 3.3 involve the degree of non-joint production as

well as returns to scale properties at the firm level. An understanding of these

conditions is of importance in attempts to extend dynamic capital theory models

to a disaggregated equilibrium framework. (See, for example, Bewley [1979].)

Furthermore, the results that relate the returns to scale and joint production

properties at the firm level to the concavity properties of the aggregate technology

are also of independent interest (see Theorems 5 and 6). The last section of the

paper briefly discusses the relation between capital deepening (the response of

steady state capital stocks to changes in the discount rate) and the stability of

equilibrium.

2. THE MODEL

Consider the following optimal control problem:

(2.1) Max U(z, k)e-tdt

subject to

* Manuscript received November 15, 1979; revised December 24, 1980.

1 This is a revised version of the paper titled "Note on the Saddle-point Stability and

Regularity of a Steady State in Multi-Sector Optimal Growth Models."

2 We thank Professor W. A. Brock, E. Burmeister, M. Magill and Mr. 0. Ichioka for valuable

comments. We are heavily indebted to the detailed comments of an anonymous referee. For

any remaining errors we are entirely to blame.

275

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276 J. BENHABIB AND K. NISHIMURA

Z dk t k, (z, k)LE B, k(O) = ko

di

where B is a compact and convex subset of Rm x Rn and R+ = {x e Rn/x 0}.

U: B-*RO is a concave twice differentiable utility function where t is time, z(t)

is the vector of investment levels, k(t) is the capital stock vector, and 3 is the

discount rate.

We define a modified Hamiltonian system:

HO = U(z, k) + qz.

Let H =maxz HO. Assuming an interior solution, this yields

(2.2) HO = OHO(z, q, k) = 0

az

or

-au + q = 0.

Let 02H0/az2=HOH?. We assume the following:

(Al) Hz is non-singular.3

Under (Al) and by the concavity of HO we obtain, by the implicit function the-

orem, z = z(q, k) where z(q, k) is twice differentiable in (q, k) for z, q, k >0. Thus

we can define H=H(q, k). Applying the Maximum Principle

q - aq _ H(q, k)

Oki

(2.3)

kj= H(qv k1).,n

DEFINITION 1: A non-trivial steady state of a dynamical system (2.3) is a

solution of

aq - aH(q, k) = 0

Ok

(2.4) OaH(q, k) _ 0.

aq

Denote it (k(b), q(6)). The matrices defined below will be used throughout the

paper. The Jacobian matrix of equations (2.4) will be

bi - Hkq Hkk

Hqq Hqk

where

3 If (Al) is not satisfied the control vector z might not be uniquely determined, from the maxi-

mization of the Hamiltonian. In that case the function H(q, k) would not be well-defined.


STABILITY OF EQUILIBRIUM 277

___ F J 2H 1

Hqq = and Hqk =Hkq.

The Kb-matrix is defined as follows:

Hk k 21

-_a_- I Hqq

The matrices below were used by Magill [1977] to study the local stability of the

reduced system which is derived by using properties of the value function.4

R =qkH' 1 + H-jHqk

Ma - (6I - Hkq)'H-J + Hkk(6I - Hkq)

We note that -Hkk and Hqq are positive semi-definite (see Brock [1978]). We

assume the following:

(A2) Hkk is non-singular.

In the following sections we study the local stability properties of the steady

states and relate them to technology.

3. THE STABILITY OF A STEADY STATE

3.1 Local saddle point stability.

DEFINITION 2. The matrix A is positive (negative) quasi-definitive if A+A'

is positive (negative) definite.

I We can reduce the dynamical system (2.1) in the 2n dimensional space to a system in the

n-dimensional space. Let W(k,) be a value function derived by solving (2.1), that is

W(k,) =M- Max5 e -s U(k(s), k(s))ds

k (s)

s.t. k(t)=k,

(k(s), k(s)) E B.

Assume the value function is twice differentiable. Since q(t)=aW(k,)/ak and Wkk =a I W/ak2

is negative semi-definite when U is concave and B is convex, we obtain the reduced system

k= aH (q(kt), k(t))

aq

or

4=q_ 8aH (q, k(q)).

The latter is obtained by solving q = Wk(k) for k provided Wkk is nonsingular.



THEOREM 1. Let Aj, j= l,..., 2n, be eigenvalues of J(3) and R(Aj), j= ,..., 2n,

be real parts of 21. If one of the following holds, then R(,j) is positive for n

eigenvalues, and R(Xj) is negative for the remaining n eigenvalues.

(i) = O.

(ii) Hqk = [].

(iii) K6 is positive definite.5

(iv) HqkHjJk is positive quasi-definite and (6I-Hkq)Hj is positive quasi-

definite.

(v) HqkH-k is negative quasi-definite and (6I-Hkq)H-1 is negative quasi-

definite.

(i) is well-known. (ii) was used by Scheinkman [1978] to prove the global

convergence of optimal solution paths to a steady state. The first part of con-

dition (iv) implies that Ma is negative-definite and the second part of (v) implies

that R is positive definite. Both of these conditions have been discussed by

Magill [1977, p. 187]. However we do not use the valuefunction or assume its

differentiability (see footnote 4). Thus to study convergence to the steady

state we do not have to assume the existence of an optimal path. In fact our

conditions can be used to establish the existence of an optimal path since con-

vergence to the steady state assures that the usual transversality conditions hold

for that path and it is indeed optimal. Condition (v) above is a new stability

result.

We will now prove two lemmas which will be used in the proof of Theorem 1.

LEMMA 1. -Hkk and Hqq are positive definite.

PROOF. Since -Hkk is positive semidefinite and by (A2) it is non-singular, it

must be positive-definite. We prove the positive-definiteness of Hqq. From

(2.2) we have Hzzdz+H Hqdq=O for fixed k. Under (Al) this yields [dz/dq]=

(H?z)-IHOq Using (2.2) we find HOq =I. Calculating Hq from the Hamiltonian

we obtain

Hqq = AZ = H0 -'HO = HO -1

Hq q Oq Zz zq Z Z

Since H2 -' is non-singular by Al, Hq is positive definite. Q. E. D.

The crucial result used in the proof of theorem is the following:

LEMMA 2. Let C=AB where A is symmetric and B+B' is positive definite.

Let c+, cO, c denote the number of positive, vanishing, and negative real parts

of eigenvalues of C and let ad, ao, a_ denote the number of positive, vanishing,

and negative eigenvalues of A. The co =a+, co=ao, and c =a.

PROOF. See Theorem 5 in Wielandt [1973]. Q. E. D.

I Note that positive definite kI implies positive definite -Hkk and Hqq.



PROOF OF THEOREM 1. (i) This is the special case of (iii).

(ii) For a proof see Scheinkman [1978].

(iii) Let B(b) = J(6) 0L I Then B + B' = 2Q(b) is positive definite. B [3 Oj-

J(6) also holds. Since det AI - 0 I is reduced to (A + 1)n(Q -)n, the theorem

follows from Lemma 2.

(iv) Let B= ,(I IHk7)H~ q Hql HjIand A= CH9 0] Then B+B' is

positive definite and A has n positive eigenvalues and n negative eigenvalues.

Since BA = J(5), (ii) is proved.

(v) B defined in (iv) becomes negative quasi-definite. The same argument

as (iv) is applied to (- B), (- A) and J; then the theorem follows from (- B) (- A)

=J. Q.E.D.

Even if we show the existence of the stable manifold with dimension n, this

does not directly imply the existence of a price path corresponding to any capital

stock even in the neighborhood of a steady state (see McKenzie [1963] and

Scheinkman [1976], p. 20). However (Al) above, which implies the non-

singularity of Hqq, assures the existence of such prices.

Condition (iii) of Theorem 1 is independent of the Hqk matrix and is always

satisfied for sufficiently small S if Hqq and Hkk are non-singular. On the other

hand condition (ii) is given only in terms of Hqk and is independent of 6. Under

the condition given below we obtain a theorem establishing the dependence of

local stability on the matrix Hqk and suggesting the existence of a wide class of

problems with a unique, totally unstable steady state. Note that the following

condition (B) is satisfied in problems with a quadratic Hamiltonian:

(B) There exist positive values K, e and 3 independently of 6 e [3, 0o] such

that the following relations are satisfied at steady states:

IHqiqjj K HkikJI< K, j=1,...,n.

THEOREM 2. Assume JR(f3j)I? > 0for 6 larger than some 6. Let R(flj) be

the real parts of the eigenvalues of Hqk. If condition (B) is satisfied and R(flj)>O

for j=1,..., s, then J(6) has na+s eigenvalues with positive real parts and ne-s

eigenvalues with negative real parts for 6 sufficiently large.

PROOF.

1 [ -~I- -H k - Hkk

Hqq Hqk

If 6 is sufficiently large, sign J(6) =sign0O I ] The same relation holds for

any of the other principal minors. Consider

f(5) = det [I - J(6)] = A2in + a2n1A2n1? +. + aA + ao.



Let us define the matrix A = [ 7] Coefficients aj are sums of principal

minors of J(6) which are approximated by the principal minors of A. Hence

f(6) may be approximated by g lj=det [21-A]. Solutions of g()=0 are

(1,..., 1 I; fl /3). If R(b) is positive for j=1,. . s then g()=0 has n+s

positive roots and n - s negative roots. Since f() =0 has roots with the same

sign pattern as g(A)=0, the theorem follows. Q. E. D.

The following corollary is immediate:

COROLLARY.6 Let condition (B) and (RXflj)>>0 holdfor 6 larger than some

value 6. If Hqk is negative (positive) quasi-definite at the steady states fr

66 [6, ooj, then (2.2) is locally saddle point stable (totally unstable) for suf-

ficiently large 6.

This corollary gives interesting information about the relation between the Hqk

matrix and the stability of the steady state. If the condition (iv) of Theorem 1

holds, then R=HkHI~ + H4Hqk is positive definite. This is never true if Hq

is negative quasi-definite (since H-4 is positive definite.) The positive definiteness

of R would be more likely to hold when H,, is positive quasi-definite. We suspect

that if Hqk is positive quasi-definite the system is likely to be saddle point stable

when the discount rate is small, and if Hqk is negative quasi-definite, it is likely to

saddle-point stable if the discount rate is large.

3.2 Implications of the K6-condition. The K6-condition has been widely

used to establish stability properties of the optimal path. McKenzie [1976] and

Magill [1977] have shown the relation of the K-condition to the value-loss

approach for the study of stability. However, the implications of the K6-condition

for technology and the utility function have not been fully established. In order

to clarify the structure that the K-condition imposes on the utility function and

technology, we reformulate the problem 2.1 so that the production technology is

explicitly introduced in the disaggregated form. Note that the formulation below

allows joint-production.

(3. 1) Max e3'U(co(t), c(t))dt

subject to (with time subscripts suppressed):

Lo f0(Yooj Y1}**_) Adz1 K10 .. 1 KnO)

Ls ? fs(Yn,, y1,n*, Yn, K1I ,... Knn)

j=0

6 See Benhabib and Nishimura [19791 for a construction of a unique totally unstable steady

state and the limit cycle.



Kij < kil }

S

I Yu Yi, 01 11 ... ,

ki yjl - gki -ciZs*'

J=o Co, =0,l.,,

y0 =C0,

k() = k.

fi is the J-th production function that relates the minimum amounts of labor

required to produce the outputs (Yoj, Yj,..., Y,,j) using the capital stocks

(K1J,.,, Kn1). g is the depreciation plus population growth rate, .y is a pure

consumption good and Yl*,,, y,n are assumed to be either pure capital goods or

consumable capital goods. Hence y, k e Rn = {x e Rn I x ?0 ) and c e RI', where

rn!n. If i>m, c=O0. 6 is the discount rate where 6>g?0. Notice that in

the above formulation we assume free disposal. Note also that total labor is

already normalized to be one.

The disaggregated form of production sets in (3.1) may be rewritten using the

social production function, co= T(y, k). The social production function may be

obtained as a maximum value of co given each (y, k) as follows:

Max Z Itj

i=1

subject to Li > fi(Y01,,,,, Y4 K11..., K,,), j = 0..., s

S

j=0

S

E Kijf : ki** n*

j=0

j=o

Then the problem (3.1) is equivalent to the following*

Max 5 -&, U(c(t), c(t))dt

(3.2) subject to

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

k(t) yt) - gk(t) - c.

U: R?++' 1R+ is assumed to be a concave twice differentiable utility function

where t is time, k(t) is the capital stock vector, y(t) is the output vector, (c0(t), c(t))

is the vector of consumption levels. T(y, k)It R x Rn is also assumed to

concave, twice differentiable function that describes the efficient production sur-

face of the technology. (For a long but straightforward proof of the differentia-



bility of T(y, k) under nonjoint production see Benhabib and Nishimura [1979,

appendix 11]. The same method of proof can be used in the joint-production

case.) In what follows we will exploit the restrictions imposed on T(y, k) by the

structure of technology. All variables are in per capita terms.

We define a modified 11amIltonian system:

(3.3) HO - U(T(y, k), c) + q(y - gk - c).

Let H max HO where z (c, y). Assuming an interior solution, this yields

(3.4) HO - aIf?(z, q, k) o

au a z

CCo ay, i

au

( qi j = 0.. ~

Let a2H01/z2 = We assume the following:

(Al)* H?Z is non-singular.

We have to modify the proof of Lemma I in order to guarantee the non-

singularity of Hq,. We have, from the proof of Lemma 1, [dz/dy] = (HO,)I HOq.

Using (3.4) we find HOq ll On-":t where In is the (n x n) identity matrix and

On__ is the ((m) x (n - m)) zero matrix. Calculating Hqq from the Hamiltonian we

obtain H: acl/q + ay/Oq 7 K IK L -d ] L j [Hfj - I L - It j

n-tn dq tO-OJ n

To see that Hqq is non-singular note that the (n + m)-vector s = LI?2nix #0

qq I~~~~~~~~~~~~~~~~~~n

for any n-vector x#00. Thus x'Hqqx=s'[HLH i's> for any x O since [HOJI I

is negative-definite by the concavity of HO and (A1)*.

THEOREM 3. Let (At)* hold. If the K6-matrix is positive-definite then if

either

(i) the utility function is strictly concave, or

(ii) UO0OO and the utility function is concave,

the Hessian of the social transformation function T(y, k) has at least rank

PROOF OF THEOEM 3.

(i) Suppose the K6-matrix is positive definite. Then we can show that -Hkk

and Hqq must be positive definite, To see this suppose I1kk is positive semi-definite.

Consider -4 J =Ez2XiHX -X21HkkX2+bXX2. Choose x2 such



that Hkkx2=O and x'x2<O. Then for 3?0 and sufficiently small lei, Lex',

-- x] Q ?0 X2 < . This contradicts the positive-definiteness of Q. The argu-

ment is the same for Hq.

(ii) We now consider the matrix Hkk. Since H? =O along the optimal path,

Hdz+H zkdk=O for fixed q. By assumption (Al)*, H?_ is non-singular. Thus

[az/ak] =(Hz)-'H9k. We also h kave Hk=aHO(z(q, k), k, q)/ak=H Hz[Oz/ak] +

HOk. Substituting for [az/ak] we obtain Hkk =Hkk-HOZHz?- 'Hok. Let the

Hessian of the Hamilton ian be M-= KH0z JJk7 and let P'= [ ...I.. 07?. |

Tliceti

0 -H''z L) HH'K H I

- P'MP L -HkH.HgHk L H

0 kk + HO HOZ 0z g Hkk

Since H?- is non-singular and has rank m+n by assumption (Al*), M has rank

in + 2n-G if and only if Hkk has rank n-G.

(iii) We now establish the relationship of M to technology and the utility

function.7 Let

M HZz Hz~k 7 - UCUOP UcOW" r Uo o -

HO 0~~~I

HZH 0 -ap/aj; -ap/ak

,. . . . . . . . . . . ... . . . . . . . . . . . . . .'- -- . . . . . . . . . . . . . . . . . . . . . . . - - - - . . . .

WUoc:UoowVp':U0O1vw'- _ wlayJ aW/ak

where p= -aT/0yi, wi- T/1ki and Uij 02 UIOCiCj. If U00#0, M can be

expressed as

L0c U U-PI W

UUaw/ay Uaw/ak j

The first matrix Oll the right hand side has rank one while thae ranlk of the second

depends onl the r-anks of the two diagonal matrices. =C[Uc - (1/ U0) UoUco]

I For one of the few early attempts that tries to relate the Ks-condition to technology and

preferences see Magill [1977, Section 6, Example 2].



will have full rank m if the utility function is strictly concave since Det [UOO UOc]

=U00 det [Ucc-(1/UOO)UOcUcO] while the rank of N= FO p/ay -ap/ak7

Lawlay aw/akj

depends on the degree of joint production (see Burmeister and Turnovsky [1971])

and the returns to scale properties of technology (see the discussion following

Theorem 4). Then the rank of M cannot exceed the rank of & plus the rank of

N plus one.

If U00=0, M can be written as

r C U U0CPpUoc', ' U000

M - = | --------,,, ----- ---------- + U 0 -------- -

p luco, l

and it is easily shown that the rank of M is at most the rank of N plus the rank

UCC plus one, since [- Uocp'] and [U0cw'] have rank one.8

If the Ka condition holds we have shown that Hkk must have full rank n. This

implies that M has rank m+2n. From the above discussion this implies that

the Hessian of T(y, k), given by N must have at least rank 2n - I. Q. E. D.

The result of Theorem 3 can be stated independently of the optimal growth

context. Under the assumption that HO, is non-singular, we see from the proof

of parts (i) and (ii) of Theorem 3 that the K"-matrix is positive definite if M=

rH? HHk7(and therefore HOk) has full rank. Otherwise Hkk does not have full

rank and part (i) of the proof of Theorem 3 shows that the K6-condition cannot

hold. On the other hand if HO does not have full-rank the function H(q, k)=

H(z(q, k), k), and therefore the KI-matrix, is not well-defined.

It may also be useful to derive an expression for the Ks-condition in terms of

the Hessian of the pre-Hamiltonian with respect to the state and control variables,

that is, in terms of the Hessian of the integrand of the general optimal control

problem (2.1). Using the Theorem below we may directly test for the Kb-con-

dition without having to express the Hamiltonian in terms of the state and costate

variables only. Note that the resLIlt of this Theorem applies to the general

optimal control problem (2.1) and not only to the optimal growth problem (3.1)

of this section.

THEOREM 4. The K-condition holds if and only if the matrix

8Consider ZL] UCC:UOC'--UOC j where the vector _.]has zeros for its

wUc 0:

first m elements. Since [-Uocp] and [Uocw] are of rank one there are (2n-1) independent

vectors x such that Z(O/x)=O. This is obvious if one considers the augmented matrix

- U0Cp, U0Ow ] which. can only have one non-zero root. Thus the rank of Z is at most the

rank of Uc, plus one and the rank of M is the rank of U,, plus the rank of N plus one.



HO 0 6Ho

ZZ 0Zk z

kz ki HZ0Z (ohH0k + H2o)

is positive definite.

PROOFS Let Me-THo HO and P be the matrix in the proof of Theorem 3.

Let

H_ 0

0 , -Hk I 0 1

Note that P is non-singular and P't- l oAfz,,z ], Thus Ks is positive

definite if and only if P-"QP' is. But (see proof of Theorem 3)

H~~ H?~~~~ 0 6Ho

Ho 'fO Ho b

H!Z L 2 _ 2

L H?~~k + 6~ H0' (Hozk +Hz)

Q.E.D.

Note that the matrix M is also the Hessian of the integrand U(z, k) of the general

optimal control problem (2.1).

3.3 Discussion of Theorem 3.

a) The case of constant returns to scale. The above theorem shows the

restriction on technology imposed by the Ks-condition. For instance, if all

goods are produced non-jointly tinder constant returns to scale, the Hessian of

T(y, k) of dimension (2n x 2n) has at most rank n (see Samuelson [1966]).

Burmeister and Turnovsky [1971] have defined the degree of joint production,

R, for an economy where goods can be partitioned into R non-overlapping subsets

and where goods in each subset are produced jointly. This results in (n +1)

goods being produced with n +1 factors by R industries. Burmeister and

Turnovsky have shown that if we define the degree of joint production by R,

the Hessian T(y, k) has at most rank 2n +1- R (taking into account that we

normalized by labor). For the K6-condition to hold, our theorem shows that R

must be less than or equal to two.

If all goods were consumable and there were no pure consumption goods,

then the analysis above is somewhat simpler. The modified Hamiltonian given

in (3.3) becomes H=U(c0, c)+q(y-gk-c)+qo(T(y, k)-gko0-c0). The

matrix M in the proof above is also simplified:



M -~~~~0 '-~ U .0..

M=L Oki

where UcC now has dimension (n + 1) by (m +1). The maximal rank of AM is

(in+ 1) if the utility function is strictly concave. For the K'-condition to hold,

the (2n x 2n) lower diagonal matrix of M, which is the Hessian of the Hamiltonian

with respect to (y, k) must have rank 2n, since rank M=rank Ki. But this

means that the Hessian of T(y, k) has rank 2n and the results of Burmeister and

Turnovsky cited in the earlier paragraph imply that R must equal one for the

KR-condition to hold. This result is even more restrictive than the case proved

in Theorem 3. By removing the pure consumption good and introducing a

consumable capital good in its place we may in effect increase the dimension of the

flat. This is because the transformation between the use in consumption and the

use in production of this good is now linear.

b) The case of decreasing returns to scale. When all goods are produced

under decreasing returns to scale it may be shown that the function T(y, k) is

strictly concave. On the other hand, if only two or more goods are produced

non-jointly under constant returns this result fails. We proceed to demonstrate

these results.

(A3) Let (K... Kimjm) be the set of inputs used in the production of good j.

Then good j cannot be produced without Ki i- I,..., m;.

(A4) Each good is produced non-jointly and Hessians of the production

functions are negative-definite.

(A5) For none of the production processes, the marginal products of all

the factors are simultaneously non-positive.

Assumption (A3) rules out corner solutions for the input coefficients for given

factor prices and specified output. Note that it does not require all factor inputs

to be necessary for the production of a good but only a subset of them. The

imposition of Inada-type conditions, for example, would assure that this assump-

tion is satisfied. Assumption (A4) implies that goods are produced under de-

creasing returns to scale.

Under (A3), (A4) and (A5) it is shown in the appendix that prices can be written

as p= P(w, y) and [ap/ay] = [10/aw] [OW/aY]k-fixed + [a/aY]w -f ixed The Hessian

of T(y, k) can now be written as

N=L ifyaw 7 raw Jk a y -fixed K- .

Noa I L h s a o -

Now consider pre-multiplying N by the quasi-triangular non-singular matrix



L I --- I ] This yields the quasi-triangular matrix Le /?]0

which is non-singular provided [a/aY]w-fixed and [aw/ak] are non-singular.

[aw/ak] is non-singular under (A3) since for fixed outputs y-, there exists a differ-

entiable inverse function k=k(w, y-) for w(k, Y). To see this note that in the

appendix we show that K1J(w, y)'s are differentiable functions. Given the fixed

output levels, the total input requirements are determined and we obtain the

differentiable functions kj(w, -) for j =0, 1,..., n. [ap/Oy]w-fixed is also non-

singular, as shown in the appendix, provided all goods are produced wit/i de-

creasing returns to scale. Thus the matrix N, which is the Hessian of the concave

function T(y, k), has full rank. We have then the following Theorem:

THEOREM 5. Let (A3), (A4) and (A5) hold. Then the Hessian of T(y, k) has

full rank.

Theorem 5 establishes thefull rank of the Hessian of T(y, k) under non-joint

production and decreasing returns to scale in every industry. We can now

sharpen the above theorem by establishing another result. We will show that if

two or more goods are produced non-jointly the Hessian of T(y, k) cannot have

full-rank, no matter how other goods are produced. We therefore replace

assumption (A4) by (B4) which requires non-joint production for only a subset

of the goods and assures that the production isoquants are strictly convex to the

origin, so that factor prices uniquely determine the optimal per unit input co-

efficients:

(B4) Let s goods (2?s?n) be produced only non-jointly and let their pro-

duction functions be homogeneous of degree one, twice differentiable and have

bordered Hessians whose principal minors alternate in sign, with the i-th principle

minor having sign (- I)i.

Under assumption (B4) it is well-known that prices, which are equal to unit

costs, are functions of factor prices alone. Let, without loss of generality, the

O-th (the numeraire) good and r-th good be produced under constant returns.

We have, Po/wo = P0(1, w), PrwO= Pr(1, w) and therefore Pr/P0 = pr(w). Con-

sider now the function Pr(y, k), which is the r-th element of the vector p(y, k).

We have, for all j,

apr = E ( and

(3.5)

aPr =Z Ad)Pr"\t(aWi

ak i\awJ\ak1 J k

Now premultiply the Hessian of T(y, k), the matrix N, by the vector (0,...., 1,..., 0;

apriaWi... apr/aWn) where the number "1" is the r-th element. By equations (3.5)

this yields the null-vector. Furthermore, for every additional good also produced

non-jointly there is another independent vector which yields the null vector when



it pre-multiplies N. Note again that this result is independent of the structure

of production of the remaining goods. In particular we are allowing joint-

production and decreasing returns in the production of those goods. We now

express this result in the following theorem

THEOREM 6. Let (A3), (B4) and (A5) hold* Then the rank of the Hessian of

T(y, k) cannot exceed 2n + -s.

When combined with Theorem 3, the above theorem further specifies the limits

under which the K6-condition can be applied.

c) Number of goods relative to the number of factors. Introducing more

than one pure consumption good does not create problems as far as the matrix

algebra is concerned but since the control variables exceed the state variables, the

controls are not uniquely determined at each instant. Hz, is singular and z

cannot be determined as a function of state and costate variables. In other words

we have more goods than factors and the production possibility frontier, T(y)

with k fixed, has flat surfaces. Even if some selection criteria for choosing

appropriate controls (outputs) were adopted, our preliminary results indicate

that the restrictive joint-production conditions for the K-condition to hold

cannot be relaxed.

4. AN EXAMPLE OF A SADDLE-POINT UNSTABLE ECONOMY THAT

EXHIBITS CAPITAL-DEEPENING

Capital-deepening as defined by Burmeister and Turnovsky [1977] measures

the change in the value of the steady state capital goods vector, evaluated at the

original steady-state prices. Formally an economy exhibits capital-deepening

if the quantity Echo pi(dki/db) is negative. Of course it is always negative for

single capital-good models, Levhari, Liviatan and Luski [19741 investigated

the relation between capital deepening response and the stability of the steady

state, They gave an example of a stable steady state where capital deepening

does not hold, i.e., where YXio pi(dki/db) is positive. An interesting question is

whether capital deepening response, or the stronger assumption that dk1/db<0

for all i, implies saddle-point stability under plausible assumptions.9 Magill and

Scheinkman [1977] have shown that when Hqk is symmetric, capital deepening

response does imply saddle-point stability. However this cannot be true for the

general case. Below we provide an example of an economy that is totally unstable

and where dk/db5<0 for all i.

In an earlier paper, we provided an example of a saddle-point unstable economy

with one consumption good and two capital goods using a Cobb-Douglas tech-

nology and a utility function that is linear at the steady state (see Benhabib and

Nishimura [1979]). The Cobb-Douglas technology that yields the unstable

9 This question was posed by Brock and Burmeister [1976].



steady state for the rate of discount 6 = 0.4 and population growth rate g = 0.1 is

given below.

2

(4.1) = hi IH Kp;

where

Z a 1= 0, 1, 2

j~=0

and

bo= 10.6462 oo =0.8627 age =0.0032 0 *=0.1341

b1= 1.5878 a Q=0.9437 at = 0.0534 120.0029

b2= 3.0555 20 =0.2304 a21 =0.6848 %2-0.0858.

The steady state values, for g = 0.1, are given by

(C, ks, ?2) = (9.9169, 0.06272, 0.7061)

(wO, w1, wI) = (8.6539, 3.2097, 1.8770)

(POv PI, P2) = (1.000, 1.4194, 3.7540)

where c, k1, k2 are the per capita consumption and capital stocks, wo is the wage

rate, W1, w2 are factor rentals, po is the price of the consumption good and Pt, P2

are the prices of capital goods. We normalized by the price of consumption good

and set po=1.

To see capital deepening we first note that

(4.2) Ay + a0c = k

where A is the matrix of capital inputs to the capital goods and a0 is the vector of

capital inputs to the consumption good, y is the vector of outputs of capital

goods and k is the vector of capital stocks. At the steady state, y=gk. Differ-

entiating (4.2) we obtain

dk- (Ig -ap-I g !YA-- k + ca (4.3)dk .(I -gA -6aopt1gLJ)+ i9

(4,3) ~ ~ d6d6 db

where we use the well-known result dc/d6=(r-g)(dk/dJ) (see Burmeister and

Turnovsky [1972]). The values of the elements of dA/dJ and da0/db can be

calculated using the Allen partial elasticities of substitution, oik .' (See Benhabib

and Nishimura [1979, appendix 4]). Tedious calculations yield

(c/1 dk\__

y-dk t -- ( 0.2137 - 2.4089).

I For a Cobb-Douglas technology at I for isk and Act can be calculated from IZKkao

0 where K, is the proportion of the cost of factor k to the total cost of producing good J.



Setting w0= 1, we obtain (pr, P2)=(0.7417, 0.4337) and 6,pi(dkidb)=dcldb

- 0.4866.

Note that in the above example not only do we have capital deepening response

but the property that both steady state capital goods decrease with 6. This

result tells us that asymmetry of Hqk matrix could make the steady state totally

unstable independently of the capital deepening property. (For the symmetric

case see Magill and Scheinkmnan [1979]). This possibility has also been pointed

out by Magill [1979, section 6] in a general context.

New York University, U.S.A.

University of Southern California, U.S.A.

APPENDIX

In this appendix we show that under decreasing returns to scale relative prices

can be expressed as functions (locally) of factor prices and output levels, that is

as p = P(w, y). Furthermore we show that [afP/YLw-fixed is non-singular under

the assumption (A3), (A4) and (A5) given in section 3.3(b).

Let Y, and Kij be the i-th output and the j-th input to produce i-th output.

Y0 is the consumption good and Kio is the labor input to the i-th good. Consider

the following neo-classical technology:

Yi = fi(Kio, Ki,..., Kin), I = 0, 1,..., n.

To obtain the social production function c= T(yI,..., Yn, ko, k1,..., kn) we set

c = Maxf 0(KOO, Ko I,., Ko)

subject to yj = f i(Ko, Kil,..., Kin)

n

ko= E Ki?

i=O

kj = Kij.

We restrict (y, k)=(y1,..., Yt, ko, kl,..., kn) to be non-negative.

First order conditions for the above problem are given as follows:

pJft - wt = 0, i, t = 0, ,..., n

ft Yt = ?, t =0, l, ..... I n

(Zl)

ko- Ki 0,

i=O

ki - Kid .j = , ... n

wic

where po =1. These conditions can be used to derive the differentiable social



production function c0= T(y, k) used throughout the paper (see Benhabib and

Nishimura [1979, appendix II]). Using the implicit function theorem we will

derive the functions P(w, y), Kij(w, y), k(w, y) and y(p, w). To simplify notation

and without loss of generality, we assume that all inputs are required for the

production of every good. Otherwise we would appeal to assumption (A3) and

a reduction in the number of first-order conditions would be required. The

Jacobian of the equations (ZI) with respect to (Kij, p, k) is given by

. '..0.......0 0 0 Po.)~hh-hh- '

., F

R t 0

.0..p,,f F,, 0 R P

Q= .... .. ~~~~~~~~~~~............

0 F . .......... Fn' 0 0 S Z

-I1 ............ -I 0 I

where [f rj] is the Hessian matrix of the production function ft and

= [ ......"0 J 0 . ..07

- ......f. 06 .....

i-1 n 1

If Q is singular, there exists a non-zero vector y'=(yl, Y2, y3) such that Qy=0.

In particular this implies that

(Z2) - 0l

Ry,+1 + Y2=0

- 11

(Z3) [0, F...F FIjyj = 0.

Premultiplying (Z2) by y' and (Z3) by y' and then adding, we obtain y'Ry =0.

This is impossible if all production functions are produced with decreasing returns

to scale since [ftj], t=0, 1..t.i would all be negative definite. Thus Yi=0. But

-0

then the term #1 |Y2 0 if Fi#0 for all i, that is, if none of the production pro-

LF,i

cesses has all the marginal products of its factors equal to zero. This is ruled out

by (A5) in the text. Thus Y2 = 0. But from the last row of Q we then have Iy3 = 0,

which can only hold if y3=0. Therefore Q is non-singular and by the implicit

function theorem there exist differentiable functions k = k(y, w), p = P(y, w) and

Kij = Kij(y, W).

Consider now a variation in y compensated by w so that dk =0. We obtain

dk=[ak/aw]dw + [a0/Dy]dvy=0, or [dWI(Yy]k-fixed=-[ak/aw]-1 [0k2/ay]. The

invertibility of [Ok/aw]y-fixed can be obtained from the existence of differentiable



inverse functions w(k, y) (see Benhabib and Nishimura [1979, appendix II])

since k(w, y) is also differentiable. We can now express -OIT(y, k)/(ay k)=

[Ep/ak])yfixed by using the functions p(w, y) if we make sure that any variation in

y is compensated by w to keep k=[1, kl,..., k,, fixed. (This also normalizes by

labor.) Differentiating, we obtain:

JP 4 OW + _^L

[ ~ k-fixe [@W] [aY- k-fixed + a'.

_ iend _ W - _ rlfixedelfsd

[_ ]yfxed OW Ok i-fixed,

Finally we consider the rank of EOP/OY] w-fixexd Since P(w, y) is differentiable,

if we can show that there exists a differentiable inverse function -(p, w), the

[OP/OY]wfixed cannot vanish. Consider the Jacobian of equations (Zi) with

respect to (Kij, y, k):

R, ~~0

M=LV....j.. .m.. .... t

O F,... F I O

-I ....... -I , I _

Since M is quasi-diagonal, it is non-singular if R and fj j are non-singular.

Thus R must be non-singular. As discussed before, this requires decreasing

returns to scale in every industry.

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