JOUKNAL
OF ECONOMIC
37, 7698
THEORY
( 1985)
The Existence of Optimal in Optimal Economic with Nonconvex R. E. Depurtmen~
of Mathemutkv, Fort Collins.
Consumption Policies Growth Models Technologies
GAINES Colorudo State Colorado 80523
Universiry,
J. K. PETERSON Depurtment
of Mathrmarical and Computer Michigan Technological Universily, Houghton, Michigan 49931
Received
December
I, 1983; revised
February
Sciences,
6, 1985
An existence theorem for a class of continuous time infinite horizon optimal growth models is developed. The underlying technology set is not assumed to be convex, instead the “slices” of the technology set corresponding to a fixed capital stock vector are assumed convex and compact in the consumption and net investment variables. This allows consideration of the case of increasing returns to scale. Existence of an optimal capital stock and consumption policy is proved directly without consideration of the underlying Hamiltonian dynamical system that arises from applying Pontryagin’s maximum principle. Journal of Economic Li/eru/uw Classification Numbers: 110, I1 I. 1 1985 Academic Press, Inc.
In this paper, we consider a problem of optimal economic growth within the general framework outlined by Cass and Shell [6]. That is, we consider the infinite horizon continuous time economic growth model -x
maximize
u(c(t)) e p’ dt,
(0.1)
i 0
subject to
(4th =(t), -k(f)) /G(t)=z(t),
t > 0 a.e.,
k(0) = I?, 76 0022-0531/85 Copyright AlI rlghrr
$3.00
a 1985 by Academic Press, Inc of reproduction in any form reserved
E 5,
t >, 0,
(0.2) (0.3) (0.4)
EXISTENCE
AND
NONCONVEX
TECHNOLOGY
77
where u: [O, ‘a) -+ [O, nj)
is concave and continuous;
CER ke R”
is the consumption flow is the net investment flow is the capital stock
9 c R x R” x R”
is the technology set.
TE R”
and
The maximum is taken over the class of measurable consumption functions, absolutely continuous capital stocks and measurable investments. In order to treat the case of increasing returns to scale, we will not assume .F is a convex set. Instead we will only require that the slices F(k) = ((c, :)I((; z, -k) 1 E .F be nonempty, convex, and compact valued for all k 3 0. We will prove, subject to the assumptions of Section II, that this problem has a solution on [0, GCI). The proof does not rely on any necessary conditions for the infinite horizon model (see, e.g., Halkin [ 141, Brock and Haurie [ 131, and Haurie [ 121). Instead, the existence of an optimal capital stock and consumption plan follows from a direct approach. We show that the set of “admissible” functions k; i.e., the set of functions satisfying the constraints, is relatively compact in the topology of uniform convergence on compact subsets of [0, m). We then establish lower semicontinuity of the Lagrangian functional (3.7) asociated with the problem to obtain existence. The Lagrangian formulation is obtained using the infinite penalty framework of Rockafellar [4, 7, 81. The theory of normal integrands and measurable selections (see Rockafellar [4] and Castaing [5]) plays a vital role in the existence proof. We prove a preliminary global existence result for the differential inclusion .t(t,~G(x(t))
a.e., X(O) = .Y,,
where G(k)=
(:IthereisacERwith(c,:)EF(k)i
using the set-valued topological degree of Webb [I]. The reader should contrast this work with those by Rockafellar [ 111 and Gaines [9, IO] for the convex technology case. The hypotheses imposed in Section II are considerably less restrictive than those imposed in the convex case. However, no information is obtained in the current case concerning the “dual” problem of price behavior. In a sequel we will consider the dual price structure in the nonconvex case.
78
GAINES AND PETERSON I. PRELIMINARIES
We use the following notation: 1. .Y= (.u ,,..., x,,)E R” with 1x1 = (C;!=, $)“‘. For any fixed T, 2. L,,[O, T] = 1-x: [0, T] -+ R”I?c is Lebesque measurable on [0, T]f. 3. Lj,[O, T]={.YEL,,[O, 4.
T]JJ;/x(f)~dt
L,:CO,Z-1= {x&,CO, Tll II.d, -=Im},
where /I.~ll, =esssup((x(r)l IO
on [0, T] i with
/I.K I/ = max ( / .u(t)l IO d t < TJ, C,,= 1-y: [0, a) + R”1.x is continuous on [O, cc)) 6. A,,[O, T] = 1-y: [0, T] -+ R” I .Y is absolutely continuous on [0, T] ] with II .K)IA = 1-u(O)1+ Jg l i( dz, A,, = [.Y: [0, cc) -+ R” 1.Yis absolutely continuous on [0, co) 1.. 7. If A E R” is compact and v E R”, the distance from y to A is defined by (1: A)=min{/.u-yI IxEA). 8. If A, Bc R” are compact, the distance from A to B is defined by d(A, B)=max(max,..,p(y, A), max.., p(x, B)}. 9. If .YE R”. we define for all Y> 0,
If A ER”, we define for all r > 0, B(A;r)=
{y~R”lp(y,
A)
We include certain basic facts and definitions. DEFINITION 1.1. Let X be a Banach space and let I2 c X be open. The multifunction G: 52 + 2’ is upper semicontinuous (USC) at x if for any E> 0 there exists 6(-u, E) > 0 such that
G(B(x; 6(x, E)) c B(G(x); E).
G is USCon Q if G is USCat each x E 52. The definition of USCon d requires obvious modifications. Equivalently, G is USCat XE 52 if for all sequences (.u,,),l-= , and ( y,,),;“= , with x, -+x and y, -+ J’ such that y, E G(x,) for all n, we have .I’ E G(x).
EXISTENCE AND NONCONVEX TECHNOLOGY
79
1.1. Let X he a Banach spaceand D c X he open and bounded.
THEOREM
Assume G: D + 2.’ satisfies
(1. I )
For all .KE Q, G(x) is a closed, convex, and nonempty
subset
of x; (1.2) G is USCon Q; ( 1.3 ) G(D) is relatively compact in X; (1.4) 0&(1-G)(x) for all .YE%~. Then there exists a topological degree, denoted D,,.[I-- G, Q I], for the degree of G relative to Q and 0 that satisfies: (1.5) If OJIG, Q, 0] #O, then there exists .YE Q such that .K E G(x); (1.6) If H:Qx [0, 11-2,’ satisfies (l.l)-(1.3) and 06 I- Ht.3 t)) for all .YE dQ and for all t E [0, 11, then D,,.[I-
(1.7) Proof.
H(., l), Q2,Ol = D,, [I-
H(., O), Q2,0];
D,,.[I, 52, 0] is defined and nonzero if 0~52.
Webb [ 11.
THEOREM 1.2. ( The Dunfard-Pettis criterion ,for relatively weakly compact subsets c?f Lj,[O, T], where 0~ T-C CE is ,fixed.) ScLj,[O, T] is relatively ujeakl?,compact lf and only $for any E-C0 thereis a C?(E) > 0 s&l that {j’ E c [0, T] is a measurable set btith meas < C?(E), then I6 ju(t)l dt
Proof:
Dunford and Schwartz [2]
LEMMA 1.1. Let (.Y,,)c Lb[O, T] und assumethere esists WEL![O, T] such that .r,, -+ II‘ weakly. Then ,for each positive integer j, there are positive integers N,, k, and u,finite set afconstants (a,, ,..., fl,x,) such that /3;,20 for l
Pror$
This is essentially Mazur’s lemma. See Ekeland and Teman [3,
P. 61. DEFINITION
1.2. Let I be an interval of the real line R. Let F: I --+2R”
satisfy ( 1.8)
F(t) is a nonempty
closed subset of R” for each t E I;
80
GAINES AND PETERSON
( 1.9)
For any closed subset C E R”, the set
is a Lebesque measurable subset of 1. Then we say F is a measurablemulttfunction on I. A complete discussion of measurable multifunctions and measurable selections can be found in Rockafellar [4] and Castaing [5]. The sets C in ( 1.9) may be taken to be compact instead of closed if desired, see Rockafellar 14, Proposition IA, pp. 160-1611. A typical selection result can be stated as follows: THEOREM
1.3. Let F: I-+2R’
sati.$~~ ( 1.8). Then, the following
are
eyuivalent: ( 1.10)
F is measurable;
( I. II) (Castaing representation). There is a countable (or finite) where P is the inde.u set, such that firmilr (.Y;lit p c L,,[I], F(t)=cl(x,(t)IiE P}f;7r all tel.
Rockafellar [4, Proposition IB, pp. 161-1631. Note that (1.11) immediately implies there is at least one function .YE L,,[Z] such that x(t) E F(t) for all t E I. In our work, we will require certain smoothness in the optimal control problem. The appropriate amount involves the concept of normal integrands. See Rockafellar [4] for details. Proqf:
DEFINITION
1.3.
W: [0, T] x R” -+ R u { + co} is said to be a normal
integrand if (i) W( t, .) is lower semicontinuous in x for each t E [0, T] (ii ) W( ., ) is measurable with respect to the o-algebra generated by products of Lebesque measurable sets of [0, T] and Bore1 sets in R”. If W satisfies (i), (ii), and (iii) W( t, I) > --CT, for all (t, X) E [0, T] x R” and there is at least one (i, .U) such that W(t, X) < a, we say W is a proper normal integrand. Finally, if W satisfies (i), (ii), and (iv) W(t, ) is convex in x for each t E [0, T], we say W is a convex normal integrand.
81
EXISTENCE AND NONCONVEX TECHNOLOGY II. THE STATEMENT
OF THE PROBLEM
We now consider a problem of optimal economic growth within the general framework outlined by Cass and Shell [6]. That is, we consider the problem SC u(c(r))-“‘lit;
maximize
(2.1)
s0
subject to (c(t), z(t), -k(r))EY
for t 3 0;
(2.2)
R(t)=:(t)
for almost all t 3 0;
(2.3)
k(O)=L, k(r)>,0
for f 3 0,
(2.4)
where II: [o, 1-C)+ [o, a)
is concave and continuous;
c’E R”
denotes consumption
z E R”
denotes the rate of net investment;
k E R”
denotes capital stocks;
.P
denotes the set of all feasible triples (c, Z, k) which we call the technology set.
Note u(L’) = c is a common For fixed k30,
choice in optimal
define F(k)=
or utility;
economic growth models.
((c,:)I(c,:,
--k)~Yl.
We assume
3 is a closed and nonempty subset of R x R” x R”.
(2.5
F(k) is a nonempty, convex, and compact set.
(2.6
F(k) is upper semicontinuous
(2.7
If ,420 and (c.:)~F(k),
for k 3 0.
then 06cand
where d: [0, cc ) + [0, cu) is continuous
c+(:Id&IkI)
(2.8
and satisfies 4(O) = 0.
There is a i E (0, 1) such that for all k 3 0 z + i.k 3 0 if (c: :) E F(k).
(2.9)
82
GAINES AND PETERSON
There is a positive constant N such that if 1k 13 N and k > 0, we have for all (c, z)~F(k), (2.10)
-.k
b
Ii> 0.
(2.11)
Some of the assumptions (2.5)-(2.11) have an economic interpretation which are deserving of comment. By only requiring that the cross sections of ,Y, F(k), be convex for all fixed k b 0, we allow for increasing returns in production, a case suggested by Cass and Shell [6, p. 353 to be “an important area in the future development of economic dynamics.” Condition (2.8) merely says that production of investment and consumption goods is limited by the availability of capital. Condition (2.9) says that net investment cannot be lower than the loss due to depreciation. Condition (2.10) says that when capital exists in sufficiently large quantities, the loss due to depreciation exceeds production.
III.
THE INFINITE-PENALTY
FORMULATION
OF THE PROBLEM
We will put the problem (2.11)-(2.14) into a more convenient form using the infinite penalty method of Rockafellar. (See Rockafellar [4, 8, 91 for a detailed exposition.) Define K: [0, co) x R” x R” x R” x R + R u { + co} by K( t, k, z, v, c)= -u(c)e--“’
=+a2
ifk>O,(c,z)EF(k),andv=z, otherwise.
(3.1)
LEMMA 3.1. K is a proper normal integrand on [0, CC) x R” x R” x R” x R. Futher, K( t, k, ., ., .) is convex in (2, v, c) for each (t, k) E [0, co) x R”.
Proqf:
We leave this to the reader.
LEMMA
3.2. Let L: [0, co) x R” x R” + R v { + CC} be defined by L(t,k,v)=inf{K(t,k,z,v,c):(z,c)ER”xR}.
(3.2)
Then, L(t, k, v) > -CQ always and is a proper normal integrand that is convex in v. Further, if L( t, k, v) is ,finite, there exists a (c, v) EF(k) such that &(t, k, u) = -u(c) e “‘. Proof:
We leave this to the reader also.
83
EXISTENCE AND NONCONVEX TECHNOLOGY
The original
problem
(2. I )-(2.4) is then equivalent
to
2. - inlimum
ii 0
- u(c( t)) r -‘I” r/t /(2.2)-(2.4) hold and (~,-)~L[O,cx,)xL,,[o,cc),k~A,,
(3.3)
I
or - inlimum
z K(r, k(t), I;(t), It(t), c(t)) dt + l(k(O))l c E L[O, co), k E A, , ii 0 I (3.4)
where I: R” x R” +Ru{+cII)
is defined by l(a) = 0
ifa=k,
=+a, By standard results of Rockafellar - infimum
(3.5)
otherwise.
[7, 81, we know (3.4) is equivalent to
x L(t,k(t),~(t))dt+I(k(O))(kEA* i!’ 0
(3.6)
We will define @,.,: A,, -+ Ru { + a5 ) by @L.,(k) = IL ur, k(t), ~W) d + ww)). 0 Our problem is therefore equivalent
(3.7)
to
-infimum{@Jk)lk6,4,,).
IV. EXISTENCE AND A PRIORI BOUNDS FOR FEASIBLE CONSUMPTION-INVESTMENT PATHS LEMMA 4. I. def kf by
Let k E C,, satisfii’ k(t) 2 0,fbr all t 2 0. Thenf‘:
f(t) is (1 twtictvptj’. [O, ,x’ ).
conipact.
and
[0, CC) + 2R”
= 44t))
closed-valued
(4.1) measurable
multifunction
on
84
GAlNES AND PETERSON
Proqf: It suffices to show thatfis measurable on [0, rj). Since F is USC for all k 3 0, it is clear ,f is USCfor all t 3 0. Let A c R” be closed and consider j
(A)=
{tE[O,
cr_,):J’(t)nA#@j.
Let ,f’+(A)=
(tE [0, nj):,f(t)CA}.
Clearly, [0, m)\f’ (A)=,ft([W”\A). Let tef+(R”\A). Then,f(t)clW”\A. Since KY\ A is open, there is an open set W such that f(t) c WC IW”\A. Thus, by USCof,f; there exists a neighborhood U,,. of t ,f(U,,)s
WCR”\A
implying
U,,.Gf + ([W”\A). We conclude j” measurable.
(IW”\A) is open and hence f‘~ (A) is closed. Thus, f is
Q.E.D.
It follows immediately by Theorem (~,z)EL[O, fx,)xL,,[O, z) such that (I.(t), =(t))EF(k(t)) Define the multifuncton
1.3 that there is at least one for all t > 0.
(4.2)
G: R” -+ 2Rnby
G(k)={:/thereisa~~Rwith(c,z)~F(k)}.
(4.3)
LEMMA 4.2. Let g: [0, a] + 2R” he defined by g(r)= G(k(t)), where k E C,, with k(t) 2 0 ,fi)r all t > 0. Then g is a nonempty, compact, convexvalued measurablemultifunction on [0, a).
Proqj:
Follows from Lemma 4.1 and Theorem 1.3.
Consider the differential
Q.E.D.
inclusion
.+(t)EG(x(t)),
x(0) = k
(4.4)
for .YE A,, and t > 0. We will show there is a solution x to (4.4). LEMMA 4.3. !f’k E C,,[O, T] such that k(t) > 0 and k satisfies (4.4), then fbr all t 3 0,
EXISTENCE
(i)
k(r)>&
(ii)
lk(t)/
AND
NONCONVEX
TECHNOLOGY
85
“>O; N);
where N is the constant ,fiom (2.10). Prooj-f:
For each t 3 0, there exists a number c, such that (c,.tG(t))EF(k(t)).
(Note that it is a matter for separate proof to show that the function defined by c(t) = cI may be chosen so that it is measurable.) By (2.9) there is a E.E (0.1) such that L(t)+ik(t)>O.
The boundary condition
(4.5)
k(0) = k together with (4.5) implies k(t)>&e
“>o
(4.6)
for t 3 0. Choose T> 0. By continuity of k, there is a I, E [0, T] such that I&to)1 =max{Ik(t)l IKE [0, T]j,. Assume Ik(t,)l >max(IEl, N)=t. Let I k(t)l’ = u(t). Since u(0) < <‘, our assumption implies t, E (0, T]. The continuity of u implies there is an q > 0 such that if t,,- r\ < t < z,,, then u(t) > j”. Hence, by (2.10) k(t)4(t)
(4.7)
for t,,-tf U(r,,) for t, - q < t < t,. This contradicts the definition of t,. We conclude 1k( t)l d s” on [0, T]. The arbitrariness of T implies the validity of the lemma on [0, co). Q.E.D. Let .Y,,E R”. We define T: C,,[a, h] + 2”nr”.h1 by T(s)=
iwEA,,[a,h]l3=ELj,[a,t]
withz(t)EG(x(t))
(4.8 1
a.e., and o(t) = x,, + ‘z(.s)d.s j
(ii)
4.4. (i ) I/’ s(t) B 0, T(x) is nonempty, closed, urzd convex. Jf’sZ c {SE C,,[LI, h]: .u(t)aO) is open and bounded, then T(o) is
relatiorl~~ compact in C,,[a, h].
(iii)
T is upper semicontinuou.son {.K E C,,[a, h]: x(t) 3 0).
Proof: (i) To show T(.u) is nonempty, note there is a measurable selection (c(r), :(t))eF(.y(t)) and by (2.8) we have
I:(t)l Sd(I.Qf)l).
(4.9)
86
GAINES
AND
PETERSON
Thus, z EL/,[u, h] implying T(x) is nonempty. The convexity of T(x) follows from the convexity of F(X) for each x. Finally, we show T(x) is closed in C,j[a, h]. Let (w,,): c T(x) such that H’,?+ M’ in C,,[a, h] norm. Then there is a sequence (Z,); such that for all Il.
Z,,(t) E G(x(t)) a.e.;
(4.10)
W,,(r)=x,+J’Z,(s)ds. N
(4.11)
By (2.8), we conclude ( Z,,(s)/ d &(x(s)]) for all n. So by Theorem 1.2, there is a ZE &,[a, h] such that Z,, + Z weakly in LA[a, b] and applying Lemma 1.1 we see there is a sequence (p,);=, with Z&) = 2 P$i,+i(f) !=I with Cfk j,,, = 1. p,,, > 0, and K,, < K,, , for all n and 0 6 i< k,. Further, in LA[a, h] norm
z,, + z and Z,(f) -+ art
pointwise a.e. in [a, b].
Let I?,,( f ) = x,, + 1’ Z,,(s) ds N
and m(t) = x, + 1’ Z(s) ds. u Clearly, IV,, + @‘in C,,[a, h] norm since W,, -+ W is C,[u, b] norm. Thus, W(f) = 6’(t) for all 1. Since z,, --f Z pointwise a.e., given E> 0, there is an N(E, t) such that n >, N(E, t) implies d-Z(r), W(f))
G I Z(t)-
2,,(t)l +p(.%(~),
W(t))
a.e.
a.e.
6 42 + d-%,(r), G(-Gt))
Now Z,,(t) E G(x(z)) a.e. and G(.u(t)) is convex. Thus, T,,(t) E G(x(t)) a.e. and we have P(Z(~)+ G(.~(t))
a.e.
EXISTENCE
AND
NONCONVEX
TECHNOLOGY
87
The arbitraryness of E> 0, then implies p(Z( t). G(.u( t)) = 0 a.e. and we conclude from the closedness of G(s(r)) that
Z(r)~G(.df))
a.e.
WE
a.e.
Thus, T(s(t))
(ii) Let (IV,,);=, c r(Q). Then there exist sequences (x,,),:=, (Z,,);=, such that (x,,);-=, =Q and
(4.12)
W,,(t) =x0 + j’ Z,,(s) ds
Z,,(t) E G(.K,,(~))
Now by assumption,
and
(4.13)
at.
Q is bounded, thus there is a B such that (4.14)
II -y,,II d B for all n. It follows by (2.8) that
I W,(t)1 = Iz,,(t)1 d 4(B). It is then easy to check that (IV,,):=, is a uniformly bounded equicontinuous family in C,,[a, h] and so by the Arzela-Ascoli theorem, we may assume without loss of generality that there is a WE C,,[a, h] such that
in C,,[a, h] norm. It remains to show WeA,,[a,
h]. Since
IIz,,(Oll d d(B)+ by Theorem I .2 and Lemma 1.l, (Z,,);~= , is relatively weakly compact in L![a, h]. Letting k,, (4.15) Z,(t)= 1 B,,iZE;,+i(f) ,=I rn,,( t) = s,, + j’ Z,,(s) rls, t,
(4.16)
we may assume without loss of generality that there is a Z E Lj,[u, h] such that
2,, + z Z,(f) + Z(t)
in L![u, h] pointwise a.e. in [a, h].
(4.17)
88
GAINES
AND
PETERSON
Let R(t)
= x, + j’ Z(s) ds.
0
(4.18)
Clearly, (4.17) implies IV,, + IV in C,[a, h] norm. Since W, -+ W uniformly, it is clear that Cf;- , fl,,, WKn+, -+ W uniformly also. Now
Hence, m,, + W uniformly
also implying
W(t) = m(t) on [a, b]. Thus,
(4.19)
W(r)=s<,+j’z(s)ds.
(iii) Let (s,,,),;= , and (M’,,);= , satisfy x, + x and w, -+ w in C,,[cr, h] with \v,,,E T(.u,?,). By assumption x,30 for all m implying, of course, s > 0. We must show 1~E T(x). It is easy to show the upper semicontinuity of F implies the upper semicontinuity of G. Hence, given e and t, there is a 6 (depending on E and t) such that
Since sY --f .Y uniformly
we see for sufficiently large m, G(.~,,,if) E N,:(W(t))).
By assumption, k,(f)
a.e., so
E G(x,( r))
a.e.
+,,,(t) E N,:(G(x(t))) for sufficiently large m. Since x,, + x uniformly, such that
(4.20)
we see there is a constant B
for all m. Thus, by (2.8), for all m,
EXISTENCE
AND
NONCONVEX
TECHNOLOGY
89
and
By a now standard argument, we see (k,,,),:,= , is relatively compact in L![a, h] and hence there exists a I E L![a, h] and convex combinations of the functions ti,,,,, denoted Z,,,, such that Z,,, + z in L,!,[u, h] and pointwise a.e. on [u, h], with k”,
The convexity of G(.u(t) and (4.20) then imply
for sufficiently large m. The closedness of G(.u( t)), the arbitraryness of E> 0, and the pointwise convergence E,,, + z then imply z(t)~G(.u(t))
a.e. on [a, h].
The arguments in the proof (ii) then show that njt) = X, + j:, z(s) ds and thus \t’ E T(s). Q.E.D. LEMMA 4.5. Let p E (0, 11. Let T,, he d T,,(x) = ( 11’E C,,[u, h] 1w(t) = x,, + p J;, Z(J) ds, z E Lj,[u, h], and z(s) E G(s(s)) u.e. 1. Jf’ x(t) 30, XE C,,[u, h], and XE T(.u) then x(i) 3 x,,e ” und Ix(t)1 d
max[Isi.
N].
Pror?fI
This is identical to the proof of Lemma 4.3.
LEMMA
4.6.
There is u sohtion
~(2; a, h, x,,) to
.t(t)~G(x(t)) .K(u) = x,, > 0 in A,,[u, h]. Q = {sEC,,[u, h]:.u(t) > (1 -c).\-,,e- “, IX(t)1 < (1 +c) Proof: Let max( I s,, / , N) for c sufficiently small. By Lemma 4.5, .Y& T,,(x) for /1 E (0, 1 ] and s sf (752. Moreover, it is immediate that x k T,(x) for XE X? since T,,(.Y) = s,,. Observing that T, satisfies the conclusions of Lemma 4.4 and using Theorem I. I,
90
GAINES AND PETERSON
Thus, there exists xesZ such that XE T(X). Thus x(t) = x, + I;, z(s) d.5 for some ZE L![O, T] with z(~)E G(.u(t)) a.e. The conclusion then follows immediately. LEMMA 4.7.
There is a solution x(t, k) to i(t) E G(x( t)) x(0) = k
in A,,. ProoJ: Define x(r; k) to be x(t; 0, 1, k) for t E [0, 11. Extend this solution to [0,2] by using x( 1; k) as the value at t = 1; i.e., define x(t; k) = .Y(t; 1, 2, .u( 1; 0, 1, k)) on [ 1, 21. Repeating this process, x(t; k) can be extended to [0, + co). Q.E.D.
V. THE EXISTENCE OF AN OPTIMAL CONSUMPTION POLICY
The results below rely heavily on Rockafellar [4] and are included for the convenience of the reader. Recall from Eq. (3.2) that L(t,k,v)=inf(K(t,k,z,v,c)~(z,c)~R”xR}. L(f,k(t),I;(t))=inf~K(t,k(t),z,I;(t),(’)I((.,;)~F(k(t)),z=~(t)j.
(5.1)
Define B: [0, 01) + 2R by B(t)=
{cl(c,l;(t),
-k(t))EF}.
(5.2)
We see that
Following Rockafellar [4, Corollary 2B], since K is a proper normal integrdnd it is easy to see that q: [0, cc ) x R --f R E { + co } defined by q(t, c) = K(t, k(t), b),
&t), c)
is also a proper normal integrand. Finally, from the definition of L, B, and q, we see that L(t.k(t),k(t))=inf{q(t,c)lcEB(t)}.
(5.3)
91
EXISTENCE AND NONCONVEX TECHNOLOGY LEMMA
5.1.
Let k(t) > 0 he a solution to i(t) E G(.u(t)) x(0) = F
on [0, m). For this k, B is a nonemptjs compact-tialued measurable mult~fimcton und there is a measurable function ?: [0, wJ) + R such that P(t)EB(t) a.e. with L(t,k(t),k(t))= -u(?(t))e--“‘a.e. Proqf: From (2.6) it is easy to see that B is compact-valued. From the definition of G, we know that for almost all t E [0, w ) there is a c, such that (c,, k(t), -k(t)) E B. Thus B is nonempty a.e. on [0, co). The proof that B is measurable follows from Theorem 1.3 and Lemma 4.1 and relies on the Castaing representation for F(k(t)). Letting k be as in Lemma 5.1, let
IM(t)=inf{q(t,~)IcEB(t))
(5.4
=L(t,k(t),k(t)),
and M(t)=
(5.5
~~EB(t)lq(t,c)6m(t)).
By Rockafellar [4, Theorem 2k], we see m and A4 are both closed-valued measurable multifunctions. Hence, there is a measurable function ?: [0, cxj) + R such that c’(t) e M(t) a.e. on [0, w);
implying i’(t)e B(t)
and
q(t,?(t))
Thus, L(t, k(t), I;(t))=
K(r, k(t), k(t), R(t), (‘(1)) = --u(t?(t)) e-f”.
Q.E.D.
(5.6)
LEMMA 5.2. Let CJ= (k E A,,] @Jk) < x: ). Then CJ is nonempty and rclr1ticwl~~c’0mput.t in A,, in the topolog~~of‘ un[fbrnl convergence on compact .sdwrts of’ [O, CC). Further, if‘ k E o, then:
(i) (ii)
X-(t)>k; “>O, [k(r)1
and N] -<.
Proof: By Lemma 5.1, if k is given by Lemma 4.7, then G,,,(k) < ‘x. Thus (T is nonempty.
92
GAINES
AND
PETERSON
Consider (I?,,);~=, c 6. By (2.8) and Lemma 4.3 it is clear that I/k,, // < < = max( I,6, A’) and )k:,(t)/ < 4(r) implying I k,,(r) -
k,,(f)1 d
(5.7)
for all n. Thus (k,,) c 0 is uniformly bounded and equicontinuous. By the Arzela-Ascoli theorem, we conclude there is a subsequence (k,J and k E C,,[O, xl) such that k,,,, + k uniformly on compact subsets of [IO, G). That k E A,, follows from identical arguments to the proof of (ii) in Lemma 4.4. Q.E.D. LEMMA
5.3.
Let A,:= (k: lk-kl
GE, k>O).
Then,forany f~ [o, a),
,,9,, Co n epi L( t, k, .) c epi L( I, l,
).
(5.8
4 Proqf:
Recall epi L( r, k, .) = ((r, y) E R x R” 1L( 1, k, y) 6 z }. Let (a, DIE W=
n Co u epi L(t, k, .). / >(I 4.
(5.9
Then, we may assume (CI, u) E w U,4, n, epi ,!,( t, k, .) for all m. For convenience, let J= (dimension of R” x R) + 1. By Caratheodory’s theorem and the definition of m IJ,.,, “, epi L(t, k, .),
where for all m and n and for all 1 < id J, /7:::>0 (5.11) i,
lc: = ’
with (5.12) (5.13) (5.14)
EXISTENCE
AND
NONCONVEX
93
TECHNOLOGY
Since L( 1, k;;;, ~1;;;) is finite, it is easy to show there exist numbers that
c:: such
and L( f, k;;;, w;;) = -u(c;;;) By
USC
of Fat & given N,:(F(l)),
of the kz’s,
(5.16)
there is a neighborhood
FW,(&) and by the definition
P “‘.
N6(&) such that
G N(F(k)) there exists kin(c)>0
II k::: - k II 6 6
such that
for all m 2 M,(E).
Hence, (~‘a;, M$.) E N,.( F(k))
(5.17)
for all m 3 M,(c).
The compactness of F(c) further implies there is a sequence (T;;, 12;) c F(k^) such that for all m > MO(~), (5.18)
l(CY Thus, for all fir 3 MC)(c),
By convexity
of F(K),
and by the compactness (P.~,,‘)E F(k) such that
Hence, for sufficiently
of F(E),
we
may
assume
there
exists
a
large n, say II 2 NO(a), (5.19)
64?
‘37 ‘1-7
94
GAINES
AND
PETERSON
By (2.8), for all n and m and for all 1 d i 6 J, (5.20)
o~c:G4(Ik;I). But since k;;;. E A ,!,,,, we see for all n, m and 1 < i < J,
lk:I 6 IL1 + 1. Hence,
and therefore we may assume without loss of generality that there is a cm such that (5.21)
Consider I(?,
u)-
(P,.f”)l
6 (P, u) - i p;;(c;, i= I +
w;)
i 63 c;, WZ) - (dMm,frn) . i= I
Since o= lim,, C;‘=, ,!I;;.w;., there is an N,(E) 1: NO(s) such that for all n b N,(E, n) and m 2 MO(c), I (c”‘, u) - (cP,f”)l
< 2E.
(5.22)
Since (ht, .f”‘) c F(k) f or m > M&E), again by the compactness of F(E), we may assume without loss of generality that there is a (d,f)~F’(/?) such that lim (P,,f”)
= (d,f).
m
Similarly,
(5.23)
by (5.20) and (5.21), O~c”‘qb(IRI
+ 1)
(5.24)
implying we may assume there is a c such that lim, cm = c. Note 0 d c 6 & (R( + 1). Equations (5.23) and (5.24) in conjunction with (5.22) then imply I(C? u)-
(d.f)l
f4E
9.5
EXISTENCE AND NONCONVEX TECHNOLOGY
and by the a arbitraryness (c, u) E F(k). Further,
of E> 0 we conclude
(c, I:) = (cl,,f). Thus,
- i fl;;)u(c$) e (” 6 i &;;y;; ,=I i= I and letting II + ~8, the continuity
of u implies
- u(c”‘) e “’ < 2.
Letting m + lx, the continuity of u again implies we conclude -u(c) e I” d c( and thus by definition L( t, k, v) = -U(C) - 1” < c(. We conQ.E.D. clude (CCu) E epi L(t, &, .). LEMMA 5.4. Let [a, h] c [0, CG).Let p,, +p weakly in Lk[a, h], k, -+ k pointwise on [a, h] nlith k,,(t) > 0 on [a, h] and k(t) > 0 on [a, h]. Then
lim 46 k,,(t), p,,(t))> L(t, k(r), p(t)) I,.- c
a.e. on [a, h].
(5.25)
Proof: If lim,,, ,~ L(t, k,,(t), p,,(t)) = +CG, (5.25) is trivial. Hence, we may assume that lim,,, J L(r, k,,(t), p,,(t)) = CI,< CG. By passing to a subsequence, if necessary, assume that lim,, _ r L( t, k,,(f), p,(t)) = CI,. By Lemma 1.l, since p,, -+ p weakly in L,‘,[a, /I], there exist constants M,, k, and (p,, ,..., /I,/i, i such that /I,, > 0, 2:~~ jji = 1, M, < M,, , , and we may assume ii,(r) = % B,; Pn,,+,(d i=
(5.26)
I
satisfies I?,-P
1?,(f) +p(r)
in Lfl[a, h]
(5.27)
pointwise a.e. on [a, /I].
Now
(L(t, k .~,+,(f). P.w,+;(~)), ~,~,+,(f))~epi L(t, k,,,,+,(t), .)
(5.28)
implying i At)* P,bt,+,(t)LP,(t)
>
ECO 3 epi L(r, k,,+,(I);).
(5.29)
,=I
Thus, we see
,(I, k II,1Af )qPM,+,(t)),P,(I) ECOU epiL(t,k,(t),.h 1 ‘,3M,
96
GAINES
AND
PETERSON
Since k,,(t) +k(i) pointwise, given E>O there is a positive integer N(r:, () such that n 3 N(c, t) implies ) k,,(t) - k( t)l N(r:, t) such that .j>, N,(E, t) implies M,> N(E, t) and ,<, &WA,,
&)r~,w,+i
(l)),P,(1)
1
~co()epiL(t,&t),.), .4,
where A,:= (kl [k-k(t)1
GE, k>O}.
We conclude that for almost all TV [u, h],
M,+i(f)),p(t)EEIJepiL(hk(t),,). ,,li~~~ 2 P,;L(t,k~~+i(t)r~ i
i=l
.4,
The arbitraryness of s > 0 implies lim 2 B,,Ur, kh,,+i(t)rp ( i,- L ;=,
E fi CS U epi UC 4th .). I:> 0 .%
M,+,(f)),~(f)
By Lemma 5.3, we conclude
!
lim
i
BijL(r, k,! + Jr), p h,,+i(f)),~(t)
Eepi UL k(f), .I
” - -’ i = I
implying UI, k(rLp(t))< Since lim,,
, lim - 7 ,=2 I Bj;L(l, kM,+r(f)r PM,+,(t)).
,~ L( t, k,(t), p,(t)) = a, < co, we see ki lim )I -
%
C , =
PijUt,
k,,+,(t)
PM,+i(t))=at.
,
Thus, Ut, k(r), p(f)) d nlim + Ix, Ut, k,(t), p,(f)).
Q.E.D.
Let CI= iniimum and let (k,,);;= , be a minimizing
{ Q,,,(k)1 k E A,} sequence; that is,
cx= lim sPL,,(kn). t1
(5.30)
EXISTENCE
AND
NONCONVEX
97
TECHNOLOGY
By Lemma 5.2 and (5.6)-(5.8). we see X-C CD, k,,(r)>ke“>O, j/k,,/1
II t I
a.e. in [u, h].
(5.31 )
Recall from (5.6).--(5.8) that @,.,,(k,,)=il’
-u(c,,(t))e
where (c,,(1). k,,(t))~F(k,,(r)) a.e. ~(Ik,,(1)l)~~(max((~/, N)). Hence, IUt,
k,,(t),i,,(t))l
and
By the Lebesque dominated
“‘elf,
by
Lemma4.3,
“‘<(P(max(iR/~
O
~)e
1”.
convergence theorem, we conclude
^I L(r, k,,(r), d,,(t)) tir < lim i ’ ,C.(f,k,,(r), k,,(t)) r/r J(, ,,lim - I ,i - ., (, or lim .r ’ L(r, k,,(t), i,,(t)) ()
,,-I
rt~j-~.
I)
L(r, k(r), i(t))
dt.
Since k,,(O) = k(0) = & for all n, we have ,Ilim - ., @,.AX-,,I3 @,,(k) or ci 3 G,.,(k).
But z = infimum(@ ,.,,(k)l kE A,,). We conclude aL,,(k)= finite. there is a measurable t,(t) such that Cf.=
, I0
- u( c( t)) e “’ dt;
(c’(r), ht)k
F(k(l))
X-(O) =x’ k(r)>0
cc. Since ds,.,, is
on [0, Y, ).
a.e. On [o, CO));
(5.32)
98
GAINES
AND
PETERSON
Thus there is a solution to problem (2.1)-(2.4). We have proved the following theorem: 5.1. Jf‘the technology to the optimal consumption
THEOREM
solution
set sati.~fies (2.5)-(2.10), problem (2.1 )-( 2.4).
Then there is u
REFERENCES I. J. R. WEHH. On degree theory for multivalued mappings and applications, Boll. C/n. Mat. I/u/. 9 (1974). 137-158. 2. N. DGNFOKI) ANU J. T. SCHAKTZ, “Linear Operators, I,” Wiley, New York, 1964. 3. I. EKELANU ANI) R. TEMAM, ‘Convex Analysis and Variational Problems.” North-Holland. Amsterdam. 1976. 4. R. T. RIKKAFELLAK, Integral functionals. normal integrands. and measurable selections. irt “Nonlinear Operators and the Calculus of Variations,” Lecture Notes in Mathematics Vol. 543. Springer-Verlag, Berlin. 1976. 5. c‘. CASTAING. Sur les multi-applications measurables. Ret). Franpise Ir~form. Rech. Oper. I (1967). 91-127. 6. D. CASS AND K. SHELL, The structure and stability of competitive dynamical systems. J. Ewn. Thcor~ 12 (1976). 31-70. 7. R. 7. ROCKAFELLAK. Duality in optimal control, in “Mathematical Control Theory,” Lecture Notes in Mathematics Vol. 680, Springer-Verlag, Berlin, 1978. 8. R. T. ROCKAFELLAK, Conjugate convex functions in optimal control and the calculus of variations. J. hfrrth. ANuI. App’pl. 32 (1970) 174-222. 9. R. E. GAINES. Existence and the Cass-Shell stability condition for Hamiltonian systems of optimal growth. J. G,on. Tl~rorr 15 (1977) 16-25. IO. R. E. GAINCS. Existence of solutions to Hamiltonian dynamical systems of optimal growth. J. EUJI~. Theory 12 (1976), 114-130. I I. R. T. RWKAFELLAK. Saddle points of Hamiltonian systems in convex Lagrange problems having a positive discount rate. J. &on. Theory 12 (1976) 71-I 13. 17. A. HAUNT.. Optimal control on an infinite time horizon: The turnpike approach, d. Math. Eum. 3 (1976). XI -102. 13. W. A. BKOCK ANII A. HAUKIE. An existence of overtaking optimal trajectories over an infinite time horizon. t+for/~. Oper. Rex. 1 ( 1976). 337-346. 14. H. HALKW, Necessary conditions for optimal control problems with infmite horizons, Ew~mm~/riur 42 ( 1974 ). 2677272.