Weakly maximal stationary programs for a discrete-time stochastic growth model

Japan J. Indust. Appl. Math., 10 (1993), 471-486

Weakly Maximal Stationary ProgrRms for a Discrete-time Stochastic Growth Model 1,2

George PANTELIDES and Nikolaos S. PAPAGEORGIOU 3

National Technical University, Department of Mathematics, Zografou Campus, Athens 15773, Greece

Received November 20, 1992

In this paper we establish the existence of a weakly maximal program fo ra stationary growth model with uncertainty. Our approach uses techniques from functional analysis as well as the existence of a good program. Then in the last section, we produce general conditions on the data that guaxantee the existence of a good program.

Key words: stochastic growth model, stationarity, program, weakly maximal program, good program, average turnpike property

1. I n t r o d u c t i o n

In this paper , we s t u d y the p rob lem of the exis tence of weakly m a x i m a l pro-

g ra tas for a s t a t i o n a r y g rowth model wi th uncer ta in ty . The s t a t i o n a r i t y of our mode l

is mani fes ted by the fact t h a t the p robab i l i t y d i s t r i bu t ions of the p roduc t i on tech-

nologies and the u t i l i t y funct ions, are t he same a t each t ime per iod . Our work here

ex tends the resul ts of R a d n e r [13] and D a n a [3], who cons idered ¡ d imens iona l

models , wi th more res t r ic t ive hypo theses on the da ta . A comprehens ive in t roduc-

t ion to the sub jec t of s tochas t ic mode l s of economic dynamics can be found in the

recent book by Ark in -Evs t i gneev [1].

2. T h e M o d e l

F i r s t we will descr ibe the general mode l and then we will i n t roduce the s ta t ion-

a r i ty assumpt ions . Let (~ , X, fe) be a comple t e p robab i l i t y space. We in te rp re t ~2

as the set of all poss ib le s t a t es of the env i ronment , ~ is the col lect ion of all poss ib le

events and fe(.) is the co r respond ing p r o b a b i l i t y d i s t r ibu t ion . Our p lann ing hor izon

is No = {0, 1, 2 , . . . } ; i.e. our mode l is d i sc re t e - t ime and inf in i te-hor izon (or in the

t e rmino logy of Maka rov -Rub inov [10], p. 98, a "model of the second type" ) . The

unce r t a in ty unde r ly ing the growth process , is desc r ibed by an increas ing sequence

{~n}n_>0 of comple t e sub-a- f ie lds of E such t h a t V~>o E n = ~ . As usual , the sub-a- f ie ld Xn represen t s the in fo rmat ion avai lab le at t ime n.

Our c o m m o d i t y space is a separable , reflexive Banach space X , pa r t i a l l y or-

dered by a closed, convex cone X+; i.e. x < y ir and only if y - x C X+ . We

1 Revised version. 2 Research support from the Creek Ministry of Industry and Technology. 3 Presently on leave at Florida Institute of Technology, Department of Applied Mathematics, 150

West University Blvd., Melbourne, Florida 32901-6988, USA.

4 7 2 G. PANTELIDES and N.S. PAPAGEORGIOU

will assume that in tX+ ~ 0. Recall tha t commodities are defined as physical goods, which may ditfer in the location or t ime in which they are produced or consumed, or in the s tate of the environment in which they become available. If we permit an infinite variation of any of the above characteristics, then i t i s natural to model the commodi ty space by an infinite dimensional, linear space. Given an input vector x E X, if the state of the environment is w C $2, then our production possibilities at t ime n, are given by the set Fn(w,x). So the preceding states of the economy, do not determine uniquely the technologically possible future states. This is consistent with the nature of general economic processes. Hence, GrFn(w, .) = {(x ,y) E X x X : y �9 Fn(w,x)} (i.e. the graph of the multifunction Fn(w, .)), describes the set of all pairs of possible capital t ransformations between times n - 1 and n. The uncertainty in the production technology, is manifested on the fact that the set Gn = {(w, x, y) �9 $2 x X • X : y �9 Fn(w, x)} �9 En x B ( X ) • B ( X ) , with B ( X ) being the Borel a-field on X.

The social welfare (utility, profit) achieved at each t ime period n, is described by a utility function Un : $2 X X • X --+ R. Again, the uncertainty in the utility is expressed through the hypothesis that the function un(- , - , ' ) is ~2n • B ( X ) • B(X)-measurable . We interpret un(' , ", ") as a measure of the utility achieved by the economy, when the state of the environment is w, �9 /2 and the input-output pair is (x,y) �9 X • X.

A "program" is a discrete-time stochastic process {Xn}n_>0 such that xn �9 L oo (En, X) . A program { x,~ } n> 0 is said to be feasible if Xn+ 1 (w) �9 Fn+l (w, Xn (W)) ~t- a.e. By S(Y0) we will denote the set of all feasible programs, emanat ing from the initial capital stock T0(') �9 L~r X). Since our horizon is infinite and our utility is not discounted, to avoid introducing stringent hypotheses tha t will guarantee convergence of the infinite series, in evaluating different programs, we adopt the "weak maximali ty criterion" first introduced by Brock [2]. So a program {x~ �9 S(~0) is said to be "weakly maximal" if and only if for every other {Yn},~>0 �9 S(~0), we have

N

lim E ( J n + l ( Y n , Y n + l ) - Jn+l(x~ 0 Xn+x) ) _< 0 N ~ o o n=0

where for every (v ,w) �9 L~ X ) • L ~ ( X n + I , X ) , J,~+l(v,w) = f~ Un+l (W, V(W), w(w))d#(w). Sometimes this criterion is called the "overtaking op- t imality criterion" (in the optimal control l i terature the name "weakly overtaking criterion" is also used). This definition implies that for every {Yn}n>0 g S(~0)

N given any e > 0 and any No, we can find N > No such that Y~-n=O Jn+l(Y,~,Yn+l) - N e < E n = 0 J n + l ( x ~ 0 mn+l). Hence the performance index of any other program

{Yn}n>0_ �9 S(~0) can not overtake the performance index of {ron}n_>0 . 0 Now we are ready to introduce stationaxity in our model. So we assume that

there exists a map r : ($2, En) --+ ($2, ~ n - 1 ) which is one to one, onto and both r , r -1 axe measurable. Furthermore, we assume that for each n > 1 and each A �9 E n - l , we have t z ( r - l (A) ) = #(A); i.e. T(.) is a measure preserving trans- formation. Hence the measure spaces ($2, ~7,~, #) and ($2, ~n-l,t*) are isomorphic

Weakly Maximal Programs 473

under t ransformation T(-) ana T(E,~) = ~ n - 1 while #(T(B)) - - - - #(B) for all B e ~ ~ (i.e. (f2, T(Zn), #) is equivalent to (f$, E,~-I , ;~)). We can think of 7(.) a s a t ime shift operator. If f : f2 --+ X is a Z:~-measurable function, we can find g : f2 ~ X, a ~n_ l -measurab le function such that f = g o Ÿ Then note that g = f o ~--1. Thus any f : f2 -~ X, Gn-measurable, can be described a s f = g o T n for some g : f2 --* X, G0-measurable. Furthermore, exploiting the fact that T(.) is measure preserving, for every C E B(X), we have

# ( f - 1 (C)) : #((g o Ÿ (C))

: ~ (Ÿ o (g o < ~ - ' ) - ~ ( c ) )

= # ( ( g o T n - 1 ) - l ( c ) ) . . . . . /.�91 (C)).

Therefore we see tha t f and g are essentially the same function, except that f ( . ) is shifted in a later t ime period.

The stat ionari ty will also be present in the technology multifunction and the utility function. So for each n > 1, we have

Fn(o) , ") ~- F I ( T n - I ( � 9 1 1 6 2 ")

and Un(W, k, V) = 1.ll ( T n - 1 ((.o), k , V).

Thus everything is described in terms of the corresponding quantity at t ime n = 1, by employing the time-shift operator T(.). Hence F1 and ul are the prototype technology and utility maps respectively, which determine all future such quantities. In the sequel, to simplify our notation, we will simply write F1 = F and ul = u.

To particularize the above abstract setting, we will describe a simpler version of our model, variants of which appear often in the economic li terature and which illustrates all the hypotheses made earlier.

The decision-maker in the economic process is faced with different types of uncertainty arising from various sources (uncertainties about financial data, geo- graphical uncertainties, uncertainties about the technology, climatic uncertainties, etc.). We will incorporate in our model, those uncertalnties which arise similarly in successive periods and which have a probabil i ty distribution (stationarity of the uncertainty factor). The uncertainty is represented by the occurrence of some state of the environment st at t ime t independent of the planner 's action. This state of the environment can be a vector in R m (with each component measuring the available stock of a resource involved in the economic process) o r a function (providing information about certain data) or even more generally, a binary relation (describ- ing a preference among commodities). So it is natural to assume that the space of states of the environment Ek, is a separable complete metric space (i.e. a Polish space), which is independent of t ime (i.e. Ek : E). Let S : I]-oo<k<+c~ Ek be the set of doubly-infinite sequences s : {sk} - oo < k < +oo, with sk representing the state of the environment at t ime k. Let ~ be a a-fielcl generated by the cylinder sets on S (i.e. is generated by sets of the forro YI-oo<k<+~ Ak, where Ak = E for all but finitely many k and with Ak being a Borel subset of E). It is natural to

474 G. PANTELIDES and N.S. PAPAGEORGIOU

describe the sequence of s ta tes of the env i ronment as a s tochast ic process on some probabi l i ty space (S, E , #). The first works on the subject considered sequences of independent and identical ly d is t r ibuted r a n d o m variables (see Arkin-Evs t igneev [1] and the references therein) , while Radner [13] considered a s t a t iona ry Markov process with given t rans i t ion probabil i ty. More generally, we can assume tha t we have a probabi l i ty d is t r ibut ion #(.) on S, wi thout par t icular iz ing the s tochas t ic process. This incorporates the above two special cases, v ia Kolmogorov ' s t h e o r e m (the case of i.i.d, r andom variables) and via the Ionescu-Tulcea theorem (the case of Markov process). The shift ope ra to r T : S --~ S is defined by (Ts)k = Sk+l and we assume tha t it is measure preserving (i.e. for every A �9 E, #(T(A)) = #(A) = # ( T - I ( A ) ) ; note tha t T 1 is measurab le too). If T mis the m th i terate of T, (Tms)k = Sk+m. Given any funet ion f ( - ) on S, we define T m f by Tmf ( s ) = f (Tms) . Note t h a t in the i.i.d, r a n d o m variables se t t ing the shiff t r ans fo rmat ion T is in fact ergodic (i.e. the invariant a-field is trivial). Let Z n be the a-field genera ted by all pas t histories up to t ime n (i.e. E n is genera ted by the cylinder sets 1 - I - ~ < k < + ~ Ak, wi th Ak = E for k > n). If f ( . ) is a E0-measu rab le function, then the sequence of r a n d o m variable fk = f o T k is a s t a t iona ry r a n d o m process.

We also want to emphas ize t ha t until now all works on s tochast ic growth theory assumed a finite d imensional c o m m o d i t y space. Here, following the recent t rend of "Equi l ibr ium Theory" we allow our c o m m o d i t y space to be infinite dimensionaI.

Now we are ready to introduce the detai led m a t h e m a t i c a l hypotheses on the d a t a of our model.

H ( F ) : F : $2 x 2 x \ { 0 } is a mul t i funct ion such t ha t (1) (w, x) ----* F(w, x) is ~'1 x B ( X ) x B ( X ) - g r a p h measurable ; i.e. G r F =

{ (w ,x ,y ) �9 $2 x X x X : y �9 F(w,x)} �9 E1 x B ( X ) x B(X) , (2) for every w �9 $2, G r F ( w , . ) = { (x ,y ) �9 X x X : y �9 F(w,x)} is

closed and convex, (3) there exists M > 0 such t h a t for every (w,x) �9 $2 x X, ]F(w,x)] =

sup{I]y]l : y �9 F(w,x)} < M, (4_) if y �9 F(w,x) , x' >_ x and y' < y, Hx']l,]]y'I] -< M , then y ' �9

F(.~,x'), (5) for every w �9 12, there exists (x ,y ) �9 G r F ( w , . ) such t ha t y - x �9

int X + (recall t ha t we hav, e assumed tha t in tX+ ~ 0).

Note tha t hypothes is H(F)(_2) is very c o m m o n in economic models . I t says t ha t if Yl and Y2 are two produc t ions possible for the pair (w, x) C 12 x X, then so is their weighted average kyl + (1 - k)y2 with k �9 [0, 1]. In fact if we also assnme t h a t 0 �9 F(w, x) C $2 x X (which in concrete economic te rms means" t ha t the possibil i ty of inact ion is always present) , then hypothes is H(F)(2_) implies t h a t if y is a possible ou tcome for the pa i r (w,x) �9 12 x X, then so is ky for every k �9 [0, 1]; in o ther words, nonincreasing re turns to scale prevail (i.e. for any possible y one can arbi- t rar i ly decrease the scale of operat ions) . For a detai led discussion on the economic implicat ions of this hypothesis , we refer to T a k a y a m a [15], pp. 265 and 277. The closedness hypothesis on F(w, x) is a p r imar i ly m a t h e m a t i c a l requirement , which


however is consistent with economic principles. Hypothesis H(F)(3) simp]y says tha t the production possibilities are bounded, while hypothesis H(F)(4_) is known in the economics l i terature as "free disposability hypothesis" (see Takayama [15], p. 54). It says tha t it is possible for the producers to dispose of some (or even all) of the commodities. Finally, hypothesis H(F)(5) says that for every state of the environment w, there is at least one expansible stock. Note that in the context of a stochastic model with uncertainty represented by (S, E, #) as described above and with one sector and neoclassical assumptions (see Takayama [15]), the technology characteristics of our system will be given vŸ a single-valued production function f : X • E -~ X and the evolution of the process will be described by the stochastic

difference equation xk = f ( xk -1 , sk-1).

H(u) : u : /2 • X • X ~ R --- R U ( - c o ) is an integrand such that (1) (w, x, y) --* u(w, x, y) is E1 / B ( X ) • B(X)-measurable , (2) for every w E /2, (x, y) --* u(w, x, y) is u.s.c, and concave, (3) for all (x ,y) E L~ (Zo , X ) / L ~ ( X o , X ) , J ( x , y ) = f~u (w ,x (w) ,

y(~(w)))d#(w) is finite, (4) for M1 w E /2, u(w, x, y) is increasing in x and decreasing in y.

Hypotheses like H(F) and H(u) are more or less standard in growth models; see for example, Takayama [15], p. 490.

3. Three Resu l t s from Funct ional Analys i s

In this section for the convenience of the reader, we recall two well-known results from functional analysis that we will need in the sequel and we state and p r o v e a third one tha t we believe is of independent interest.

The first result is an abstract Kuhn-Tucker theorem, originally due to Hurwicz [7], (Theorem 5.3.1), that can also be found in Arkin-Evstigneev [1], The- orem 2, p. 180. Assume tha t Y is a Banach space partially ordered by a closed, convex cone K with in tK ~ ~, and Z another Banach space. Assume that C C Z is nonempty and convex.

PROPOSITION 3.1. If f : C --~ R and g : C --~ Y are concave maps, g(C) n in tK ~ 0 (Slater's condition) and the maximum of f ( . ) on C M {z C Z : g(x) >_ O} is attained at some z ~

then there exists p E K* = {y* E Y* : y* (y) > 0 for all y E K } such that

f ( z ) § p(g(z)) ~ f ( z ~ for aU z E C.

Ir is well-known tha t the dual of the Lebesgue-Bochner space L ~ ( Z n , X) is much bigger than L l (Xn , X*). However, from the economic point of view, a functional p E L~(Z,~, X)* can be regarded as a price system only if p E Ll (~n , X*). The theorem that will help us overcome this difficulty i~ the well-known Yosida- Hewitt-Levin theorem originally due to Yosida-Hewitt [16] for R-valued functions and later extended to vector-valued functions by Levin [9].

476 G. PANTELIDES a n d N.S. PA~'AGEORGIOU

A functional p E L ~ (I2n, X)* is said to be "singular with respect to q if there exist sets {Am}m>l C ~n such that: (i) Am+l C Aro for all m > 1, (ii) /~(Am) J. 0 a s m --* 0% and (iii) p(x) =P(XAmX) for all m _> 1 and all x(.) C L ~ 1 7 6

PROPOSITION 3.2. Every p E L ~ ( ~ n , X ) * has a unique decomposition p = pa + p~, where p~ is absolutely continuous with respect to #(.) and pS is singular with respect to ~(.). Furthermore [[p[[ = ][P~[I + [[P~[]"

REMARK. In fact, the result is true even if X is any Banach not necessarily separable and/or reflexive. In this case, the absolutely continuous par t p~ corre- sponds to a function v E LI(~,~, X*. ) ; i.e. v(.) is w*-measurable, meaning tha t for every x E X , w ~ (v(w),x) is X,~-measurable and [[v(-)N C L I ( ~ ~ ) + . For details we refer to Levin [9]. In our case because X is separable and reflexive, so is X*. Hence, X* . = X* and by the Pettis measurabil i ty theorem (see Diestel-Uhl [4], Theorem 2, p. 42) measurabili ty and weak measurabili ty coincide.

Recall that given a dual pair (V, W) of locally convex spaces, the Maekey topology on V, T(V, W) , is the topology of uniform convergence on w(W, V)-compact and convex subsets of W; i.e. v~Z-~v if any only if w(v~) ~ w(v) uniformly as w runs through any fixed w(W, V)-compact and convex subset of W (here w(W, V) stands for the V-topology on W defined by the pairing (V, W); so if Y is a Banach space, V = Y* and W = Y then w(W, V) is the weak topology on Y). The next result relates convergence in measure with convergence in the Mackey topology T(L~ X), L I ( Z ~ , X*)). In what follows by ~ we will denote the convergence in #-measure.

PROPOSITION 3.3. If {gm}~>l is a sequence in L ~ ( Z n , X ) such that Iz

IIgmlto~ <- 71 and gm---*g a s m ---* c% T then gm---*g as m ---* c~, with 7 being the Mackey topology on L~176

induced by the dual pair (L~176 m X), L1(22,~, X*)).

Pro@ Let K C L I (Z=, X*) be weakly compact and convex. Then from The- orem 4, p. 104 of Diestel-Uhl [4], we know tha t in particular K is uniformly inte- grabte. Hence, the set {[lv[t. IIkll : [Ivlloo < ~, k �9 K} is uniformly integrable in L~(I:,~). Therefore given e > 0 there exists h �9 L I ( S ~ ) + , h > 0, such that for all [[v[[oo < r~ and all k �9 K we have

f{ IIv(w)ll �9 I]k(w)[[d#(w) < e II~ll'llall>h} -- 2

To simplify our calculations, assume without any loss of generality that g = 0. Hence we see that ir is enough to prove tha t there exists mo huch tha t for m _> mo we have

llg~llllkll<h~ IIg~(~)ll" IIk(~)lld.(~) _<

for all k �9 K. Since K is bounded, without any loss of generality, we may assume that fr2 [[k(w)lldtz(w) -< 1 for all k �9 K. Because {w �9 ~2: h(w) = 0} = N~>0{w �9


s : h(w) < A}, there exists V > 0 such t ha t f{h<~} h(w)dtt(w) < ~. Also choose

6 > 0 such tha t i f A E Xn, #(A) < 6, thenfAh(w)d#(aJ) <_ 6" L e t 0 < 0 < min (~ ,6 ) .

Since by hypothes is g~---*g = O, there exists mo _> 1 such t ha t for all rn > mo, we have #{w E s : h(w) >_ % [[gm(w)[[ _> 0} _< 0. Hence for m _> mo and k E K we have

{I]a~ll-Ilkll<h}n{h>_'Y}n{lla~ll_>O} IIgm(w)ll " IIk(w)lld#(w)

< f h(w)d#(w) < e_. - J{h>~}n{llg,~ll_>0} - 6

Also s i n c e / . IIk(w)lld,(w) < 1 for ah k e / 4 , we have

f~ IIgm(~)ll" IIk(~)lld,(~) < 0 < ~ IIg,~ll<0} - - 6

So finally we get for m > mo and all k E K

([Ig.,[[-I[kl,<h} [lg'~("~)II llk(w)[Idp(w)

= ~{{IIgmII.IIkIl<h}n{h>_'7}n{llg~lI>_O} I l g m ( w ) l l ' I I k ( w ) l l d , ( ~

+ f IIg-,(w)ll " IIk(~o)lld.(w) J( Ilg,~ II'llkll<h}n[{h<'Y}u{ Ilgm I1<0}]

C s C - < g + g + g = ~ .

Therefore for all m > mo and all k E K we have

L ]]gm(W)]] " ]]k(w)]]d#(w) <_ e

T ==~ gm---~g.

Q.E.D.

4. Existence of Weakly Maximal Programs

In this section, using the model presented in Section 2, we prove the existence of weakly max ima l programs.

So let K --- { f E L l ( ~ o , X ) : IIf(w)II _< M #-a.e.}. Then f rom the Dunford- Pe t t i s theorem (see Diestel -Uhl [4], T h e o r e m 1, p. 101), we get t h a t K is a weakly compac t subset of L~(Xo, X) . Consider the mul t i funct ion F : K --~ 2K\{$} defined by

F(x) = {y E L I ( L 0 , X ) : y(T(w)) E F(w,x(w)) tt-a.e.}.

Clearly F( . ) has weakly compac t and convex values (note tha t the nonempt i - ness of the values of F( . ) is an immedia te consequence of hypothes is H(F)(1) and

478 G. PANTELIDES and N.S. PA15AGEORGIOU

Aumann's se[ection theorem (see Arkin-Evstigneev [1], Theorem 1, p. 166)). In what follows, we consider the set K equipped with the relative weak-Ll(S0, X) topoIogy. Recall that if 1I, Z a r e Hausdorff topological spaces a multifunction G : Y ---, 2z\{0} is said to be upper-semicontinuous (u.s.c.), if for all C c Z closed, G-(C) = {y �9 Y : G(y) N C r 0} is closed.

In the sequel, if Y is a Banach space and A C y , by a(., A) : Y* --~ R = R U {+oz} we will denote the support function of A; i.e. a(y*, A) = sup{(y*, a ) : a E A} for every y* ~ Y*.

PROPOSITION 4.1. Ir hypothesis H(F) holds, then F(.) is u.s.c, on K.

Pro@ Since K is weakly compact in L I ( ~ o , X ) , from Theorem 7.1.16, p. 78 of Klein-Thompson [8], we know that in order to establish the upper-semicontinuity of F(.), ir sut¡ to show that F(.) has a closed graph in K • K and because of the Eberlein-Smulian theorem, it is enough to show that GrF is sequentially closed in K x K (recall K is considered with the relative weak-L~(E0, X) topology).

w w L1 So ]et (xn, Yn) �9 G r F and assume that xn--*x, yn---*Y in (~0, X). Then from the definition of F(-) and by denoting by (., .) the" duality brackets for the pair ( L ~ ( Z t , X ) , L I ( S I , X * ) ) , we have for every (A,x*) �9 ~ l x X*

= sup[(XAX*, z o -r): z e F(xn)] (the support function of F(x,~))

s <_ ~up [fA(~*,(~ o~-)(~))d.(~) : ~ ~ C(~,~) 1

L SUd [(x*, v): v �9 F@, ~~(~))1 d.(~)

(using Theorem 2.2 of Hiai-Umegaki [6])

Because of hypothesis H(F)(2), we see that z --~ a(x*, F(w, z)) is concave and u.s.c. So from Rockafellar [14], we have

lira fA a(X*, F(w, xn(w)))d#(w)

-< /A cr(x*, F(w, x(w) ) )dtt(w) = a(XAX*, F(x) P T)

s y(T(~)))d.(~) < fA ~(x', F(~, x(~)))d.(~). Since A ~ E 1 w a ~ arbitrary, we deduce that

(x*,y(v(w))) <_ a(x*,F(w,x(w))) for all w E ~2\N(x*), #(N(x*)) = O.


Let {x*}n_>l be dense in X* and set N = Un>l N(x~). Clearly # (N) = 0 and for all w �9 f2\N, we have

(x*, y(-r(w))) <_ o-(x~, F(w, x(w))), n >_ 1.

Let z* �9 X* and consider {X*}m~_l a subsequence of {x*}n>_l such tha t x*A*z * in X*. Recalling that r F(w,x(w))) is continuous, we get tha t for all w �9 ~2\N, # (N) = 0

(z*, y(~(~))) < r F(~, x(~))).

Since z* �9 X* was arbitrary, we deduce that

y(~(~)) �9 ~(w, x(~)) ,-a.e.

establishing the upper-semicontinuity of the multifunction F(.). Q.E.D.

Using Proposition 4.1 above and the Kakutani -KyFan fixed point theorem, we see that the set L -- {x E K : x E F(x)} is nonempty, w-compact and because of hypothesis H(F)(2), convex too. In fact, it is evident tha t L i s w*-compact in

L ~ (L0, X). Consider the following optimization problem:

--- sup[J (x ,x ) : x E r (x ) ] -- s u p [ J ( x , x ) : x E L] (SP)

(recall J(x, y) = f~ u(w, x(w), y(~(w)))d#(w)). From Theorem 2.2, p. 61 of Rockafellar [14], we know that J(- , -) weakly u.s.c.

on K x K and so by Weierstrass' theorem, we conclude that (SP) has a solution x* E L. Observe that because of hypotheses H(F)(4_) and H(u)(4) , the stat ionary problem (SP), is equivalent to the following optimization problem:

= s u p [ J ( x , y ) : y E F(x), x < y].

S o i f Z = L ~ ( L o , X ) • 1 7 6 1 7 6 Y = L~176 X), C = G r F and g : C --. Y is defined by g(x, y) = y - x, then we have a problem similar to the one considered in Proposition 3.1. Furthermore, note that Y = L~176 X) is partially ordered by Y+ = L~(Lo ,X)+ = L~(Lo, X+) and since by hypothesis intX+ ~= O, we have

intY+ ~ 0.

PROPOSITION 4.2. Ir hypotheses H(F) and H(u) hold, then there exists p E Li (L0 , X*) such that

J(x,y) + ( p , y - x) <_ J(x*,x*) for all (x,y) E GrF.

Proof. Let H(w) = {(x,y) E GrF(w, . ) : y - x E intX+}. From hypothesis H(F)(5) we know that for all w E f2, h(w) r O. Let {x*},~>l be a sequence of

480 G. PANTELIDES a n d N.S. PAPAGEORGIOU

nonzero vectors in X~_ such that {x*}n>_ 1 = X~ . Then H(w) = [ ~ n > _ l [ { ( x , y ) :

(x~, y - x) > 0} A GrF(w, .)]. Note that beeause of hypotheses H(F)(I_) and (2), w --~ GrF(w, .) is a multifunction with nonempty, closed and convex values, whose graph belongs in t i x B ( X ) x B ( X ) . Thus G r H = { (w ,x , y ) �9 T2 x X x X : (x, y) �9 H(w)} E Z1 x B ( X ) x B ( X ) . Apply Aumann's se]ection theorem, to get x ' , y ' : I2 ~ X , Xl-measurable funetions sueh that (x ' (w) ,y ' (w)) �9 H(w) for all w �9 f2. Set x = X'OT -1 and y = y 'o~ --1. Then (x,y) �9 L~176 X ) x L~r X) and y - x �9 intL~176 X)+. So all the hypotheses of Proposition 3.1 have been satisfied. Apply that result, to find p �9 L ~ ( E 0 , X)~_, p ~ 0 such that

J(x , y) + p(y - x) < J(x*, x*) for all (~, y) �9 C r r .

We claim that in the above inequality, we can replace p by its absolutely continuous part pa E LI(E0, X*)+. To this end, let {Aro}m>1 C Zo, Aro C Aro+l, m > 1, tt(Am) $ 0 and for every m > 1 and every x E L ~ 1 7 6 we have ps(XAmX ) = pS(x) (i.e. {Am}m>l is the E0-sequence supporting the singular part

X* of p; see Proposition 3.2). Set xm = XAg, X + XA~X* and Ym ---- X A ~ Y q-XAm �9

Clearly for every m >_ 1, ym E F(Xm) and ~ Xm--+X, yni--+y. Invoking Proposition 3.3, T T L ~ we have that xm--~x and y~-*y in (~U0, X). Then by Proposition 3.2

J(xm, U~) + (p~, y~ - xm) + pS(ym _ X~) _< j (x* , x).

But p S ( y ~ _ Xm) = ps(XA,~ ( Y m - Xm)) = 0 and furthermore from Theorern 22, p. 61 of Rockafellar [14], we know that J ( . , . ) is 7--continuous on L~176 X ) • L~176 X). Thus

J ( ~ m , Yr~) -~ J ( ~ , Y) a~ ~ -~ oo.

Also (pa, ym _ Xm) --* (pO y _ x). So in the limit a s m --* c~, we get

J ( x , y ) + (p'~,y - x) <_ J(x*, x*) for a11 (x, y) E GrF.

Therefore p~ E Li(X0, X*) is the desired priee system. Q.E.D.

Using this priee system, we can define the following value-loss funetion

~ ( x , y ) = 9 - J ( x , v ) + ( p , x - y ) fo revery (x,y) E G r F .

Following Gale [5], we say that a feasible program {Yn}~_>l is "good" (Takayama [15] uses the term "eligible"), if and only if

N

lim Z ( J k + l ( y k , y k + l ) -- V) > - o o N---* oo k=0

or equivalently tha t there exists/3 E R such that for all N _> 1, we have

N

k=0


Because of the s ta t ionar i ty of our model , we have

Jk+l (Yk, Yk+l) = J(Yk o r -k, Yk+l o T - k - 1 ) .

Eventua l ly we will show tha t the max imiza t ion of the value loss funct ion -~ ( . , .) over the family of good p rograms will p roduce for us the desired "weakly max ima l p r o g r a m ' . For this, we will need the following average turnpike p rope r ty t ha t good p rograms have:

PROPOSITION 4.3. Ir hypotheses H(F) and H(u) hold and {Xn}n>l i8 a good program,

then {~~ = ~~176176 } has weak-limit points in L I ( ~ 0 , X ) n + l " n > O

and each such point is an optimal stationary program.

Pro@ First note tha t because {x,~}n>0 is feasible (being a good p rogram) and because of hypothes is h(F)(2), we have t ha t (~,~+~ y~~k=0(xk'~ o T-k)(w),

1 x--n+l z 1 ~-~n+l T _ k nH-1 ~--~k=l ~ x k 0 T - - k ) ( w ) ) ~_ G r F ( w , .) #-a.e. , and note tha t h~ f Z..~k=l X k 0 =

1 ~--,n+ 1 1 1 n+l z-.k=0 xk o z - k - ,~+lx0 and ; T f x 0 --~ 0 as n --* c~. Also because of hypotheses

H ( F ) ( 3 ) { 1 n r _ k } ~~7 Y~~k=0 xk o = Xn is relat ively sequential ly weakly compac t n_>0

in L l(X:0, X) . Thus by passing to a subsequence if necessary, we m a y assume t h a t _ w ^ , 1 n T _ k _ _ 1 n T _ k xn---~x in L I(L'0, X ) . Clearly then ~ E k = l X k o n + l E k = 0 X k o - -

1 w ̂ , ,~:_lXO---,x in L I ( ~ 0 , X ) as n --~ oe. Thus apply ing Theo rem 3.1 of [11], in the l i m i t a s n -+ oe, we get

(~*(w), ~*(w)) e conv w-l im (xk o r-k)(w), \ n • • k=o

1 ,~+1 ) )

n + 1 ~ ( ~ ~ o ~--~)(~ k = l

G G r F ( w , - ) #-a.e.

Also f rom Jensen ' s inequality, we have

j ( 1 ~ k=0

1 n+l ) n + l E x k ~ - J(x*,x*)

k=l

1 > - - - n + l

J(xk oT -k,xk+z o r - k - l ) -- J(x*,x* > - n + l

k = 0 "

the last inequali ty being a consequence of the fact t h a t {x,~}n>0 is a good program. ( 1 n Ÿ 1 x-~~-}-I ) Now recall t ha t l i m J h~f ~ k = 0 x k o , n+l Lk=I xk o z - k <_ J (~* ,~* ) (see

Rockafellar [14]). So in the limit as n --* oc, we get

J(~*, ~*) > J(x*, x*).

But recall t ha t x* E L ~ ( ~ ' 0 , X) is a s t a t iona ry ot)t imal p r o g r a m and we al- ready establ ished earlier in the proof t ha t ~* is a s t a t ionary feasible p rogram. Hence J(~*, ~*) = J(x*, x*) = V; i.e. ~* is s t a t i ona ry opt imal . Q.E.D.

4 8 2 G. PANTELIDES and N.S. PAPAGEORGIOU

Another auxil iary rersult that we will need to establish our ex is tence theorem is the following:

PROPOSITION 4.4. If hypotheses H(F) and H(u) hold and {x~},~>0, {Yn}~>o ate two good programs whose Cesaro means {xn}n_>0 and {Yn}~_>0 have the same weak limit point x*,

then N

l im E (J(x~ OT-n,X~+l OT -'~-1) -- J(Yn ~ ~ T- '~ - I ) ) N--~ oo n = 0

N

-< ~ E (~(y~ ~ ~ - ' , y~+~ ~ T-"-~) - ~(~~ ~ T-~, ~~+ ~ o ~ - ~ - 1 ) ) .

r t = 0

Proof. N o t e that

N

n = 0

N T - n - 1 ) - - V <p, X n o T - - n - - x n + l o T - - n - - l > --__ E [ J ( X n ~ x n + l O

n 0

- J (Y~ o T- '~ , Y,~+I ~ - n - a ) + V + (P,Y~ ~ - ~ -- Y~+I ~ - n - l )

- - <p, X n 0 T - n - - X n + 1 0 T - - n - - l > - - <p, Yn o T - n - - Y n + l O T - - n - - l > ]

N = E [~(Yn O T - - n , Y n + l ~ ~ xn+l O'T--n--1)]

n : O

+ ~,yN+I oT - N - t - x N + i oT-N-I>.

Therefore, we get that

N

lim E [J(x~ o T-'~,X~+I OT - ~ - 1 ) -- J(Y,~ ~ ~ T--~--I)] N---+co n=O

N

_< l im E [~(Y'~ o T-'~, Y,~+I o T ; '*-1) - ~(x,~ ~ - '~ T ~Xn+l 0 T--n--li ] N~c~ n=O

+ l i r a (P, YN+I ~ - - X N + I oT--N--I> �9 N.---+oo

B u t from Cesaro's inequal i ty and using Propos i t ion 4.3, s ince {xn}~>0 and {Y~},~_>o have the same weak limit point , we get

l i m (p, YN+I o T - N - 1 -- XN+ 1 o T - N - l ) N~oo

N + I < l im 1 E - - N ~ o r N + 1 (P, Yn o T - n -- X,~ o T -'~ } = 0.

n ~ 0


Thus the inequality claimed by the proposition follows. Q.E.D.

Note tha t the left-hand side of the inequality in Proposition 4.4, is the limit inferior involved in the definition of the weak maximali ty criterion. This them sug- gests that ir is appropriate to consider the following optimization problem:

sup [8(~) : x �9 s ( ~ 0 ) ]

N where O(x) = limN-.oo •k=O --~(xk o T -k, Xk+l o T - k - l ) and as before S(50) is the set of feasible programs emanat ing from 50. Note that -~ (xk OT -k, Xk+ 10T - k - 1 ) <~ 0 and so the above limit is well-defined. Also from Propositions 4.3 and 4.4, it is evident that a weakly maximal program, if it exists, can be found among the good ones provided the s ta t ionary problem has a unique solution. Thus we need the following hypothesis:

H0 : there exists a good p r o g r a m {Xn}n~_O emanating from T0, and that L i s a singleton.

REMARK. If U(W,., .) is strictly concave, then L is a singleton; i.e. the stationary problem has a unique solution.

THEOREM 4.1. If hypotheses H(F), H(u) and Ho hold, then there exists a weakly maximal program.

N Proof. Let ON(X) ---- ~k=o --~(xk oT-k,Xk+l OT--k- -1 ) , X �9 S ( X 0 ) , From The- orem 22, p. 61 of Rockafellar [14], we know that ON(') is w*-u.s.c, and concave on S(~0). Furthermore as N --~ oc, 8 N .~ 8. Thus 8(.) is w*-u.s.c, and concave on S(5o). Also note that S(50) C_ 1-Ik>o K o T k and the latter is w*-compact in Hk>0 L~176 (~k, X) (Tichonov's theorem). Therefore by Weierstrass' theorem, we get that- there exists x ~ 0 = {Xn}n>0 �9 S(~0) such that

sup [0(y) : y �9 s ( ~ 0 ) ] = 0 (x~

First we claim that x ~ is a good program. To see this, let {xn},~_>0 be the good program emanating from 50, postulated by hypotheses Ho. Then given e > 0, there exists No such that for N _> No, we have

N ( 0 ~ ~ ' / O o ~ O , 0 o ~ ~ / ~)+~ j ( ~ o o ~ - ~ , ~ ~ + ~ o + (~, ~ ~ + , _ ~ ~ _

n~O

N ( - ~-n-1) o~-~-1 E J ( x n o T n Xn+l o ~ ~ , X n + l -- Xn OT - n

n:O

~) N

n z 0

484 G. PANTELIDES and N.S. PAPAGEORGIOU

N

E I _ T _ n _ l ) _ ~) > ~J(xoo~ o, xo+lo n=0

_ (p, ~ o + , o T - ~ - I - - ~ ~ + ~ o T - ~ - ~ > - - ~

>_ f l - 4 M I I P l I L ~ ( E o , X ) - - e

and from this we immediately conclude tha t o {x,~}n>0 is indeed good. X o Next we will show tha t { n},~>0 is the desired weakly maximal program. Sup-

pose not. Then there ex i s t e > 0, No ~ 1 and y 6 S(~o) such tha t for all N > No, we have

vN(x ~ + ~ < VN(y)

where VN(x ~ N 0 T - k - l ) lV = ~-~~k=O J( x~ o T -k ,Xk+l o and VN(y) = J(Yk "r-k, E k ~ O 0

Yk+l o - r - k - l ) . Hence 0 < ~ _< v N ( y ) - v N ( ~ ~

Since 0 {x~}n_>0 is good, from the above inequality we deduce that {Y,~}n_>0 is good. Fhrthermore, because {x~ maximizes 0(.} over S(50), given 6 > 0 there exists No such that for all N _> No we have

N

Z : [ ~ ( x 0 o _~ 0 T - k - l ) o T , X k + l o _ ~ ( y k o T - - k , y k + l Ÿ < 6 k=0

N

=:~ E [ V J ( x ~ o -k o 7 - k - 1 ) z o . r -k-1 - - T , X k q _ 1 0 - - <19 , kq-1 o - - XOk O T - k >

k=O

- - V + J(Yk o "r -k, Yk+i o T - k - i ) -~- ( p , Y k + l 0 T - k - 1 - - Y k 0 T - - k } ] <~

~ <_ (p,--xON+l o T - N - 1 + Yn+l o T--N--l> + 6.

As before using Cesaro's inequality and Proposition 4.3, in the l imi tas N --* oo, we get

O < e < 5 .

Since 6 > 0 was axbitrary, let 6 I 0. Then 0 < ~ < 0 a contradiction. Therefore {x~ is the desired weakly maxlmal 'program. Q.E.D.

5. G o o d Programs

From Section 4, we know that the existence of a weakly maximal program depends on the existence of a good program, because it is among good programs that a weakly maximal one can be found. In this section, we p r o v e a result on the existence of good programs. For this we will need the following hypothesis, which says tha t the initial capital stock is expansible.

H i : there exists w 6 L ~ 1 7 6 such tha t (w o 7)(w) 6 F(w,~o(W)) /~-a.e. a n d w - xo 6 intL~176 X)+ .


THEOREM 5.1. I f hypotheses H ( F ) , H(u) and H1 hold, then there exists a good program {xn}~_>o emanating flora 5o.

Proof. Choose A E [0, 1) such tha t (1 -A)x*-FASo _< w. T h e n for n _> 0 define x,, = (1 - An)x * o T n + A"50 o T n E L~176 X) . We claim tha t {xn},,_>o is a feasible

p rogram emana t ing from 5o. Since A ~ -- 1 and T o ---- id, we have tha t Xo -- 5o. Also note tha t because of the convexity of G r F (see hypothesis H(F)(2)) , we have

((1 - ~-)x* o p + ~~~ o r ~) o r(~)

�9 F(rn(w) , (1 - )~n)(x* o rn)(oj) -F/~n('x 0 0 7n(�91 #-a.e.

= F (r" (w), xn (w)) = Fn+l (w, Xn(W)) #-a.e.

Furthermore, note t ha t since by the choice of A �9 [0, 1), (1 - A)x* + A~0 _< w, we have

( (1 - A~)x * o r " + A~w o r " ) o r

_> (1 - An)x * o r n + l -t- .~n ((1 - )~)x* + A~o) o r n + l

= X* 0 r n+1 -- An(x * 0 r n+ l ) -[- )kn(x * 0 T n + l )

-- An+I (x * 0 T n+ l ) -t- An+I(x" 0 0 T n+ l )

---- (1 - An-F1)X* 0 T n+l -F An+lx 0 0 r n-F1 = Xn_F1.

So using the free disposabili ty hypothesis H(E)(4_), we get

Xn+ l (W) �9 Fn+l(W, xn(w)) ~- F(Tn(W) , Xn(W)) #-a.e.

=> {Xn}n_>0 is indeed feasible and xo = 5o. Next we will show tha t {X~}n_>O is in f a c t a good program. From Proposi-

t ion 4.2, we have

<P, X n + I 0 T - n - I - - $ n OT--n> <_ J ( x n 0 T - n , • n + l OT - n - l ) - - V ,

But note that

0 T - n - 1 (p, Xn_F1 - - X n o r - n )

= <p, ((1 - ~ n + l ) x * 0 T n + l - - ) ~ n + l x 0 0 T n + l ) 0 T - n - L

- ( (1 - A n ) x * o ~-~ - A'~~o o r n) o T--n>

---- <p, (1 - /~n+l)x* - - ) k n q - l X 0 - - (1 - An)x * + A"~o>

= <p, Ah(1 - A)(x* + Xo)>

o - o r - n - : ) - ~') >_ ~ , x * + 5 o > ( 1 - A) E Ah n_>O n>O

= (p,z* +~o> > - o c

486 C. PANTELIDES and N.S. PAPAGEORCIOU

::V {x,~},~>0 is a g o o d p r o g r a m . Q . E . D .

REMARK. In [12], we c h a r a c t e r i z e d a w e a k l y m a x i m a l p r o g r a m w i t h a s y s t e m

of s u p p o r t pr ices a n d a w e a k t r a n s v e r s a l i t y cond i t i on . T h e m o d e l c o n s i d e r e d in [12]

is no t necessaxi ly s t a t i o n a r y .

Acknowledgment. T h e a u t h o r wishes t o express his g r a t i t u d e to t h e two

a n o n y m o u s referees w h o s e m a n y r e m a r k s a n d c o n t r u c t i v e c r i t i c i sms h e l p e d i m p r o v e

t h e c o n t e n t of th i s p a p e r .

R e f e r e n c e s

[1] v . Arkin and J. Evstigneev, Stochastic Models of Control and Economic Dynamics. Trans- lated and edited by E.A. Medova-Dempster and M.A.H. Dempster, Academic Press, London, 1987.

[2] W. Brock, On existence of weakly maximal programmes in a multisector economy. 1%ev. Econom. Stud., 37 (1970), 275 280.

[3] R.A. Dana, Evaluation of development programs in a stationary economy with bounded primary resources. Mathematical Methods in Economics (ed. J. LSs), North-Holland, Ams- terdam, 1974, 179 205.

[4] J. Diestel and J.J. Uhl, Vector Measures. MathematicaI Surveys, Volume 15, Amer. Math. Soc., Providence, 1%hode Island, 1977.

[5] D. Cale, On optimal development in a multisector economy. Rey. Econom. Stud., 34 (1967), 1-18.

[6] F. Hiai and H. Umegaki, Integrals, conditional expectations and martingales of multivalued functions. J. Multivariate Anal., 7 (1977), 149 182.

[7] L. Hurwicz, Programming in linear spaces. Studies in Linear and Nonlinear Programming (eds. K. Arrow, L. Hurwicz, and H. Uzawa), Stanford Univ. Press, Stanford, California, 1958, 38-102.

[8] E. Klein and A. Thompson, Theory of Correspondences. Wiley, )�91 York, 1984. [9] V. Levin, On the Lebesgue decomposition for functionals on the vector-function space

L~(X). Fu~~ctional Anal. Appl., 8 (1974), 314-317. [10] V. Makarov and A. Rubinov, Mathematical Theory of Economic Dynamics and Equitibria.

Springe~-Verlag, New York, 1977. [11] N.S. Papageorgiou, Convergence theorems for Banach space-valued integrable multifunc-

tions. Internat. J. Math. Math. Sci., 10 (1987), 433-442. [12] N.S. Papageorgiou, Existence of support prices for a discrete-time growth model wtih un-

certainty. Preprint. [13] 1%. Radner, Optimal stationary consumption with stochastic production and resources. J.

Econom. Theory, 6 (1973), 68 90. [14] R.T. 1%ockafellar, Conjugate Duality and Optimization. Regional Conference Series in Ap-

plied Mathematics, Volume 16, SIAM, P~hiladelphia, Pennsylvania, 1973. [15] A. Takayama, Mathematical Economics. Second edition, Cambridge Univ. Press, Cambridge,

1986. [16] K. Yosida and E. Hewitt, Finitely additive measures. Trans. Amer. Math. Soc., 7"2 (1952),

45~6.

Documents

Weakly maximal stationary programs for a discrete-time stochastic growth model