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On the existence of weakly maximal programs for a multisector economy with consumption
On the Existence of Weakly Maximal Programs for a Multisector Economy with Consumption Nikolaos S. Papageorgiou*‘** National Technical University Department of Mathematics, Zografou Campus Athens 157 73, Greece
ABSTRACT In this paper consumption hypotheses
we consider
over on the
an infinite data,
a growth planning
we prove
model horizon
that
there
for a multisector of continuous exists
economy
time.
a weakly
Under
maximal
with mild
feasible
program.
1.
INTRODUCTION
In this paper we consider an infinite horizon, continuous time economic growth problem and we establish the existence of a weakly maximal program (path). Our model also involves consumption. The question of existence of weakly maximal solutions (also known as weakly overtaking solutions), was considered in the past by Peleg [l], Brock and Haurie [2] and Carlson and Haurie [3]. From these works, Peleg considered a discrete time reduced (i.e., no consumption), infinite horizon growth model and proved the existence of weakly maximal stationary programs. On the other hand, Brock and Haurie and Carlson and Haurie concentrated their attention on control systems and Brock and Haurie proved the existence of a weakly maximal trajectory (weakly overtaking, in their terminology), by examining the associated Lagrange problem. Carl-
*Research supported by NSF Grant DMS-8802688. **Presently on leave at the Florida Institute of Technology, Department of Applied Mathematics, 150 West University Blvd., Melbourne, Florida 32901-6988.
APPLZED MATHEMATICS
AND COMPUTATION
48:177-186
0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010
177
(1992)
0096-3003/92/$05.00
178
N. S. PAPAGEORGIOU
son and Haurie, on the other hand, are primarily concerned with the derivation of necessary and sufficient conditions for overtaking optimality and corresponding turnpike theorems. Our work in this paper is closer to that of Peleg,
2.
THE
hut our model
is more general
than his.
MODEL
We consider
a model
of economic
growth
with m-commodities,
so our
state space is Rm. As we already mentioned in the introduction, our planning horizon is infinite, i.e., is R,. Our economic process is described by a technology set G z W” x W m x W”. If (k, u, c)EG, then this means that if at a time instant capital k is available, then a level u of investment rate and a level c of consumption rate can be realized. We are also given a function u: W”XRmXW”-+” LJ describing the social utility (welfare) for every combination (k, u, c) of capital-investment rate-consumption rate. An admissible program (path) is a pair of absolutely continuous functions
k: R++WT and c: R++W” such that (k(t), k(t), b(t))~G a.e., k(O)= k,, c(0) = co with k,, ca~W+m and 0 0. Recall that by a well-known
k(s), capital
theorem
of Lebesgue,
the absolutely
continuous
functions
c( *) are almost
everywhere differentiable. Here k(t) denotes the at time t, k(t) is the investment rate and c(t) the rate of the initial endowment of The vector k o E WT represents
stock
consumption.
bundle allocated to capital, the vector CLEWS is the initial commodity consumption and r]&$T is for every time instant t > 0 an upper bound for the consumption vector c(t), which is assumed to be nonnegative in W”. In what follows we will denote the set of admissible programs by F. We can also think
of c(t)
as the
resource
stock
and
-c(t)
as the
resource
depletion rate. Since our time horizon is infinite to avoid issues about the convergence of integrals, which require restrictive hypotheses, we adopt as our criterion in evaluating admissible programs the weak maximality criterion. So a program (k, C)E F is said to be “weakly maximal,” if for any other admissible
program
(k’, c’) (i.e.,
(k’, c’)EF),
-
:li
~b[u(k(t),i(t),e(t))-u(K’(t),ic’(t),6.jf))]
we have
dt>O
This means that for every E > 0 and every b, > 0 we can find b > b, such that 1: u(k( t), k(t), k(t)) dt > 10”u(k’( t), k(t), C(t))] dt - E. In the terminology of [2], the program (k, C)E F is weakly maximal, if it is not overtaken by any other admissible program.
179
Weak1y Maximal Program Our goal is to establish
the existence
To this end, we will need problem.
of a weakly maximal pair (k, C)E F.
the following
hypotheses
on the data of our
G c W” x W” x W” is nonempty,
HYPOTHESIS (Hl).
closed
and con-
vex.
If k&3:,
HYPOTHESIS (H2). 4( Ilk 11)with 4: R++W+
HYPOTHESIS(H3). (k, u, c)eG,
(k, u, C)EG implies
continuous,
There
exists
monotone
Xe(0,
that (lvI[ + 11cII < and 4(O) = 0.
1) such that for every
k~R;l,
if
then v+ AkeRy.
HYPOTHESIS(H4).
There
exists a positive
constant
M > 0 such that if
IlkI1 2 M, k&2?, th en for all (k, u, c)EG, we have (k, u) < 0 (here (m, *) denotes the inner product in Rm). These hypotheses represent the technological constraints in our economy and are
more
or less
standard
in problems
of growth
theory.
So
hypothesis (H2) means that the investment and consumption rates are determined by the availability of capital. Hypothesis (H3) says that net investment cannot be lower than loss due to depreciation and finally hypothesis (H4) implies that if capital exists in sufficiently large quantities, then loss due to depreciation exceeds production.
HYPOTHESIS(H5). (i) (ii)
There
exists (I,
0, O)EG such that few:
u(k, 0, c)
and
k = x.
This hypothesis implies that utility level U = u(& 0, 0) can be maintained forever with no investment or consumption change if capital x is available. So capital g is a substitute for investment and consumption. Then hypothesis (H5) (i) says that utility level U cannot be exceeded without any investment, whereas hypothesis (H5) (ii) guarantees the uniqueness of capital stock Z.
180
N. S. PAPAGEORGIOU
Following there
exists
[2] we say that an admissible
program
(k, C)EF is “good,”
if
r9~W such that
jy[u(k(t), k(t), c(t))+] for all b > 0. This is equivalent
&
dt>e
to saying that
/+[u(k(t),ic(t),k(t))-ii]
dt>-a~.
0
b+m
In the terminology “eligible.”
of Takayama
There
HYPOTHESIS(H6).
[4, page 5891, such a program
exists at least one good program
is called
(k, c)EF.
This hypothesis implies that the underlying technological basis is rich enough to sustain a utility (welfare) level close to U. The next hypothesis is also about the richness
of the technology
set of our economy.
HYPOTHESIS(H7). For every peRrn \ {0}, sup{( p, u): (k, u, c)EG} Finally we will need a hypothesis about the utility function.
> 0.
HYPOTHESIS(H8). u: Rrn x W” x W” -+ R is a positively homogeneous, continuous concave function such that there exist a, b, CE L’(W+, RI”‘) =
L’,(W+ ) for which denotes
the concave
/a” u*( a( t), b(t), c(t)) dt > - 00, where u*( *, -, *) conjugate (i.e., concave Fenchel transform) of u( *, -,
-) (see 151). 3.
MAIN RESULT We will start with an auxiliary
feasible
result
that gives us some bounds
for the
programs.
LEMMA 3.1.
If hypotheses
suchthatllkIIm, llkllo~
( Hl)-(
H4)
llcll,, llCllo< M,
hold, then there exists M, > 0 for all (k, c)EF.
Weakly Maximal Program PROOF.
Let
181
(k, C)E F. Then by definition (k(t),
k(t),
we have a.e.
i(t))~G
Then by hypothesis (H3), we can find he(O, 1) such that i<(t)+ Ak(t) 2 0 a.e. * k(t) 2 k,eeht 2 0 in W”; i.e., k(t)&iT for all HEW,. Next let b > 0. Since k(-) is continuous for all b > 0, we can find tO~[O,
b] such that jlk( to) 11= max[jIk(t)II: te[O, b]]. We claim that Ilk(t,)II < max[ Ilk,, 11, (I111, M] = y. Suppose not. Then llk( to) )I > y and because of the continuity of k(m), we can find 8 > 0 small enough so that y < Ilk(t) II for all te[ t, - 19, t,]. But then since kc*) is R y-valued from hypothesis (H4), we have k(t))< 0 a.e.
(k(t),
on [t, - 0
*%llk(t)ll”
Ilk(to)II < Ilk(to- 0) II
* contradicting for all te[O,
the choice
b]. Since
on
of toEIO, b]. So indeed llk( to) II < y a IIk( t) II < y was arbitrary we deduce that
beR+
Ilk llmGY Then from hypothesis
(3.1)
(H2), we have
Ilk.
llac.~~(Y)
whereas since 0 < c(t) < q for all t&d+, Euclidean norm, we have
(3.2)
from
the
monotonicity
of the
lb IImGII1 II GY So from
Equations
(3.1),
(3.2),
and
(3.3)
(3.3) we see
IIc llapIlk IlaD < MI, where M, = max(y, NY)). Now we are ready to state and prove our main existence of a weakly maximal program.
that
Ilk II_,
theorem
Ili( II,__,
about
the
N. S. PAPAGEORGIOU
182
THEOREM 3.1. Zf hypotheses weakly maximal program.
(Hl)-(
H8)
hold,
then
there
exists
a
Let (k, C)EF be the good (eligible) program postulated by (H6). Using Lemma 3.1 and the continuity of u(. , . , . ), we get
PROOF.
hypothesis
that there exists
M, > 0 such that for all b > 0 we have
I;/-bu(k(t), k(t), b(t)) 0
M,
d+
whereas clearly 11l/ b/,b k(t) dt 11< M,. Then from the Heine-Bore1 rem we know that we can find (A*, &)&I x Ry such that for sequence
theosome
b, -+ 00 we have
$Lbnu(k(t),
i(t),
b(t))
dt-+P
and
1
s h
k(t) dt+^k
as n+a.
b,O Furthermore,
3.1 we have
from Lemma
and
Il$Lbnd(t)dtl(=
Because
of the convexity
\I+%) -co\1
b,
< 2M,
as n-+oo.
.r+O
of the technology
set G, for every
have
k(t) dt, $
s”‘ k(t) dt, ; Jbn6(t) ” 0 n 0
dt
n > 1 we
183
Weakly Maximal Program Passing (HI)),
to the
limit
as n +OJ
and since
G is closed
(see
hypothesis
we get (R, 0, O)EG.
Recall that (k, C)E F is a good program. So by definition -y&j and b, > 0 such that for all b > b,, we have
where
b, + 00 with b, 2 b,. Passing
there
exists
to the limit as n+ 00, we get
Note that because of the positive homogeneity and concavity of the inequality, we utility function (see hypothesis (H8)) and using Jensen’s have
;J,4”ikit), i(t),
k(t)) dt
k(t)&, Exploiting
the continuity
of u( -, -, a), in the limit as n + 00 we get
P
K =
So
{(A, y)mxWm: A< u( x, y, z), (x, y, +G}
Clearly, because of hypothesis (Hl) and (H8), K is a convex set, whereas because of hypothesis (H5), we have (U, O)+ K. So we can apply
184 the
N. S. PAPAGEORGJOU Minkowski
R m \ { (0, 0))
separation
theorem
forall(X, we can
y)~K.Ifp=O,thenwehave(p, find XER”’ such that (x, hypothesis (H7). Thus p z 0. Divide X+( Then
of convex
sets
and get
(CL, ~)ER
x
such that
for this
y)
6, y)
~?EW” consider
y~R”forwhich
y, z)EG. But then by p z 0 to get
this
contradicts
fi = p/p.
the following
minimization
inf[J~[“-Ujr(t),ijt),~(t))-(B,i(t))]dt:
problem
(j,w)eF]=m”
0
Let {(f,, w,,)}, > 1 E F be a minimizing sequence for the preceding variational problem. From Lemma 2.1, we know that {f,),~i{f,},~i { ri~~}“,i E Lz(R+) are all relatively w*-compact sets. Since L’,(R+) (the on the previous predual of Lz(R + )) is separable, the relative w*-topology sequences is metrizable; see [6], Theorem 1, page 426. So by passing to a subsequence
if necessary,
we may assume
f,, “-‘f, f,, sf and f, zf, f, $:f and ti,,
that
L”,(W+). So for every b > 0, we have L’,[O, b] (recall L”,[O, b] - Li[O, b]). Thus by using Mazur’s
in
[6] Corollary
14, page 422) and by passing
and since G is convex, b> 0 was arbitrary, we w)EF.Also because of know that the integral w*-1.s.c.
to a subsequence
ti,, 2 ti 2 2i, in
lemma (see if necessary
we get (f(t), j(t), ti~(t))~G a.e. on [0, b]. Since get that (f(t), f(t), ti(t))~G a.e. in R+. Hence (f, hypothesis (H8) and [5], Theorem 21, page 59, we functional (a’, b’, c’) -+ Joau( a’(t), b’(t), c'(t)) dt is
on L”,(W+)x
Lz(W+)x
L”,(R+).
Hence
m,=hn/m[u-u(f”(t),~“(t),ti”(t))-(c,f;”(t))]dt 0
~S~[ii-~(f(t),i(t),~(t))-(li,i(t),]dt 0
-m,=Srn[ii-u(f(t),i(t),~(t))-(B,i(t))]dt 0
185
Weakly Maximal Program In particular there exists b,
this implies
that for every
(g,
T)EF
and for every
E > 0,
> 0 such that
+eJb[u(g(t), tip)> i(t))-~(r(t)~f(t),~(t))]dt 0
(6 f(b)- g(b))
P-4) for all b > b,. Let ( g, r) = (k, C)E F be the good program is postulated by hypothesis (H6). Then we have
whose existence
/b[u(f(t),f(t),ti(t))--ii]dt+E 0 ZSb[~(kjt),L(t),djt))-ii]dt-(li,f(b)-k(b)) 0
for all b 2 b,. But since ) ( 6, f(b) - k(b)) 1 < IIfiI(. 2 M, and (k, C)E F is a good program, then so is (f, W)E F. We now claim that (f, ZD)E F is in fact the desired weakly maximal program. Suppose not. Then there exists &a > 0, b, 2 0 and (h, Z)E F such that
for all b 2 b,. Then since (f, W)EF is a good program, from inequality (3.4) above we have
(6, f(t)-
h(t))+
so is (h, Z)EF and
E
a/)(+)>+)> i(s))ds-~~(~(s).f(s),~(s))ds
N. S. PAPAGEORGIOU
186
for t 2 max(bO, b,) = b,.
Hence (fiJ-(t)-h(q)+E>GJ
*
-
‘(e,f(+qs))ds
s
0
““(e,f(s)-h(s))ds+E(t--bz)~Eg(t-h2) s 0
*tfi>
1
--
i!
l ~;(s)ds-;y),,
7
s b,
t-b,
(~,f(s)--h(s))ds+~EBt~O
t-b,
0
Recall that since (f, w), (h, Z)EF are good programs, we have +L as t-w. So in the limit as t+oo, we get l/t/;_/+)& 1/t/,“q+s E > Ed. But E > 0 was arbitrary. So &o = 0 a contradiction. Therefore we conclude that indeed (f, W)EF is the desired weakly maximal program. REFERENCES B. Peleg,
A weakly maximal
W. Brock over
an
golden-rule
Economic Reu., 14:574-581
International
and A. Haurie, infinite
time
program
On the existence
horizon,
Math.
for a multi-sector
economy,
(1973). of overtaking
of Operations
optimal
trajectories
Research,
1:337-346
(1976). D. Carlson Berlin,
and A. Haurie,
A. Takayama, bridge,
Znfinite Horizon Optimal Control, Springer-Verlag,
1987.
Mathematical
Economics,
Cambridge
University
Press,
Cam-
1985.
T. Rockafellar, Conjugate Duality and Optimization, Regional Conference Series in Applied Mathematics, Vol. 16, SIAM Publications, Philadelphia, 1976. N. Dunford and J. Schwartz, Linear Operators, Wiley, New York, 1958.
ABSTRACT In this paper consumption hypotheses
we consider
over on the
an infinite data,
a growth planning
we prove
model horizon
that
there
for a multisector of continuous exists
economy
time.
a weakly
Under
maximal
with mild
feasible
program.
1.
INTRODUCTION
In this paper we consider an infinite horizon, continuous time economic growth problem and we establish the existence of a weakly maximal program (path). Our model also involves consumption. The question of existence of weakly maximal solutions (also known as weakly overtaking solutions), was considered in the past by Peleg [l], Brock and Haurie [2] and Carlson and Haurie [3]. From these works, Peleg considered a discrete time reduced (i.e., no consumption), infinite horizon growth model and proved the existence of weakly maximal stationary programs. On the other hand, Brock and Haurie and Carlson and Haurie concentrated their attention on control systems and Brock and Haurie proved the existence of a weakly maximal trajectory (weakly overtaking, in their terminology), by examining the associated Lagrange problem. Carl-
*Research supported by NSF Grant DMS-8802688. **Presently on leave at the Florida Institute of Technology, Department of Applied Mathematics, 150 West University Blvd., Melbourne, Florida 32901-6988.
APPLZED MATHEMATICS
AND COMPUTATION
48:177-186
0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010
177
(1992)
0096-3003/92/$05.00
178
N. S. PAPAGEORGIOU
son and Haurie, on the other hand, are primarily concerned with the derivation of necessary and sufficient conditions for overtaking optimality and corresponding turnpike theorems. Our work in this paper is closer to that of Peleg,
2.
THE
hut our model
is more general
than his.
MODEL
We consider
a model
of economic
growth
with m-commodities,
so our
state space is Rm. As we already mentioned in the introduction, our planning horizon is infinite, i.e., is R,. Our economic process is described by a technology set G z W” x W m x W”. If (k, u, c)EG, then this means that if at a time instant capital k is available, then a level u of investment rate and a level c of consumption rate can be realized. We are also given a function u: W”XRmXW”-+” LJ describing the social utility (welfare) for every combination (k, u, c) of capital-investment rate-consumption rate. An admissible program (path) is a pair of absolutely continuous functions
k: R++WT and c: R++W” such that (k(t), k(t), b(t))~G a.e., k(O)= k,, c(0) = co with k,, ca~W+m and 0
k(s), capital
theorem
of Lebesgue,
the absolutely
continuous
functions
c( *) are almost
everywhere differentiable. Here k(t) denotes the at time t, k(t) is the investment rate and c(t) the rate of the initial endowment of The vector k o E WT represents
stock
consumption.
bundle allocated to capital, the vector CLEWS is the initial commodity consumption and r]&$T is for every time instant t > 0 an upper bound for the consumption vector c(t), which is assumed to be nonnegative in W”. In what follows we will denote the set of admissible programs by F. We can also think
of c(t)
as the
resource
stock
and
-c(t)
as the
resource
depletion rate. Since our time horizon is infinite to avoid issues about the convergence of integrals, which require restrictive hypotheses, we adopt as our criterion in evaluating admissible programs the weak maximality criterion. So a program (k, C)E F is said to be “weakly maximal,” if for any other admissible
program
(k’, c’) (i.e.,
(k’, c’)EF),
-
:li
~b[u(k(t),i(t),e(t))-u(K’(t),ic’(t),6.jf))]
we have
dt>O
This means that for every E > 0 and every b, > 0 we can find b > b, such that 1: u(k( t), k(t), k(t)) dt > 10”u(k’( t), k(t), C(t))] dt - E. In the terminology of [2], the program (k, C)E F is weakly maximal, if it is not overtaken by any other admissible program.
179
Weak1y Maximal Program Our goal is to establish
the existence
To this end, we will need problem.
of a weakly maximal pair (k, C)E F.
the following
hypotheses
on the data of our
G c W” x W” x W” is nonempty,
HYPOTHESIS (Hl).
closed
and con-
vex.
If k&3:,
HYPOTHESIS (H2). 4( Ilk 11)with 4: R++W+
HYPOTHESIS(H3). (k, u, c)eG,
(k, u, C)EG implies
continuous,
There
exists
monotone
Xe(0,
that (lvI[ + 11cII < and 4(O) = 0.
1) such that for every
k~R;l,
if
then v+ AkeRy.
HYPOTHESIS(H4).
There
exists a positive
constant
M > 0 such that if
IlkI1 2 M, k&2?, th en for all (k, u, c)EG, we have (k, u) < 0 (here (m, *) denotes the inner product in Rm). These hypotheses represent the technological constraints in our economy and are
more
or less
standard
in problems
of growth
theory.
So
hypothesis (H2) means that the investment and consumption rates are determined by the availability of capital. Hypothesis (H3) says that net investment cannot be lower than loss due to depreciation and finally hypothesis (H4) implies that if capital exists in sufficiently large quantities, then loss due to depreciation exceeds production.
HYPOTHESIS(H5). (i) (ii)
There
exists (I,
0, O)EG such that few:
u(k, 0, c)
and
k = x.
This hypothesis implies that utility level U = u(& 0, 0) can be maintained forever with no investment or consumption change if capital x is available. So capital g is a substitute for investment and consumption. Then hypothesis (H5) (i) says that utility level U cannot be exceeded without any investment, whereas hypothesis (H5) (ii) guarantees the uniqueness of capital stock Z.
180
N. S. PAPAGEORGIOU
Following there
exists
[2] we say that an admissible
program
(k, C)EF is “good,”
if
r9~W such that
jy[u(k(t), k(t), c(t))+] for all b > 0. This is equivalent
&
dt>e
to saying that
/+[u(k(t),ic(t),k(t))-ii]
dt>-a~.
0
b+m
In the terminology “eligible.”
of Takayama
There
HYPOTHESIS(H6).
[4, page 5891, such a program
exists at least one good program
is called
(k, c)EF.
This hypothesis implies that the underlying technological basis is rich enough to sustain a utility (welfare) level close to U. The next hypothesis is also about the richness
of the technology
set of our economy.
HYPOTHESIS(H7). For every peRrn \ {0}, sup{( p, u): (k, u, c)EG} Finally we will need a hypothesis about the utility function.
> 0.
HYPOTHESIS(H8). u: Rrn x W” x W” -+ R is a positively homogeneous, continuous concave function such that there exist a, b, CE L’(W+, RI”‘) =
L’,(W+ ) for which denotes
the concave
/a” u*( a( t), b(t), c(t)) dt > - 00, where u*( *, -, *) conjugate (i.e., concave Fenchel transform) of u( *, -,
-) (see 151). 3.
MAIN RESULT We will start with an auxiliary
feasible
result
that gives us some bounds
for the
programs.
LEMMA 3.1.
If hypotheses
suchthatllkIIm, llkllo~
( Hl)-(
H4)
llcll,, llCllo< M,
hold, then there exists M, > 0 for all (k, c)EF.
Weakly Maximal Program PROOF.
Let
181
(k, C)E F. Then by definition (k(t),
k(t),
we have a.e.
i(t))~G
Then by hypothesis (H3), we can find he(O, 1) such that i<(t)+ Ak(t) 2 0 a.e. * k(t) 2 k,eeht 2 0 in W”; i.e., k(t)&iT for all HEW,. Next let b > 0. Since k(-) is continuous for all b > 0, we can find tO~[O,
b] such that jlk( to) 11= max[jIk(t)II: te[O, b]]. We claim that Ilk(t,)II < max[ Ilk,, 11, (I111, M] = y. Suppose not. Then llk( to) )I > y and because of the continuity of k(m), we can find 8 > 0 small enough so that y < Ilk(t) II for all te[ t, - 19, t,]. But then since kc*) is R y-valued from hypothesis (H4), we have k(t))< 0 a.e.
(k(t),
on [t, - 0
*%llk(t)ll”
Ilk(to)II < Ilk(to- 0) II
* contradicting for all te[O,
the choice
b]. Since
on
of toEIO, b]. So indeed llk( to) II < y a IIk( t) II < y was arbitrary we deduce that
beR+
Ilk llmGY Then from hypothesis
(3.1)
(H2), we have
Ilk.
llac.~~(Y)
whereas since 0 < c(t) < q for all t&d+, Euclidean norm, we have
(3.2)
from
the
monotonicity
of the
lb IImGII1 II GY So from
Equations
(3.1),
(3.2),
and
(3.3)
(3.3) we see
IIc llapIlk IlaD < MI, where M, = max(y, NY)). Now we are ready to state and prove our main existence of a weakly maximal program.
that
Ilk II_,
theorem
Ili( II,__,
about
the
N. S. PAPAGEORGIOU
182
THEOREM 3.1. Zf hypotheses weakly maximal program.
(Hl)-(
H8)
hold,
then
there
exists
a
Let (k, C)EF be the good (eligible) program postulated by (H6). Using Lemma 3.1 and the continuity of u(. , . , . ), we get
PROOF.
hypothesis
that there exists
M, > 0 such that for all b > 0 we have
I;/-bu(k(t), k(t), b(t)) 0
M,
d+
whereas clearly 11l/ b/,b k(t) dt 11< M,. Then from the Heine-Bore1 rem we know that we can find (A*, &)&I x Ry such that for sequence
theosome
b, -+ 00 we have
$Lbnu(k(t),
i(t),
b(t))
dt-+P
and
1
s h
k(t) dt+^k
as n+a.
b,O Furthermore,
3.1 we have
from Lemma
and
Il$Lbnd(t)dtl(=
Because
of the convexity
\I+%) -co\1
b,
< 2M,
as n-+oo.
.r+O
of the technology
set G, for every
have
k(t) dt, $
s”‘ k(t) dt, ; Jbn6(t) ” 0 n 0
dt
n > 1 we
183
Weakly Maximal Program Passing (HI)),
to the
limit
as n +OJ
and since
G is closed
(see
hypothesis
we get (R, 0, O)EG.
Recall that (k, C)E F is a good program. So by definition -y&j and b, > 0 such that for all b > b,, we have
where
b, + 00 with b, 2 b,. Passing
there
exists
to the limit as n+ 00, we get
Note that because of the positive homogeneity and concavity of the inequality, we utility function (see hypothesis (H8)) and using Jensen’s have
;J,4”ikit), i(t),
k(t)) dt
k(t)&, Exploiting
the continuity
of u( -, -, a), in the limit as n + 00 we get
P
K =
So
{(A, y)mxWm: A< u( x, y, z), (x, y, +G}
Clearly, because of hypothesis (Hl) and (H8), K is a convex set, whereas because of hypothesis (H5), we have (U, O)+ K. So we can apply
184 the
N. S. PAPAGEORGJOU Minkowski
R m \ { (0, 0))
separation
theorem
forall(X, we can
y)~K.Ifp=O,thenwehave(p, find XER”’ such that (x, hypothesis (H7). Thus p z 0. Divide X+( Then
of convex
sets
and get
(CL, ~)ER
x
such that
for this
y)
6, y)
~?EW” consider
y~R”forwhich
y, z)EG. But then by p z 0 to get
this
contradicts
fi = p/p.
the following
minimization
inf[J~[“-Ujr(t),ijt),~(t))-(B,i(t))]dt:
problem
(j,w)eF]=m”
0
Let {(f,, w,,)}, > 1 E F be a minimizing sequence for the preceding variational problem. From Lemma 2.1, we know that {f,),~i{f,},~i { ri~~}“,i E Lz(R+) are all relatively w*-compact sets. Since L’,(R+) (the on the previous predual of Lz(R + )) is separable, the relative w*-topology sequences is metrizable; see [6], Theorem 1, page 426. So by passing to a subsequence
if necessary,
we may assume
f,, “-‘f, f,, sf and f, zf, f, $:f and ti,,
that
L”,(W+). So for every b > 0, we have L’,[O, b] (recall L”,[O, b] - Li[O, b]). Thus by using Mazur’s
in
[6] Corollary
14, page 422) and by passing
and since G is convex, b> 0 was arbitrary, we w)EF.Also because of know that the integral w*-1.s.c.
to a subsequence
ti,, 2 ti 2 2i, in
lemma (see if necessary
we get (f(t), j(t), ti~(t))~G a.e. on [0, b]. Since get that (f(t), f(t), ti(t))~G a.e. in R+. Hence (f, hypothesis (H8) and [5], Theorem 21, page 59, we functional (a’, b’, c’) -+ Joau( a’(t), b’(t), c'(t)) dt is
on L”,(W+)x
Lz(W+)x
L”,(R+).
Hence
m,=hn/m[u-u(f”(t),~“(t),ti”(t))-(c,f;”(t))]dt 0
~S~[ii-~(f(t),i(t),~(t))-(li,i(t),]dt 0
-m,=Srn[ii-u(f(t),i(t),~(t))-(B,i(t))]dt 0
185
Weakly Maximal Program In particular there exists b,
this implies
that for every
(g,
T)EF
and for every
E > 0,
> 0 such that
+eJb[u(g(t), tip)> i(t))-~(r(t)~f(t),~(t))]dt 0
(6 f(b)- g(b))
P-4) for all b > b,. Let ( g, r) = (k, C)E F be the good program is postulated by hypothesis (H6). Then we have
whose existence
/b[u(f(t),f(t),ti(t))--ii]dt+E 0 ZSb[~(kjt),L(t),djt))-ii]dt-(li,f(b)-k(b)) 0
for all b 2 b,. But since ) ( 6, f(b) - k(b)) 1 < IIfiI(. 2 M, and (k, C)E F is a good program, then so is (f, W)E F. We now claim that (f, ZD)E F is in fact the desired weakly maximal program. Suppose not. Then there exists &a > 0, b, 2 0 and (h, Z)E F such that
for all b 2 b,. Then since (f, W)EF is a good program, from inequality (3.4) above we have
(6, f(t)-
h(t))+
so is (h, Z)EF and
E
a/)(+)>+)> i(s))ds-~~(~(s).f(s),~(s))ds
N. S. PAPAGEORGIOU
186
for t 2 max(bO, b,) = b,.
Hence (fiJ-(t)-h(q)+E>GJ
*
-
‘(e,f(+qs))ds
s
0
““(e,f(s)-h(s))ds+E(t--bz)~Eg(t-h2) s 0
*tfi>
1
--
i!
l ~;(s)ds-;y),,
7
s b,
t-b,
(~,f(s)--h(s))ds+~EBt~O
t-b,
0
Recall that since (f, w), (h, Z)EF are good programs, we have +L as t-w. So in the limit as t+oo, we get l/t/;_/+)& 1/t/,“q+s E > Ed. But E > 0 was arbitrary. So &o = 0 a contradiction. Therefore we conclude that indeed (f, W)EF is the desired weakly maximal program. REFERENCES B. Peleg,
A weakly maximal
W. Brock over
an
golden-rule
Economic Reu., 14:574-581
International
and A. Haurie, infinite
time
program
On the existence
horizon,
Math.
for a multi-sector
economy,
(1973). of overtaking
of Operations
optimal
trajectories
Research,
1:337-346
(1976). D. Carlson Berlin,
and A. Haurie,
A. Takayama, bridge,
Znfinite Horizon Optimal Control, Springer-Verlag,
1987.
Mathematical
Economics,
Cambridge
University
Press,
Cam-
1985.
T. Rockafellar, Conjugate Duality and Optimization, Regional Conference Series in Applied Mathematics, Vol. 16, SIAM Publications, Philadelphia, 1976. N. Dunford and J. Schwartz, Linear Operators, Wiley, New York, 1958.