An existence criterion for models of resource allocation over an infinite horizon

Economics Letters 5 (1980) 209-213 0 North-Holland Publishing Company

AN EXISTENCE CRITERION FOR MODELS OF RESOURCE ALLOCATION OVER AN INFINITE HORIZON * Michael J.P. MAGILL Uni~~ersityof Southern California, Los Angeles, CA 9000 7, USA Received

3 June 1980

This paper presents an existence criterion established in Magill (1980) for models of resource allocation over an infinite horizon. The criterion involves inequalities on parameters characterising asymptotic properties of the underlying technology and preferences. The analysis is an extension of earlier results of Koopmans (1965) and is closely related to the work of Bewley (1972).

1. Statement of problem

We consider a class of resource allocation problems over an infinite horizon Let (Z,5’) be a measurable space where, gdenotes the Lebesgue measurable sets of I and let %” = %“(I, 9) d enote the space of R”-valued (n > 1) g-measurable functions defined on (Z, 9). Let t E 7?Zn satisfy

I = [0, -).

(1) and let

x(t)

=x0

+

s 0

’{(T) d7,

tEI,

(2)

where x0 E R:. g(t) denotes a flow of output of n goods at time t, x(t) the accumulated stocks. A technology set r(t) C R: X R:, t E I, gives the set of flow outputs c(t) producible with the existing stocks x(t). We require that

[x(t), t;(r)J E r(t)

a.e.

(3)

We say that r(t) : I + R: X R: is regular if it is a closed, convex-valued measurable correspondence. E E 3fI” satisfying (l)-(3) 1s . called a feasible production program. Let $j’denote the set of all such programs. We say that I’ satisfies a growth condition * This research

was supported

by a Grant

from the National 209

Science

Foundation

SOC 79-25960.

210

M.J.P. Magill /An existence criterion for modeis of resource allocation

if there exist y, y E 7?Zand k E R: such that t 0 < y(t)
a.e.

y(r) dr < w

s

forallfEZ,

(4)

0

Ilt(t)ll G r(t)

ax.

for all < E $j ,

(5)

l(t) > ky(t)

a.e.

for some tE $j.

(6)

Let $r denote the indicator $r(x, C>t) = 0 m

function

if

(x, 0 E r(t)

if

(x, <)@ r(t)

of r

“e’

then $j’= dom \k(E) = dom [Gr[x(t), I

t(t), t] dt .

For y E ?Z satisfying (4) define the Banach spaces

V’ =L?;,, =

77E

s

'%"I llvlly,~ = h(t) r(f>ll dt
We let o(V, v’) of seminorms

A preference

where A E m,

.

denote the locally convex topology induced on 33 by the family

ordering on programs is induced by an integral functional

VU = &[E(t)~

R: XI + [--,

I

fl A(t) dt a 0 < A(t) < m a.e. denotes a discount factor. We say that u(S; t) : m) is regular if u(., t) is upper semicontinuous and concave,

dom u(., t) has a non-empty interior and g({, .) E ‘??Z,for all t E I and { E R” respectively. We say that u satisfies a growth condition if there exist upper semicontinuous non-decreasing functions 9, @I: R, j [--, -) and an upper semicontinuous concave function h : R: -+ R, satisfying h(C) 2 0 >

h(c)<0

for{EKCR:,

h(U) = M(S) >

h>O,

M.J.P. Magill /An existence criterion for models of resource allocation

211

for all (c, f)ER:

(7)

such that XI.

We say that (r, u) satisfy compatible growth conditions if s &(t) I

-m < J @[y(t) II A(t) dt , I

(8)

x] A(t) dt < 00

( E $jJis an optimal program if U(t) > U(g’) for all g’ E 9.

2. Existence criterion and application Proposition.

If (r, u) are regular and satisfy compatible growth conditions,

then

there exists an optimal program.

The basic idea of the proof is to show that U(t) and q(l) are u(V, v’) upper semicontinuous. (5) and the theorem of Alaoglu [Dunford and Schwartz (1957, p.424)] then imply that $j is ~(‘37, V’) compact, from which existence follows. As an application we consider an economy with m > 1 goods, in which program over the l= (c, v) E wnl X 3tI m d enotes the consumption-investment infinite horizon. Output devoted to investment accumulates to form a productive stock of capital, output devoted to consumption is immediately consumed so that x = (0, z) with z(t) = ze + J’, u(r) d7. A reduced form production set n(t) C Ry X RT is a set of pairs (a, b) such that b is the output producible with the stocks a. We say that the technology set I’(t) is representable in reduced form if there exists a reduced form production set II(t) such that for (x, 0 = (xl, x2,, Cl, r2) r(t)=

{(x, f>ER?”

XRf”I(x~,

i-1 +CdE

n(t),

61, Cd>O)

a.e.

For a E R” let a > 0 denote ai > 0, i = 1, .... m. We say that a production set fI C Ry X Ry is standard if (i) fl is a closed convex cone, (ii) (a, b) E TI with a = 0 implies b = 0, (iii) there exists (a, b) E n with b > 0, (iv) (a, b) E II, a’ > a, 0 < b’ G b implies (a’, b’) E n. Example. The production hedral cone

II = {(a, b)l(-a,

b) < (-A,

set in von Neumann’s model (1939) is the convex polyB) 71, 17> 0, b Z 0) ,

(9)

212

M.J.P. Magi11 /An

existence

criterion

for models of resource

allocation

where (A, B) is a pair of m X s matrices (m > 1, s > 1) satisfying (i) aii, bii > 0, (ii) for any i there exists i such that aii > 0, (iii) for any i there exists 1 such that bij > 0. (i)-(iii) imply (9) is standard. For any (a, b) E n, P(a, b) = sup b E R 1b > Pa} is called the expansion rate of the process (a, 6). P * = sup(,, b)EII ~(a, b) is called the maximal expansion rate for TI. We say that TI is productive if (a, b) E II with ~(a, b) = p* implies a > 0. Example. M’CM= {1,2,..., m} is an independent subset of goods if there exists S’ C S = {I, 2, . .. . s} such that for each i EM\M’ and j ES’, aii = 0, while for each i En/i’, bii > 0 for some j E S’. The pair (A, B) in (9) IS said to be irreducible if the set M has no non-trivial independent subsets. Gale [(1956, p.295)] has shown that if (A, B) is irreducible, then n in (9) is productive. Corollary. Let z. > 0. If the technology set F is representable in reduced form by a standard, productive set n with maximal expansion rate p*, if the discount factor is exponential A(t) = e-” , -- < 8 < 09 u is regular and satisfies a growth condition with 3

K= {~=(~l,~*)ER~‘nl~l>O}

0#/3=Gl)

@W~s~=~, =lns,

p=O,

(LO)

and if IS> PP*

(11)

then there exists an optimal program. The idea of the proof is to use an earlier result of Gale (1956) and von Neumann (1939) to show that r satisfies a growth condition [(4)-(6)]. By (10) u satisfies a growth condition. (11) ensures that (8) is satisfied, so that the growth conditions are compatible. Since r and u are regular, the result follows from the Proposition. The definition of K in (10) implies U(g) = U(c). Let m = 1, then if S
(12)

there is no optimal program. This was the case considered by Tinbergen (1960). If CT(t) denotes consumption optimal at time t for the problem on [0, T], then lim CT(t) = 0 T-t-

for all t

and

lim c#)

=m

T-tm

so that the limit of the finite horizon optimal paths is in a real sense the worstpossible path. Let /3= 1 - 77,then it can be shown that 6 + np* is a measure of the rate of impatience on a path of consumption growing exponentially at the rate p*. Since (12) reduces to

6 +qp*
(13)

M.J.P. Magill /An existcrm criterion for models of resource allocation

213

if the rate of impatience is less than the maximal rate ofgrowth p*, then the problem of resource allocation has no solution. When (13) holds if agents mimick the limit of a sequence of finite horizon problems in attempting to arrive at a solution, then they are led to a program of fruitless over-accumulation of capital. The basic economic content of the compatible growth condition (8) may be expressed in simple terms, if we recognise that this condition arises from the requirement that the preference function U(.) be upper semicontinuous in the o(V, CLI’)topology 0nV =L?y,_. Using an idea of Bewley (1972) and BrownLewis (1978) we make the following definition. If for any {, t’ E 9 such that U(t) < U($) and for < E c13there exists 7 E 1 such that U(t + tT) < U($‘) for all T> 7, where CT = {xrr,_, and where x[~,_J denotes the characteristic function of the interval [T, -), then U(.) is said to exhibit impatience on 0. It is readily shown that for any C E cz? {x[~,-, + 0 in the o(V, V’) topology as T + m. Thus if U(.) is upper semicontinuous in the @V,V’) topology on $j’, then lJ(.) exhibits impatience. The economic significance of the existence condition (8), and hence (1 l), thus lies in the requirement that the preference ordering represented by U(.) exhibit impatience in the topology of growth generated by the technology.

References Bewley, Truman F., 1972, Existence of equilibria in economies with infinitely many commodities, Journal of Economic Theory, 5 14-540. Brown, Donald J. and Lucinda M. Lewis, 1978, Myopic economic agents, Cowles Foundation Discussion Paper no. 481 (Yale University, New Haven, CT). Dunford, Nelson and Jacob T. Schwartz, 1957, Linear operators, Vol. 1, General theory (Interscience Publishers, New York). Gale, David, 1956, The closed linear model of production, in: H.W. Kuhn and A.W. Tucker, eds., Linear inequalities and related systems (Princeton University Press, Princeton, NJ). Koopmans, Tjalling C., 1965, On the concept of optimal economic growth, Pontificiae Academiae Scientiarum Scripta Varia 28, 225-300. Reprinted in: Scientific Papers of Tjalling C. Koopmans, 1970 (Springer-Verlag, New York) 485-547. Magill, Michael J.P., 1980, On a class of variational problems arising in mathematical economics (University of Southern California, Los Angeles, CA). Tinbergen, Jan, 1960, Optimum savings and utility maximization over time, Econometrica 28, 481-489. Von Neumann, John, 1939, Uber ein Gkonomisches gleichungssystem und eine verallgemeinerung der Brouwerschen fixpunktsatzes, Ergebnisse eines Mathematischen Kolloquiums 8, 73-83. Translated by G. Morgenstern as, A model of general economic equilibrium, in: John von Neumann, 1963, Collected Works, Vol. VI (Pergamon Press, Oxford) 29-37.