An optimal control problem in economics

VOL. 14 NO. 3 (1991) 537-544

AN OPTIMAL CONTROL PROBLEM IN ECONOMICS

JANNETT HIGHFILL

Department

ofEconomics Bradley University Peoria, Illinois 61625

and MICHAEL McASEY

Department

ofMathematics Bradley University Peoria, Illinois 61625

(Received February 5, 1990 and in revised form July 2, 1990)

ABSTRACT. The first problem in the economics ofnatural resources is to find therate at which to extract the resource in order to optimize its value when there are no extraction costs. It is shown that the existence ofan optimal extraction path is not guaranteed by a

utility function that is merely (strictly) concave, but that the additional requirement of "asymptotic nonlinearity" willassuretheexistenceof thedesired optimum.

KEY WORDS AND PHRASES.

Existence, optimalcontrol,nonrenewableresource.

1980AMS

(MOS)

SUBJECT CLASSIFICATION" 49A10, 90A16

1. INTRODUCTION.

The first problem in theeconomics of exhaustible natural resources is to establish the extraction rateoftheresource. This problemisformulatedin thelanguageof the calculus of variations or, more"commonly,as anoptimalcontrolproblem: chooseanextraction rate q(t)

to maximize the total discounted utility of the resource,

j.o

U(q(t))

e-t dt, subject to

o

appropriate initial conditions. The subject of this paper is the existenceofa solution to this

problem. Existencetheoremsforoptimal control problemsareabundantinthe literature,but the simplicity of the problemconsideredheredemandsasimpleanswer.

optimalsolution is notalways assured. Indeed, ifthe utility functionislinear, asolution does not exist that is "usable" for a resource extractor. Although concavity is the natural hypothesis for the utility function

U,

the existence of an optimal extraction path is not

guaranteed by the strict concavity of U. It will be shown that the additional geometric requirement of "asymptotic nonlinearity" will resolve these two difficulties and assure the existenceof thedesired optimum.

Optimal extraction problems have been discussed extensively in the literature.

Although more sophisticated models have been developed, the one here is always the first

discussed; see, for example, the text by

Dasgupta

and Heal

[1]

and the references therein.

Two recent works that include a discussion of existence for this simple resource extraction problemare Epstein’s 1983paper

[2]

and Lozada’s 1987 thesis

[3].

Both these works consider the optimal control problemfrom the point ofviewofan unboundedhorizon problemandso are more general than the paper of Yam’i

[4]

on the finite horizon problem.

More

importantly, as both Epstein and Lozada realize, the key to existence is exhaustion. The exampleat theendofsection2 will illustrate this. Thecondition of asymptoticnonlinearity introduced here is a simple geometric condition on the utility function, U(q), saying essentially that

U(q)

does not become too linear for q large. This is sufficient to guaranteeexhaustion and hence,using aresultofToman

[5],

existence isassured. It is easy to show that this condition is in fact equivalent to Epstein’s two integral conditions in his

Lemma 1, but the condition here is more easily verified and has a simple geometric interpretation. The conditionof asymptotic nonlinearity is distinct from Lozada’sconditions asexamples will show. Finally recentresults ofBotteron and

Dacorogna

[6]

in the theory of optimal foraging provideconditions similar inspirittoours. Theirresultsarecomplementary

toours inthat theirproblemhasafixed

(finite)

terminal timeandafixedterminal state.

In

section 2, notation and preliminary results will be established.

A

briefproofofa

"folk theorem" is givenwhichprovidesa simplecriterion todetermine if extractionoccurs on afiniteorinfinite interval.

An

exampleisgivenat the endof section 2ofastrictlyconcave

utility function for which the associated optimal control problem does not have a solution.

In

the second section it is shown that if the utility function is strictly concave and "asymptotically nonlinear" then an optimal solution exists. The theorem is compared with the results of Epstein

[2],

Lozada

[3],

andBotteronand

Dacorogna

[6].

2. STATEMENT

OF THE

PROBLEM AND

PRELIMINARY RESULTS.

A

utility

function

isamap U:

R +

R

i)

U is twicecontinuouslydifferentiable;

ii)

U>_0;

iii)

U

>

0;and

iv)

U"

< 0.

satisfyingthefollowingconditions:

T

I

U(q(t)) e-Stdt subject to the conditions dx

x(0)

x(T)

>

0,

>

0

d---

q’ =Xo, q

0

Since

U

is increasing and nonnegative, if an optimal path exists, the resourcewill

always be exhausted:

x(T)

0. This fact will imply the non-existence of an optimal

extractionpath for theutility function constructed at the end ofthis section. The following propositionsummarizessomewellknown necessary conditions for optimalextractionpaths.

PROPOSITION 2.1.

A)

If

x(t)

isoptimal, then lim

x(t)

0.

t--T

B)

Let q beanoptimalextraction rate. Then

i) q(t)

> 0,for 0<

< T;

q(t) =0 for

> T;

ii)

q(t)

isacontinuous,decreasingfunctionon

[0,W]

iii) q(W)

0.

Finally, it is important in problems of resource extraction to know if the time to exhaustion is finiteor infinite. This is especially critical sincethe transversality conditions

pften differ in thefinite and infinitecases.

In

the followingtheorem, it is shown that the value of the derivative ofthe utility function at q 0 providesasimple test todetermine the extent of theextractionhorizon.

THEOREM

2.2. Assumeanoptimal solutionexists. The extraction horizon is finite if andonlyifthederivative

U

isbounded

(at

q

0).

PROOF.

The Hamiltonian for theproblem is

H

U(q)

e"t

Aq

The Maximum Principle guarantees theexistenceof the constant A. Apply theMean Value Theorem to

H

as follows. Let q beafeasible extraction function (nonnegative and piecewise

continuous).

Fix t. Then

H(0)

n(q(t))

[U(0)

U(q(t))

e-t

+

Aq(t)

U’(fl)

(-q(t))

e-t

+

Aq(t)

by theMeanValue Theorem where 0<

ft< q(t)

-q(t)

U’((:I)e

-St- A

If

U’(0)

+oo

(i.e.

lim

V’(q)

+o

),

wewanttoshow thatit isnot optimalto

q---}O

+

have

q(T)

0 for any finite T. Fix T andconstructafeasiblepath q with

q(T)

> 0 but so small that

A

et

<

U’(q(W)).

Then

H(0)

H(q(W)) <

0, so by the Maximum Principle, the optimalsolutioncannot bezeroat T.

Onthe otherhand, if

U(0)

<

oo, wewant to show that the optimalsolutionwill be

zeroforlarge t. Since

U(0)

isbounded,forany extractionfunction q, the value of can

be chosen solarge that

U’(q(t))e

-t

X <

0. Thus

forq(t) >

0,

H(0)-

H(q(t)) >

0 and so

q(t)

>

0 is notoptimal.I-!

Thefinal itemofthis section is anexampleshowing that the MaximumPrinciplefails to yield the optimum extraction path for a class of utility functions satisfying assumptions

Principle yields no information, although such a function also fails tosatisfy conditions (iii)

and

(iv).

With a linear utility fimction, the present valueofutility is always increased by extracting the same quantity ofresource over a shorter time. (So the "optimal" extraction decisionwouldbe to extract allof theresourceat thefirst instant. Althoughpointmasses axe

well known to ariseas solutions tooptimal controlproblems, they are not solutions thatcan

beimplemented by a resource

extractor.)

Theproblem witha linear utility function would disappear if the feasible extraction rates were uniformly bounded. However without

introducingotherconsiderations intothe model (eg.

costs),

there are no aprioriupperbounds

onthe extraction rates which arejustified from theeconomics ofthe problem. This issuecan

also arisefor utility functions which arestrictly concave, asshown in thefollowing example.

Theexampleis similar toonebyYaari

[1964].

EXAMPLE

2.3. Let U(q) -e-q

+

q

+

1. TheHamiltonianis

H

e-6t -e-q

+

q

+

Aq.

Applyingthe Maximum Principle yields A e-6t

(e

-qq-

1)

_Solvingthisfor q gives q

ln(Ae

-6t

1).

Since q(T) 0 ifanoptimalsolution exists, itfollowsthat

A

2e-6T and q(t)

In (2e

-6(T-t)

The issuehereis the existenceofanupper bound for q. From the form of q it is seenthat theexistence ofanoptimal extractionratedependson the terminal time T. Such atime T

isfound by solving,ifpossible,thefollowing equation representingexhaustionof theresource:

T T

x

I

q(t)

dt

I

-ln(2e-(T-t)

1)

dt.

(2.1)

0 0

Since q must be nonnegative, thereisanupperboundon

T,

namely

T

_<

Tm

ln(2)/.

Thus equation

(2.1)

canbe solvedfor all stock sizes, Xo, less than somemaximum, Xm

Since

T

m is known, xm can be computed explicitly in this example by a change of

variables:

Tm

Xm

-ln(2e

-(Tm-t)

1)dr

12.

0

So for small stock sizes

(x

o

_<

xm

),

the optimal extraction path is determined by the first orderconditions.

Moreover,

the valueoftheresourceisgiven by

T T

V(x)

I

-e-q

+

q

+

)e

-t dt

I

-ln(2e

-(T-t)

-1)

e-t dt

0 0

-1

e-q(0) _q(0)

e-q(0)]

However

for x_o

>

Xm asolution

T

does notexistfor equation

(2.1).

Thus for stocksizes

greater than _Xm no solution will exhaust the resource, and so by Proposition 2.1, no

optimalsolution exists.

implies that theinitial rateofextractionis q(0)

+co.

Further,althoughanupperbound exists for the value function

V(xo)

(in fact

V(xo) <

1/t

that limit is not reached if

q(0) isfinite. Since extraction rates are,by definition, finiteeverywhere,nooptimalsolution exists incase _Xo=Xm.

3. EXISTENCE OFOPTIMAL EXTRACTION PATHS.

It is routine to check that ifthe admissible extraction functions q(t) are requiredto take on values in a boundedset

(e.g.

0

<

q(t)

<

B)

then, with the usual assumptionson

U,

thereexists an optimal path.

(See

Theorem 2 ofToman

[5].)

The precedingexample

showsthatthe boundedness requirement is nontrivial.

Moreover,

in theabsence ofextraction costs, there are no aprioriboundson the extractionrate.

However,

assuggested by Toman

[5, 7],

itmay bepossibleforbounds tobe inferredfrom other aspects of theproblem. This is done next. Thegoalis toshow that theresourcewill be exhausted. Thiswill implythat the initial extraction rate is finite and hence q(t) is bounded sincethe extraction function is monotone. Oneadditional assumption is neededon the utilityfunction

U.

The assumption is basically that the asymptotic behavior of

U

is no_At linear. This assumption is used to

guaranteethat theresourcewillbeexhausted

(i.e.

j"

q(t)

dt

Xo.)

Thisin turnprovides the requiredboundaryconditionsneededtosolvethe optimal controlproblem.

In

order to state the main theorem, two pieces of notation are needed. Let

fl

lim

U’(q),

if this limitexists; otherwiselet

fl

+co.

Similarly let a lim

U’(q).

q--O

+

q--*oo

Since U is monotone, this limit alwaysexists. Abusingnotation slightly, write

U(O)

fl

and

U(co)

c. The condition that

U

be asymptotically nonlinear is simply that lira

U(q)

cq

+co.

Thismeans that the utility functionis "not toolinear" forlarge

q--+o valuesofq.

THEOREM 3.1. Assume U satisfies conditions

(i)-(iv).

Let q(t)satisfy thefirst orderconditionsfrom theMaximumPrinciple.

In

case c

>

0, assumeadditionally that

U

satisfies lim U(q) cq

+co

Then the stock will be exhausted. Furthermore,

q(O) <

co.

PROOF.

Since

q(t)

satisfies thefirst order conditions, the Hamiltonian

H

U(q)

e-&

Aq

is maximized as a function of q. Therefore

q(t)

W(Ae

t)

where W

(V’)

"1.

There are two constants left to be determined: $ and T The terminal condition on q lim

q(t)

0 from thefirst orderconditions allows for computation of

T

as afunctionof t--T

A.

(It

is convenient for theproofto find T and

A

in this

order.)

Finally, thedomainof

W is theinterval

(c,/)

sothat c

<

A

et

<

/,forall E

[0,T]

andso

A

e [a,].

T

To

determine

A,

usethe stockconstraint: _Xo

=/

q(t)

dt. 0

T

[

W{et dt

(3.1)

Xo

0 T

Toshow that

{3.1)

has asolution

A,

observefirst that the integral

I

W(A

et dt is a

0

continuous functionof

A

Toshow that the stockcanbe exhausted,weshow that T

i)

,alim

I

W(A

et dt

+o

and 0

T

ii)

lim

_a

[

W($e

t)dt

=0. 0

Toverify

i)

and

ii)

itishelpful tochangevariables: v $et sothat T

I

W(et)dt

I

Wv(V)

dv.

0

f

w()

Assertion

i. lim _v dv

+oo.

a +

Since may be infinite, it will be convenient to fix "r E

(cr,).

The assertionis easilyverified if cr O.

In

this case

f

wlv/

v dv

_>

f

v dv

>

W(’)

dv

-, oo as A-,0.

Incase cr

>

0, use asimilarestimation,but withthe roles of

W(v)

and

1_

interchanged and apply the "asymptotic nonlinearity" hypothesis. Fix & E

(c,3)

anddefine

1

and

q,,

respectively, by

U’(I)

& and

U(qA)

.

Comparingareas under thegraphsof

U’

andits inverse

W,

weseethat

&

q

3

W(v)

i

&

W(v)

f

--I

(U’(q)-c)dq

v dv

>

v dv

>

W(v)

dv

>

1

1._(

U(q,O

trq,

U(I)- Ctl

).

Assertion nowfollows since,

q,

oo as A c and, by hypothesis, lim

U(q)-

crq q--+oo

+oo.

Assertion ii. lira

W(v)

_v dv O.

In

case

<

oo, fix

o

E

(c,)

and observethatfor

>

o

I

W(v)

W(o)

I

In case

under U and W:

I

W(v)

dv

Ao

Nowfor

A >

Ao,

observe first that

qo

I

(U’(q)-A

o)dq

0

W(v)

dv

<

oo. Tosee this, compare theareas

qo

<

[

U’(q)

dq

U(q)-U(O)

<

0

I

w(v).v

_<

A

A

A

o

Soas

A

--,

B

theintegralonthe left approacheszero.

D

W(v)

dv.

Theissueofexistence ofanoptimalsolution totheresourceextractionproblemcannow beresolved.

Suppose

aninitialstocksize _{Xo and}autilityfunction

U

satisfyingastandard set of assumptions

(i)

(iv)

are given. Under the additional assumption of asymptotic nonlinearity, an extraction function

q(t)

W(Ae

t)

has beenproduced which satisfies the necessary conditions of theMaximum Principle. Suchafunction isunique,but is itoptimal?

To

seethatitis, let

B

q(0).

Theorem 3.1 implies thatadmissibleextraction functionscan

be limited to those satisfying

q(t)

E

[0,B]

so that assumption A12 of Toman

[51

is satisfied. The other conditions of

Toman’s

Theorem 2 are easily verified and therefore an

optimalsolution exists. This discussionissurnmarized inthe followingtheorem.

THEOREM

3.2. Assume U satisfiesthe conditions

(i)-(iv). In

case c

>

0, assume additionally that

U

satisfies lim

U(q)

cq

+oo.

Then thereexistsanoptimalsolution

q---cx)

totheproblemofsection2.

Finally werelate ourresults to thosein the literature. Lozada’s thesis

[3],

of course, contains muchmore than existenceresultsfor thesimple modeldiscussed here. Howeverhis result on the problem consideredhere is basically that if a 0 or if inf

W(v) >

0 then theresource will be exhausted. So his theorem does not apply and Theorem 3.1

does)

to utility functions such as

U(q)

ln(q+l)

+

q,

U(q)

q-ff

+

q, or

U(q)

tan-lq,

while our theorem would not applyto functions such as

U(q)

,]q2_

Thusour results

aredistinctfrom Lozada’s.

Epstein’s paper

[2]

is also not principally concerned withexistence resultsforoptimal

controlproblems. Sosomeofhis conditions arephrasedtobemore applicableto histopic of analyzingrisk aversion. Lemma 1of

[2]

is the result of relevance here. The two conditions whichassureexistenceofanoptimalsolutionare

OO C1

(IC1)

I

-cU"(c)

I

-cU"(

c

U’(c)

dc oo and

(IC2)

U’(c)

de

<

oo.

Co 0

By

changing variables x

U(c)

),

it follows that these two conditions are essentially the two assertions in the proof of Theorem 3.1.

In

case a

(=

U’(e)

is positive,

(IC1)is

Lastly,

Botteron

and

Dacorogna

[6]

prove an existence theorem for the problem of minimizing

I

g(t,v’(t)) dt for functions vE

C1([0,1])

satisfying

v(0)

0,

v(1)

0

S,

and

v’(t) >

>

0. The hypotheses on g fall into 2 groups. The first group includesregularityand convexityconditionsandaremoregeneral thanourconditions.

(This

is duepartly toourmorespecializedeconomicapplications.

Our

theoremscanbegeneralized

toincludediscountingfunctions morevariedthan

e-t.)

The second hypothesison g in

[6]

is aconditionon

0g(t,y)/0y.

While similar inspirit toourconditions on

U’

theconditions arequite distinct due to the fact that theproblemin

[6]

has afixedfinal stateandafixed final time.

In

our optimal extraction problem the final time, in particular, is one of the choice variables. (Especiallyin thecase that

T

+oo,

the condition in

[6]

isinapplicable

to ourproblem.) As aresult, the conditionsof

[6]

and thecurrentpaper aredifferent andin fact are complementaryin that taken together, they coverboth the free andfixed endpoint

problems.

REFERENCES

DASGUPTA,

P.S.

and Heal,

G.M.

Economic

Theory and

Exhaustible

B,

esou’ces,

CambridgeUniversity

Press,

1979.

EPSTEIN, L.G.

Decreasing absoluterisk aversionand utilityindicesderived from

cake-eating problems,

J.

Ecoaorp,..i.c.Theory 29

(1983)

245-264.

LOZADA, G.A.

Equilibrium. in

Exhaustible

lesource

Industries, Dissertation, Stanford

University,1987.

YAARI,

M.

Ontheexistenceofanoptimalplanin acontinuous-time allocation process,

Economet’ica

32

(1964)

576-590.

TOMAN, M.A.

Optimalcontrolwithanunbounded horizon,

J. Economic

Dynamicsand

9

(1985)

291-316.

BOTTERON,

B.and

DACOROGNA,

B. Existenceof Solutions foraVariational

ProblemAssociatedtoModelsinOptimal ForagingTheory,

J.

Math. Anal.

Appl.