A general equilibrium model of international trade with exhaustible natural resource commodities

A general equilibrium model of international trade with

exhaustible natural resource commodities

Citation for published version (APA):

Geldrop, van, J. H., & Withagen, C. A. A. M. (1988). A general equilibrium model of international trade with exhaustible natural resource commodities. (Memorandum COSOR; Vol. 8804). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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Memorandum COSOR 88-04

A general equilibrium model of

international trade with exhaustible

natural resource commodities.

by

Jan van Geldrop and Cees Withagen

January 1988

Jan van Geldrop

Department of Mathematics and Computing Science

Cees Withagen

Department of Philosophy and Social Sciences

Eindhoven University of Technology

PO

Box 513

5600 MB Eindhoven

The Netherlands

In the last decade economic theory has been enriched by an abundant literature on natural exhaustible resources. It is commonly agreed upon that the origins of this (since Hotelling (1931) renewed atten-tion are rooted in the 1973 oil crisis and Forrester's (1971) book on World Dynamics and the subse-quent work of the Club of Rome. The latter type of work concentrates on global resource problems, whereas the oil crisis revealed the vulnerability of some parts of the world through international trade problems. Economic theory has been developed on both aspects. We refer to Peterson and Fisher (1977) and Withagen (1981) for surveys and to Dasgupta and Heal (1979) for a standard introduction. Here we shall be dealing with trade in exhaustible resource commodities.

The existing literature can be divided into two broad categories : one branch follows a partial equili-brium approach, the other is of a general equiliequili-brium nature. In the partial equiliequili-brium literature one studies the optimal exploitation of an exhaustible natural resource in an open economy where (world) demand conditions for the raw material are given for the optimizing economy. In this area interesting contributions were made by a.o. Vousden (1974), Kemp and Suzuki (1975), Aarrestad (1978) and Kemp and Long (1979, 1980 a,b) for the competitive case, by Dasgupta, Eastwood and Heal (1978), who deal with monopoly, by Lewis and Schmalensee (1980 a,b) for oligopolistic markets, whereas, finally, Newbery (1981), Ulph and Folie (1980) and Ulph (1982) study a cartel-versus-fringe market structure. Relatively minor attention has been paid to general equilibrium models of international trade in raw materials from exhaustible resources. Withagen (1985) surveys some studies in theis area. Kemp and Long (1980 c) present a two economy model. One of the economies is resource-rich, the other is resource-poor but has the disposal of a technology to convert the raw material into a consumer good. Each economy then aims at the maximization of its social welfare function under the condition that equilibrium on its current account prevails. Kemp and Long analyse general equilibrium under several assumptions with respect to the market behaviour of each participant in trade. Chiarella (1980) extends the analysis into two directions : first, he introduces labour and capital as factors of non-resource pro-duction (which takes place according to a Cobb-Douglas function) and, second, he allows also for lend-ing and borrowlend-ing between the countries involved. Etbers and Withagen (1984) drop the dichotomy between the economies by postulating each country to possess an exhaustible resource. The withdrawal from the resource is costly. They address the problem of existence of a general equilibrium and give a characterization. More recent papers are those by Hillman and Long (1985), Marion and Svensson (1984) and Van Wijnbergen (1985). They differ substantially from the work presented here in that they consider models with a finite (2) number of periods and assume unilateral ownership of the exhaustible resource.

The purpose of the present study is to generalize and to add several new aspects tothis general equili-brium approach. It is straightforward to see that there is a fair number of good reasonsto do so. First of all it goes without saying that a general equilibrium analysis of trade should be preferred to a partial equilibrium treatment, if only from a methodological point of view. Furthermore, it may enable us to explain the time path of a crucial variable in the theory of exhaustible resources, namely the interna-tionally ruling interest rate. In the partial equilibrium approach this variable is always assumed a fixed constant, which will tumout below to be justified only in a very special case. Second, the concise sum-ming up of the presently known models shows that the theory can (and should) be extended in a number of non-trivial ways. The assumption of unilateral ownership of a natural resource can be dropped and the assumption of a Cobb-Douglas technology, describing non-resource production possi-bilities, seems rather restrictive. Furthermore, extraction costs deserve a closer examination.

The plan of the paper is as follows. Section 2 describes the model. The central question is : do there exist prices which generate a general competitive equilibrium and, if the answer is in the affirmative, how can these prices and the corresponding commodity allocation over time for the economies partici-pating in trade be characterized ? It turns out that the literature on existence of equilibrium when an infinity of (dated) commodities is involved is not readily capable of providing the answer to the former question. Therefore we tum to an alternative approach which may also shed light on the latter question. In Section 3 we study a system whose solution is shown to be a Pareto-efficient (PE) allocation. There also properties of PE allocations are derived. Section 4 addresses the existence of PE allocations for arbitrary weighting factors (one for each economy). Section 5 deals with some comparative dynamics.

Finally Section 6 summarizes and concludes. The fonnal proofs are given in Appendix A, B and C.

(2.2)

(2.3) (2.5) 00

J

Ei(l) dl

~ So;, ;=1,2,

o

Y;(I)

=

F; (Kl(l) ,R;(t». 2. The model.

We consider two economies which can be described as follows. Economy

i's (;=1,2)

social welfare functional is given by

Exploitation is not costless. It is assumed that, in order to exploit, one has to use capital (which is per-fectly malleable with the consumer good) as an input Following Heal (1978), Kay and Mirrlees (1975) and Kemp and Long (1980 b) we postulate an extraction technology of the fixed proportionstype :

Kt(t) =a; E; (I) , (2.4)

where a; is a positive constant andKt(l) is the amount of capital used at1 by economy ;. Economy 1 is cheaper in exploitation than economy 2.

(P.2) al

<

az·

Capital can also, together with the (homogeneous) resource commodity, be allocated to non-resource production. LetR;(I), Kl(I), Y;(I) andF; denote the rate of use of the resource good, the use of capi-tal, the rate of non-resource production and the technology, respectively.

Then

00

Ji

(Ci )

=

J

e

-PI' Vi (Ci(I» dl , (2.1)

o

where 1 denotes time, Pi is the rate of time preference (p; >0), C;(I) is the rate of consumption at

time 1 and V; is the instantaneous utility function. It is assumed that Vi is increasing, strictly concave and provides a strong incentive to consume. Furthennore the elasticity of marginal utility is bounded. (P.I) V;(C;)

>

0 ,V;'(C;)

<

0 for alt Cj

>

0 ,V;(O)

=

00

v:'c.

lli(C;) :=~ is bounded.

V;

Each economy has the disposal of an exhaustible resource of which the initial (at1

=

0) size is denoted by So;.

The resources are not replenishable. Let E;(I) be the rate of exploitation of resource ;. Then it is required that

AboutF; the following assumptions, customary in models of international trade, are made. F; exhibits constant returns to scale (CRS), is increasing, concave and differentiable for positive arguments. It is furthennore assumed that each input is necessary for production. Finally, the elasticity of substitution {J.l.;) is bounded. In the sequel this set of assumptions will be referred to as P3. Apart from CRS they are rather innocuous. CRS however may seem unrealistic when labour is a factor of production. This is

SObut for the time being we shall stick to it in order to keep the model manageable.

It is customary in models of international trade to make assumptions with regard to the relation between technologies, for example in tenns of capital-intensity. We shall make such assumptions as well by using the concept of factor price-frontier (fpt). This concept will be intuitively clear but a rigorous statement can be found in appendix A.

00 00 (2.6) (3.1) (2.7) (2.8)

(2.9)

I

- J

r('t)d't x(t) :=

e

0 00 ) x(t) ( pet)

E

j (t)

+

Yj (t) )dt

+

KiO , R, (t)

+

R2(t)S; E, (t)

+

E2(t) , K! (t)

+

K~ (t)

+

KI

(t)

+

K! (t)S; K, (t)

+

K 2 (t) ,

C,

(t)

+

C

2(t)

+

K

1(t)

+

K

2(t)S; Y1(t)

+

Y2(t) . where 2) no excess demand :

3) pet)=0 when (2.7) holds with strict inequality; ret)=0 when (2.8) holds with strict inequality. Condition (2.6) needs some clarification. It is assumed that there exists a perfect world market for resource commodities which have spot pricep(t) and a perfect world market for capital services with spot price

r

(t). Since capital is also a store of value this implies the existence of a perfect world market for "financial" capital where the rate of interest isret). Hence condition (2.6) requires each economy to make plans such that the discounted value of total sales exceeds the discounted value of total expendi-tures.

(P4) The fprs have, in the strictly positive orthant, a finite number of points in common. This assumption merely states thatthe non-resource technologies essentially differ among economies.

Z; (t)

= (

Cj (t),Yj (t), Kl (t).Ri (t),Kt (t),Ei (t) ) .

) x(t) (p (t) Ri (t)

+

ret) (Kt (t)

+

K1(t»

+

Ci (t») dt S;

Let KiO be economy

i's

given initial non-resource wealth and Kj (t) its wealth at time t. A general

competitive equilibrium is defined as follows.

( K, (t), K2(t), Z, (t),Z2(t),P(t), r (t») with p (t)~ 0 and r (t)~ 0 constitutes a general competi-tive equilibrium if

1) fori

=

1,2 :Zi (t) maximizesJi (C_{i )}subject to (2.2) - (2.5) and Define

3 Pareto efficiency

It will turn out to be convenient to consider the set of Pareto-optima. In this section we shall define some properties of this set, which

carry

over to a general equilibrium. Without proof we state the first law of classical welfare economies.

Theorem 1

Let (K ,(t),K2(t),z,(t ),z2(t ),p(t ),r(t») be a general equilibrium. Then (K ,(t),K2(t ),z,(t ),z2(t)) is PE.

a

Next, consider the problem of maximizing

(3.13) (3.14)

(3.9)

(3.10) (3.11) (3.12)

J

Ei(/)dl S SiO, i = 1,2, (3.2)

o

E_i (/)~

0,

i

= 1,2, (3.3) Kt(/)=aiEi(I),

i

= 1,2 , (3.4) Yi (I)

=

Fi (Kl(/),Ri (1», i

=

1,2, (3.5)

K (I) = Y1(I)

+

Y2(I) - C1(I) - C2(I) , (3.6)

R1(I)

+

R2(I)S E1(I)

+

E2(I) , (3.7)

K (I)~ Kl (I)

+

K! (I)

+

Kf(I)

+

K~(I) , (3.8)

whereK(0) isgiven and equalsK10

+

K20and(P1),(P2) and(P3) are satisfied. Oearly, an allocation is Pareto-efficient (PE) if, for some positive ex and ~, it solves the above problem. Furthermore, if an allocation is a solution of the problem, it is PE. Remark that C1

=

0ifex:::O and C2

=

0 if~:::O. It will alwaysbeassumed that

ex>O

or~>O.

Attention will be restricted to continuousK (the "state variable") and piece-wise continuous instruments Ei ,Kl etc. These seem tobethe only classes of functions which bear economic relevance in this

con-text Discontinuity in the stock of capital would mean an infinite rate of investment or disinvestment, which cannot begiven a meaningful interpretation here, whereas discontinuities of the second kind in the instruments should be ruled out for the same reason.

This restriction enables us to invoke the Pontryagin maximum principle. Define the Hamiltonian.

00

00

H(ChC2,Kl,K!,RhR2,EhEi)

=exe

-PI'U1(C 1)

+

~e-Pz'U 2(Ci)

+~(F1(KLR 1)

+

F 2(K!,Ri)-C l-Ci)

+

01(-E1)

+

02(-Ei)

and the Lagrangean

L(C h C2,Kl,K!,Rl>R 2,El>E2,K)=H(.)

+

0'1E1

+

0'2E2

+~p(E1+E2-R I -Ri)

+

~r(K -Kl-K! -aIEl-a2Ei).

Remark that along an optimal trajectoryK (I)

>0

for all 1because otherwise consumption would equal zero which is ruled out by U;'(O)=oo. Now suppose that (K(/),U(/»):= (K(/),C_1(/),C2

(/),K1(/),K!(/),R1(/),R 2(/),E1(/),Ei/)) with K(/) >0 and U(/)~ 0 for all I solves the problem

posed above. Then there exist

non-negative constants 01and

O

2 continuous~(I)

v (I) :=(r (I) ,P (I), 0'1(I) ,0'2(I» ~ 0 which iscontinuous except possibly at points of

discon-tinuity ofU(I) such that for all 1~ 0

ae-PI' U;(Cl(/»=~(/),

~-pz'

ui

(C₂(/» = ~(/),

~(/)(P (I) - air (I»

+

O'i(I) =Oi, i =1,2 ,

--ei»

(I)/~(I)

=

r

(I) ,

Fi (Kl(/),Ri(I» - r (I) Kl(/) - P(I)Ri (/)~

These conditions

are

all straightforward (see Takayama (1974» except for (3.13) which may need some clarification. It follows from the necessary conditions that the Hamiltonian is maximized with respect to the instruments subject to (3.3) - (3.5) and (3.7) - (3.8). The conditions stated above

are

not only the necessary conditions for an optimal trajectory. It will be shown in Theorem 12 that they

are

sufficient as well. It is furthermore easily seen that the integral (3.1) is bounded from above (the proof is given in Appendix B).

The equations can be given a nice economic interpretation when

ell,

OJ.P and r

are

thought of as the shadow-prices of non-resource production, a non-exploited unit of resource i, resource input in non-resource production and capital input respectively. r andp will hereafter be referred to as the (real) shadow-price of capital and the (real) shadow-price of exploited commodities.

Much of the subsequent analysis can conveniently be illustrated graphically in (r,p)-space. See figure 3.1 below.

Figure 3.1

In view of the homogeneity ofFj • the left hand side of (3.13) equals zero for bothi. Leti be fixed for

themoment. The set of real factor shadow-prices for which (3.13) holds is convex and closed. In order for economy i to produce the non-resource commodities it is necessary that the real shadow-prices belong to the boundary of the set just mentioned. This boundary will be called factor price frontier (fpt). It is negatively sloped and hits (one ot) the axes and/or has (one ot) the axes as an asymptote. There cannot be positive asymptotes because of the necessity of both inputs. Examples of fpfs are the curves 1 - 1 and 2 - 2 in figure 3.1. See Appendix A for the derivation of these results. With the interpretation of r and p as shadow-prices. the line r

=

pfaj is the locus where shadow-profits of exploitation from resource i

are

nil.

Let us now list some properties of the solution of the system given by (3.2) - (3.17). To each of the fol-lowing theorems we add a description ofthe line of proof or merely the intuition on which the proof is based. This might be misleading for its simplicity but those readers who are interested in the formal (3.15) (3.16) (3.17)

O';(I)E;(I)

=

O. i

=

1.2.

r

(I) (K (I) -

KI

(I) - Ki (I) - K1 (I) - K~(I»

=

0 ,

P (I) (E1(I)

+

E2(I) - R1(I) - R2(I) )

=

0 .

proofs are invited togocarefully through the appendices.

Let Q'"if) bedefined as the point of intersection of the line r =pfaI and the fpf for which in that point

r

is maximal. See figure 3.1.Evidently real factor shadow-prices should allow for non-negative shadow profits from exploitation of the cheapest resource. Hence

r

(t)S

r.

[]

IJ

Theorem

4.

r

(t) > 0

for

all t ~ O. Theorem 5.

p(t) andr (t) are continuous for all t ~

O.

Theorem

6.

For the real shadOW-price trajectories there are two possibilities. 1. (r(t),p(t» =Q'",j})for all

t

~

0;

2.

r (t) <

r ,

p

(t)

>

jf for allt ~

O.

If possibility 2 occurs then

r

(t) monotonically decreases towards zero and p (t) monotonically increases to infinity

or

a given constant, namely where a fpf is tangent to thep-axis. I] If the real shadow-prices equal Q'"JJ) for an instant of time, then it follows from (3.11) that they will have these values forever. In this case the second economy will never exploit (otherwise 9₂<0). The first part of the second statement then immediately follows. The time-path of the real shadow-prices fol-lows from (3.11), (3.12) and the fact that 9; 's

are

constants. The result is in fact a kind of Hotelling rule.

Theorem 2.

The real shadow-prices move along that fpf which, given

r,

has maximalp. [] This is clear from the fact that otherwise maximal shadow-profits of one of the economies would not be

zero.

The intuition behind Theorem 5 is simple. Inspection of equation (3.11) learns that it should hold along intervals of time where C1; is zero for some

i

for then a jump of one of the shadow-prices is accom-panied by a jump of the other in the same direction which contradicts Theorem 2. As long as the shadow-price of capital services is positive there is supply of such services and hence non-resource pro-duction takes place, which necessarily requires exploitation. So then the condition with respect to C1;

holds. What one wants to exclude therefore is the possibility ofr (t) becoming zero. This can be done by showing that in order for such a phenomenon to arise the capital-resource input ratio in one of the economies becomes infinity within finite time, which is not possible with bounded elasticities of substi-tution. This type of argument has also been followed by Dasgupta and Heal (1974).

Theorem

3.

p

(t)

>

0

for

all t ~

O.

[]

Ifthe theorem would not hold, there would be no exploitation (C1i >0 from (3.11», nor non-resource production. But then capital is useless, its price zero and (3.13) is violated.

Theorem 7.

If the real shadow-prices are(rJJ) then min (PI'P~>

r.

[]

Since the capital-resource input ratio is constant in this case and the stock of the resource is finite

00

J

K(t)dt <00. This impliesK (t) ~0 ast ~00. It follows from (3.9), (3.10) and (3.12) that C; is

o

decreasing onlyif

Pi

>

r.

This is necessary to preventK (t)from becoming negative (see (3.6». We now turn to the description of commodity trajectories. Some of these immediately follow from the shadow-price paths.

Theorem 13.

Let {K t (t),K2(t),Zt (t),z2(t),P (t),r (t»)

be

a general equilibrium. Then Theorems 2 - 11 hold.

U

This result needs no further comment except possibly for the following. In partial equilibrium models involving exhausible resources it is customary to use constant interest rates. In view of Theorem 7 (and Theorem 15 of the next section) it can be doubted if this isingeneral justified.

[) Theorem 11.

Production gelS more capital-intensive over time.

This is a consequence of the decreasing real shadow-price of capital services. Finally there is

Theorem 8.

Suppose that the (prs do not coincide for r

=

plat and that if they intersect for positive real shadow-prices, the number of points of intersection is finite. Then non-resource production is always

special-ized. I]

This follows from Theorem 2

4 General equilibrium

This section addresses the relation between PE-allocations as discussed in the preceding section and general equilibrium.

Let {K t (t),K2(t),Zt (t),Z2(t),P (t),r (t») constitute a general equilibrium. Then, according to Theorem I, {K t (t),K2(t),Zt (t),Z2(t)) is Pareto-efficient. Since conditions (3.2) • (3.17) are necessary and sufficient for Pareto-efficiency it follows that all the results of the previous section, which charac-terize PE allocations, hold This is summarized in

Theorem 12.

[K(t)

,u

(t)} is PE, [)

The proof of this theorem concentrates on asserting that el>(t) K (t)

-+

0 as t -+

00:

the shadow-value of the stock of capital goes to zero. This being done, the rest of the proof is straightforward, using concav-ity of the functions involved.

Theorem9.

Exploitation is always specialized. Moreover, the second resource is only taken into exploitation after

exhaustion of the first resource. [)

The first part of the theorem follows from the fact that if there were simultaneous exploitation

r

el> and

Pel> would be constants (see 3.11) and real shadow-prices would both be increasing, contradicting Theorem 6. The second part is less straightforward but reslS on the idea that cheaper resources should be exhausted first(seealso Solow and Wan (1977». The order of non-resource production can easily

be

traced by looking at figure 3.1 as an example. One could start at point I where economy 1is producing. Over time pointSis reached. Afterwards economy 2takes over ad infinitum. The resource of economy 1gets exhausted. This happens after point

U

has been passed. Hence, eventually the second economy carries out both production and exploitation.

Theorem 10.

Consumption in each economy will decrease eventually, towards zero. If

r

(t)-+0 the share in total consumption will move in favour of the economy with the smaller ratio of the rate of time preference

and elasticity of marginal utility, at zero consumption. [)

00

J

e-n

C; (t)dt

:s

K;o.

i

=

1,2.

o

(5.3) (5.2) (5.1) (5.4) 00 00

J

e-n

C1(.(O),Ii,t) dt

=

KIO

o

J

e-r;_{C 2(.(0),Ii,t) dt}

=

_{K 20 ,} o

In equilibrium an equality will prevail since the instantaneous utility functions are increasing. These observations give risetothe following.

CalculateC1andC2from

U;

(C1) -

j@,

/Pt-r>I

-

a

'

U' (C-\

=

~ (P2-r>I

2 ZJ 1 e ,

-a

where ~(O) and

a

are fixed positive constants for the moment and 0

<

a

<

1. If

PI >

rand

P2 >

r there clearly exist.(0)>0 and0

<

Ii

<

1such that

5 Existence

or

general equilibrium

Hereto nothing has been said about the existence of a general equilibrium. This is the aim of the present section. In view ofthe two qualitatively different possible price-trajectories it seems useful to distinguish between them. We shall therefore first examine under which initial states of the economies the equilibrium interest rate is constant (f). Subsequently we shall go into the existence problem when the initial conditions do not allow for a constant interest rate.

Itshouldbe stressed here that we are only interested in equilibria which display a continuous stock of capital and piece-wise continuous rates of consumption, extraction etc. This implies that the results on existence of equilibria with infinitely many commodities, obtained by Bewley (1970 and 1972), Hart e.a. (1974), Jones (1983) and others cannotbe invoked in this case in general. The class of functions they allow for is much

larger

than the one we wish to employ and there is no guarantee whatsoever that their equilibrium has the desired continuity properties. Models of economic growth where the infinity of the horizon is explicit, have been studied by a.o. Radner (1961), McKenzie (1968) and Gale (1967). However these models do not take into account the exhaustibility of resources. Moreover, and this seems crucial, they work in discrete time, which, as is well-known, is a more tractable concept when dealing with existence problems. Finally, Mitra (1978 and 1980) considers a model that is closely related to ours, although there is no international trade aspect in it. He gives an elegant existence proof. However, he works in discrete time and assumes away time preference. One is therefore tempted to conclude that the present model cannot be set in a format which makes an application of known results possible.

The analysis will be conducted along the following lines. If there exists a general equilibrium it is Pareto-efficient. This implies that there exist a and ~ (~

=

1- a) such that the equilibrium commodity trajectories and the equilibrium price trajectories solve(3.2) - (3.17),where the additional variables (J;

and

9;

are implicitly defined. We then use our knowledge of PE allocations to obtain the existence results.

If

r

(t)

=r, the shadow-price of the resources,

9i' equals zero. This implies that the balance of pay-ments condition (2.6)can be written as

where C1and C2 are the solutions of (5.1) and (5.2).~(O) and

Ii

are

unique. LetK (t) be the solution of

K

(t)

=

fK(t)-C1

(~(O),Ii,t)

-

C2(~(0),Ii,t),

K (0)

=

KIO

+

K20 •

In view of(5.3) and(5.4)K (t)

>

O.

Let, without loss of generality, the first economy have the more efficient non-resource technology at

r.

Define

x

by r

=

FIK<X,l).Consider

I K(s)

S10 - S1(t) :=

J _ -

ds . (5.5)

o

x

+

al

The right hand side of (5.5) is total extraction of the first economy's resource before t, along the pro-posedprogram. For

K

=

KJ

+K1

=

KJ

+

aiR1

and

K

_

""E;

=X + a l ·

Now ifS1(t)S S10for all t, all the conditions for a general equilibrium are satisfied. The desired con· dition is therefore thatS10is sufficiently large relative toK10

+

K20' Hence we state

Theorem 14.

Suppose PI

>

r

and P2

>

r.

If

S

10 is large relative toK10

+

K20 there exists a unique general

equili-brium with(r(t),p (t» =

cr,p).

I]

Matters become seriously more complicated when the conditions of Theorem 14 do not hold. But we know that in this

case

the interest rate will monotonically fall and the resource

price

will monotonically increase. Furthermore the first resource will be exhausted first and the second will be exhausted in infinity.

Suppose that we fix a (and ~

=

1-a),

r

(0),91and that we let exploitation of the second resource start

when the rate of interest reaches r

*.

Obviously we must take

a

on the unit interval, r (0)

<

r,9

1

posi-tive and

r

*

<

r

(0). In addition, exploitation of the second resource must be profitable eventually :

p * -a 2r*'2

0,

where p * is the resource price which, together with r*, yields zero profits in non-resource production. Appendix C shows how for any vector of such parameters initial values

K

lO

,K

20

,SIO'S2O

can be calculated which would induce a general equilibrium. More formally, define

v

=

(ro,r*,a,9

1).

Theorem 15. Suppose

S

10

=

S

10(V),

S20

=

S20(V),K_lo

=KIO(v),

K₂₀

=

K_{20 (v) .}

then there exists a general equilibrium characterized by

r

(0)=

r

0,

r

(t)

<

0, EI(t)

>

0 as long as

r

(t)

>

r*, E

2(t)

>

0

for all t such that

r

(t)<

r*.

[J

This theorem does not solve the existence problem. The functions SiO(V) and KiO(v) are very hard to treat analytically. In fact we want them to be sufficiently surjective, which is difficult to verify. Two advantages of this approach should be mentioned. First, it is constructive. Second, it can easily be gen-eralized for an arbitrary number of economies participating in trade.

6Conclusions.

In this paper we have presented a general equilibrium model of trade in natural exhaustible resource commodities. General equilibrium has been characterized for quite general utility functionals and non-resource production functions. The model furthermore takes exploitation costs into account In these respects considerable progress has been made compared with other general equilibrium approaches. Existence of an equilibrium has been examined, departing from the characterization of Pareto-efficient allocations.

However some weaknesses of the model should be mentioned, thereby pointing out where further research could focus on. First there is the assumption of CRS in non-resource production. Second, the simplicity of the description of the extraction technology. Third, one could introduce (manufactured) substitutes for the resources. Finally there is the assumption of perfect mobility of capital goods which allows for instantaneous switches in productive activities from one economy to the other. Nonetheless

o

some positive conclusions can be drawn.

Our analysis does not justify to apply partial equilibrium models with constant interest rates to markets for exhaustible resource commodities, except for the rather

special

case where one of the resources is economically speaking abundant The results can easily be generalized into the direction of more resources with different exploitation costs and more non-resource production possibilities and hence the model offers a good starting point to study for example the world oil market which seems to become more and more competitive.

Appendix A

In this appendix some duality results

are

derived which

are

frequently used in the main text and appen-dix B. The following notation will be adopted. For a vectory

=

(yltY~ we write

Y ~ 0 if Yi ~ 0 for alli ,

Y~O if Y~O andy

*0,

y

>

0

if Yi

>

0

for alli .

Consider a concave and homogeneous production functionF (K, R) with the following properties

F (0,R )

=

F (K,0)

=

0 ,

Fk :

=

of 10K >

0

for all(K , R) >

0 ,

Fr :

=

of loR

>

0 for all(K, R )

>

0 .

Define

V :

=

{(r,p) I F (K, R) - rK - pR ~ 0 for all(K, R )~ OJ.

Clearly V is convex and closed. Furthermore(r, p)E V implies(r,P )~ O. Let

av

be the boundary of

V.

Lemma AI.

av

=

{(r, p)~ 0I(r, p) E V and there exists(K,

R)

~ 0 such thatF (K, f)-rK -pR

=

OJ.

Proof.

(r, p)E

av

~(r, p)E V and there exists a sequence_{(rll , PIl)}

e

V with _{(rll , PIl)} ~ (r, p).

(rll , P,.)

e

V ~ there exists_{(KIl , R,.)} such that

F_{(KIl , R,.) - rllKIl - PIlRIl}

>

O.

(K,., RIl )can be chosen on the unit circle, converging to

(K, R)

~ 0 with

F (K,R) - rK - pR~

O.

Since (r,p)E V this expression equals zero. Conversely, suppose (r,p) E intV and there exists

(K, R)

~ 0 such that

F (K, R) - rK - pR =

0 .

But then there exists(P,

p)

E V,close to(r, p) such that

F

(K, R) - PK - pR

>

0,

a contradiction.

LemmaA2.

p [] [] p

r

p r p

v

r i) '2~ '10

ii) if'2

='1 then'l =

0 andPI

>

O.

Lemma A3.

Suppose('hPl) E BV , ('2,Pi) E Bv and'2

>

'I.Then

i) P2~Pl

ii) ifP2

=

PI thenPI

=

0and'1 >O. Proof. This follows directly from Lemma 1.

Proof.

The proof follows immediately from the previous lemma.

LemmaA4.

(, .p)E BV " (,.p)

>

(f,/J)

=>

(f

,p)

EV .

Lemma AS.

8W

=

{(r, p)~ 0 I (',p)E

W

and there exists

(K,

if)~ 0 such that

F

j

(K,

if)

-,K -

pif

=

0 for

somei}.

Remarks.

1. It cannot be that the curves displayed have an f

>

0 or a fj

>

0 as an asymptote in view of the necessity of both inputs.

2. The above analysis obviously

bears

similarity with duality approaches in production theory. How-ever, in for example Diewert (1982) nothing is said about one of the input prices being zero. In resource theory this seems inevitable. But for positive prices the results are the same.

Next, attention is paid to the case of two production functions F1andF2with respective inputs K and

R indexed by 1 and2. VIandV2are defined analogously toV. Define furthermore

W :

=

VI

n

V2•

Proof. This is evident

:ConsequentlyV andBV

can

take the shapes

as

depicted below. . Proof. Thisis clear from the definition ofV and Lemma AI.

Finally y (t +) := lim y (t + h), y(t -) := lim y (t + h) . "J. 0 "i0 [] [] [] Lemma A7. (3.13)implies that (r,p)E

W.

Now suppose

there exists(r , p) ,(KI'R1)and(K2,R~ such that

F

dK

ItRI) - rK1 - pR1~ F1(K,

if) -

rK - pR for all(K, R)~

0 ,

F 2(K2, Rv - rK2 - pR2~ F 2(K, R) - rK - pRfor

all

(K, R)~

O.

This is equivalent to (3.13). LemmaA6.

Lemmata A2, A3 and A4 hold with

V

replaced by

W.

Without proof we state

AppendixB Lemma AS.

(Kj , Rj ) >0 for somei implies(r,p) E

8W.

Clearly Lemma A7 proves theorem 2. Lemma AS will turn out useful in AppendixB.

In this appendix the theorems of Section

3

are

proved. For convenience the assumptions of the model

are

restated

here.

"

,

" ,

Vi

c; .

(PI) Uj

>

0,Uj

<

0,Uj(0)

=

00 , 1')j (Cj ) :

=-,-

IS bounded,

Uj

(P2)

al

<

a2'

(P3) Fj (Kl,Rj)is concave and homogeneous and satisfies

Fi (O,Rj)

=

F j(Kl,O)

=

o.

Fif( :

=

aFj/aKl

>

0

for all (Kl,Rj)

>

0,

FiR :

=

aFj/aR j

>

0

for all(Kl,Rj)

>

0,

The elasticity of substitution

eIlj

is bounded.

(P4) The set ((r,p) I(r,p)> 0and(r,p)E ~Vl

n

~V2) is finite.

Assumption (P4) says that essentially the Fj's differ. In the sequel the argument t is omitted when there is no danger of confusion. The following notation willbeused.

Xj :

=

Kl/R j,

Ii

(xj):

=

F;(X;,1) . Itis easily seen that

Fif(

=

Ji',

FiR =

/i -

x;Ji',

Ii"

<

0,

/i

(0)=0 . ~j = -

Ji'

i f j -

x;/i')/Ji"x;/i .

Theorem 2 has been proved in Appendix

A.

We first present a lemma thatisfrequently used in the sequel. LemmaBl.

Proof. Suppose that there exists tI~ 0 such that r (t1)>0 and

Y

I(tI) =

Y

2(tI)

=O.

Then

KI

(tI)

=

K~ (t1)

=

0, otherwise (r,p)E Wand Lemma A7 is violated. Ifp(tI) = 0 then C1j (tI)>0 for both

i

(since 9j

=

,(t)(p(t)-ojr(t» +C1dt)~0) and E I (tl)=E2(tl)=R1(tl)=R2(tl)=0

(3.11, 3.15 and 3.7). Ifp (t1)>0 then also R dt1)

=

R2(t1)

=

0 since (r.p) E

W.

Therefore in both cases K{ (t1)

=

0, i

=

1;1..But K (tI)>O.It follows from 3.17that r (t1) = 0,a contradiction.

IJ

Theorem 3.

p

(t) >0 for all t ~ O.

Proof. Suppose there exists t I such thatp (t I)

=

O. Since (0,0)

e

Vj , i

=

1,2,

r

(t 1)

>

o.

9j ~ 0 for both

i,

hence C1j (tI)>0 for both

i

(from (3.11» implying E1(t1)

=

E2(t1)

=

R1(t1) = R2(tI)

=

O. Therefore

Yj (tl)

=

0, i

=

1,2 and Lemma Bl is violated. []

Theorem 4.

r (t)>0 for all t~ O.

Proof. The proof will be given in several steps.

1) Suppose that there existtl~ 0 and

i

such that r (tl)

=

0 and R j (t1)

>

O. Then Kl(tl)

>

0,

other-wise (r (tI) ,p(t1» E Vj in view of the fact thatp (tI)

>

O. Hence (r (t1) ,p(tI» E 5Vj (Lemma AI). But FiK

>

0 and this implies (r(t1),p(t1))

e

5Vj • Hence, r(t1)

=

0 implies

Rdtl)=0 ,

i

=

1;1..

2) Suppose that there exists tI

>

0 such that

r

(t1)

>

0, whereas

r

(0)=

O.

Then cj)(t1)~ cj) (0) in view of(3.12). Since r (tl)

>

0,Yj (tl)

>

0 for somei and therefore Rj (tl)

>

0 for thisi.This implies

that there existsj such thatEj (tI)

>

0 and

9

j

=

cj)(tl)(p (tl) - OJ r (tl» = cj)(0)(p(0) - OJr(0»

+

C1j(0) .

SoP(tl)

>

P(0) and (r(tl) ,p(tl»

>

(r(0) ,p(0». But then (r (0) ,p(0»

e

W, contradicting Lemma A7. Hencer (0)= 0 implies r (t)= 0 for all t ~ O.

3) r(0)

>

O.This is so since, if r (0)= 0,r (t)= 0 for all t (2) implying Rj (t) = 0 for all t and both

i;

henceEj(t) = 0 for all t and both

i

(3.17)contradicting(3.14).

4) Suppose there exists tl>0 such that r (tl)

=

0 and r (t) >0 for all 0

<

t

<

t1 •

a) Suppose that

r

(t1 - ) >O.

In view of the piece-wise continuity of

r

there exists 1

<

tIt close enough to tIt such that

ruJ

>O. Hence Yj (t)>0 for some

i

(Lemma Bl). Then also p Q)~ P(tl) otherwise

(r (tI) ,

p

(tI»

e

Vj (Lemma A4). Since cj)(t) is continuous by assumption we then have C1jQ»C1j(tl)~0 (j = 1,2). Therefore EIQ) =E₂Q) =R₁Q) =R₂Q)=O, contradicting

Yj (t)

>

0 for some

i.

So

b)

r

(tl - )= O.

Assumption P4 implies that there exists an interval ('t , tl) such that Yj (t)

>

0 for all t E ('t ,tl)

and just one i. Since E j (t) is piece-wise continuous for both i the interval ('t , tl) can be par-tioned in a finite number of subintervals such that Ej (t) (i = 1,2) is continuous along each

subinterval. Observe first that there is no subinterval with

E

I>0 and

E

2>0 along a non·

degenerated subinterval of it For suppose that there exist 1 and

t

with 't~ 1 <

t

<tI such that

EI(t) >0 and E₂(t)>0 for all t E

11.,

tj.

This implies that cj)(p - olr) and cj)(p - Off) are

constants for 1

~

t

~

1.

But

~

(t)

<

0 for 1

~

t

~

1.

Since 0I

*

02 cj)p and cj)r are constants, implying that (r

(i)

,p (0)>(rQ) ,p(t». Therefore (rQ) ,p

uJ)

e

W, which is not allowed by Lemma A7. So for all partitions

E

I(t)

=

0 ~E2(t)

>

O.

Hence there exists'f<tl such that

along the interval (f,tl) EI(t) =0 and E₂(t)>0, or E1(t) >0 and E₂(t) =

O.

Assume, without loss of generality, that YI(t»O and E1(t»0 for all t E (f,tl)' Straightforward

Let

cr,P>

be defined byji =

a

1rand

cr,P>

E

cSW.

See figure B1.

calculations yield

i 1/x l

=

(-a1(r - r2

)/I;

(Xl)xtJ

+

Ildl(xl)/X1 , t E ('f,tl) ,

where it should be recalled that

III

is the elasticity of substitution. If

r

(t)~ 0 for some t E ('f,t1)then

i

1(t)~ 0 because

l

(x1)<O. Therefore

i

1/x l~ Ildl (x 1)/X1 •

Since

r

(t) ~ 0 ,X 1must become arbitrarily large. But

lim11 (Xl)1xl =0

%1-+00

(]

so that

x

1is bounded on any finite interval. Theorem 5.

i) r(t

+ )

= r (t - ).

ii) P(t + )

=

P(t - ).

Proof. 9; is constant. Hence

Cf;(t - ) - Cf;(t +)

=

cjl(t)

{a;(r

(t - ) -

r

(t +

»-

(p (t - ) - p(t +

»} ,

i

=

1,2 .

a) Suppose that

r

(t1 - )

>

r

(t1

+ )

for some t1

>

O. Then there exist1. and

I,

close enough to t10 with1.

<

tl

<

1

such that

r

(0

>

r

(I).

Since

r

(t)

>

0 ,Y;(t)

>

0 for some i (Lemma Bl). Then

also p (t)~ p

(I)

otherwise (r

(I)

,p

(I» (/.

V; (Lemma A4). Therefore

Cfi(tJ>Cfi(l)~

0

U

=

1,2) and E1(t) =E2(tJ =R1(tJ=R2(t)=O contradicting that

Y; (t)

>

0 for some

i.

b) Suppose thatr (tl - )

<

r (tl

+ )

for some t 1>O. In this case the proof is analogous to the proof under a. This shows the validity of part i) of the theorem. The proof of part ii) is similar and

r

fig.Bl

In view of the assumptions made(r.P)exists. In fact

r

is the maximal feasible r. Theorem 6.

i) (r (t1) •P (t1»

=

(r,ji)for somet1~ 0~(r(t) • P (t»

=

(rif)for all t ~ O.

ii) Supposer (0)

<

r.

Then

a)

r

(tl)

>

r

(ti)and

p

(tl)

<

p

(t,) for all t2

>

tl~ O. b) lim

r

(t)

=O.

1

-Proof.

Ad

n.

1)

r

(t)~

r

for all t. For suppose that for some t1~ 0

r

(t1)

>

r.

Then Y;(t1)

>

0 for some

i

(Lemma Bl).P(tl) S

if

otherwise(rii)EBW (Lemmata A6 and A4).Since 9_j ~ 0 for bothj it follows that CJj(tl)>0 for both j, which implies thatEj (tl) =Rj (tl)= 0 for bothj

contradict-ing

Y

j (t1)

>

0 for somei.

2) Suppose (r(tI) •p (t1»= (rjf) for some t1~ O. Then E2(t1)= 0 because

if -

a2r <0 and 92~ O. But, since

r

(tI)>

o.

R1(t1)

+

R2(t1)>0 (Lemma B1). implying thatE1(tI) >O. There-fore 91= 0 and p (t) - aIr (t) S 0 for all t. If, for some t2~ 0, P (t,) - aIr(t,) <0 then

E1(ti)

=

0 (CJI(ti)

>

0). A fortiori E_2(t,)

=

O. Hence Yj (ti)

=

0 for both i, contradicting Lemma B1. SoP(t)= aIr(t) for all t ~ O. This proves the first part of the theorem.

Adin.

1) Suppose r (tl) S r (ti). r (t)

>

0 for all t (Theorem 4), hence Yj (tl)

>

0 for some i and

Yj(ti)>0 for some

i

(Lemma Bl). Therefore (r (tl).P (tl» E BVj for some

i

and

(r (t,),p (t,) E BVj for some

i.

So (r (tl),p (tl» E BW and (r (t,)oP (t,) E BW. (Lemma AS). Using Lemmata A6 and A2 we find p (t1)~p (t,). By the same argument r (t1)

=

r (t i) if and only if

p

(tl)

=

P (t,).

2) Now suppose r (tl)

=

r(t~.

OJ = cj)(t)(p(t) - aj r (t»

+

C1j (t) for all t and both j.

Because cj) is continuously decreasing and p (t,)

=

P (t~. C1j(t~

>

0 for both j. So

R1(t~

+

R2(t~

=0 and Yj (t

~

=

O.

contradicting Lemma B1. We can therefore restrict ourselves to the following case.

3) Suppose r (tl)

<

r(t~. Then p (t,)

>

P(t~ in view of 1). Then a fortiori Y1(t~

=

Y2(t~

=

O.

contradicting Lemma B1. This provesii) a).

4) Suppose

lim

r

(t)

=

f

>

0 .

t~

Then

~(t)/cj)(t) ~

-f as t

~

00 (3.12). So cj)(t)

~

0 as t

~

00. Furthermore p (t)

~

fi

>0 as t ~ 00. In the case at hand OJ >0 for both j. Hence Ojlcj) (t) ~ 00 as t ~ 00, implying that C1j (t) >0 for t large enough. Then Ej (t)= 0 for both j and t large enough, contradicting

Yj (t) >0 for all t and at least onei.

0

Theorem 7.

(r (t),p (t»

=

(f"ji)=>

PI

>

r,

P2

>

r.

Proof. 0

=

p (t) - al r (t)

>

p (t) - a2r (t). _{Hence E 2(t)}

=

0 for all t~ O. Since ~/cj)

=

-r

it follows thatcj)(t)

=

cj)(0)

e-".

Hence

U ' (C )

=.!ill.

(Pl-r)t. U' (C-'

=.!JQL

(Pri')t

1 1 ( I e , 2 2 1 ~e

.

Furthermore

K

=

KI

+

K!

+

a1(R1

+

R~ KIR =K1IR

+

K!IR

+

al ,

where R :

=

R1

+

R 2.

If

Y,

(t1)

>

0 then Kl (t 1)IR, (t1) is constant If

Y,

(t ,)

=

0 then Kl (t 1)IR, (t1)

=

O. So there are constants

b1andb2, such that

KIRSb 1+b2 +al.

Hence

00

J

K(t)dt<oo.

o

In view of the homogeneity ofFj F;(Kl,R,)=

r

Kl

+

pRj . Therefore and t K (t)

=

Koe" -

f

e,(t-s)C(s)ds ,

o

where C

=C

1+C2•It follows from

K

S fK, K ~ 0 and) K (t)dt <00that K (t) ~ O. To see this

T T

OJ

(K -

r

K)(dt)

=

K (T) - K(0) -

r

J

K dt So T T

r

J

K dt

+

K(0)

=

K (t)

+

J

rr

K -

K)

dt

o

0

The left hand side of this expression is bounded The second term of the right hand side is monotoni-o

cally increasing (since

K

~

r

K). ThereforeK (t) -+ 0 for otherwise

J

K dt would diverge. T

Now, if

PI

~

r

or P2~

r

then C(t)~

E

>0 for some

E

and for t large enoughK (t) becomes

nega-tive, which is not allowed []

As a corollary we mention CorollaryB1.

(r (t),p (t»

=

(T,p)::;>~(t)K (t) -+ 0as t -+ 00.

Theorem 8.

Suppose (r ,al r)f£ oV1noV2. Take t2~ tl' Suppose Y1 (t) >0 and Y2(t) >0 for all t

e

[tl,tz].

Then tl

=

t2.

Proof. Suppose there exist tl and t2 with t2> tl such thatY1(t)>0and Y2(t)>0for all t

e

[tlh].If

(r (t),p (t»

=

(T,if) for all t ~

0

then (r, air)

e

oW.

If(r (O),p

(0»

*"

(TJJ) then r is monotonically

decreasing. HenceP4 is violated. []

Theorem

9.

i) Taket2~ tl' SupposeE1(t) >

0

andE_2(t) >

0

for aU t E [tlh]. Then tl

=

t2'

I

ii) E2(t) > 0::;>

J

E1(t)dt = S10'

o

Ad

n.

The argument has already been given in the proof of theorem 4 but will be repeated here for conveni-ence. Suppose there exist t1and t2.with t2.>t1such that E1(t) >0 and E2(t) >0 for all t E [titt

zJ.

Then, from (3.11)and(3.15)

~(p-ajr)=aj' j=I,2, te[tl,tzJ.

Since a1

*"

a2' ~p and eIlr are constants along [t1,tz]. But ell(tz) <ell(t1) and therefore

(r (tz},p (tz))

>

(r (tl),p (tl»' So (r (tl),p (tl» f£ W (Lemma A4), which is not allowed (Lemma A7). Adin.

Suppose there exists t1such thatE2.(t1)>0 and

t1

J

E1(s)ds < SIO. o 00 a)

J

E1(s)ds < SIO.

o

In this case

a

1

=

0, implying (r (t),p (t»

=

(TJJ) for all t ~ O. Therefore <12(t) >0 for aU t as well as

00

18

-So

(]

eIl(ti) (p (ti) - air(ti)~

ell

(II) (p (11) - atr (tl»,

eIl(tl) (p(tl) - a2r(tl»~ eIl(ti) (p(ti) - a2r(ti). E1(ti)

>

0,

E2(ti)=

0,

(II(ti)=

O.

Hence

9

1

=

eIl(ti) (p (ti) - air (ti),

92~ eIl(ti) (p (ti) - aff (ti).

b)

J

E

t(s)ds

=

SlO'

o

There exists an interval ['tJ,'ti! , tI~ 'tt<'t2, with Et(t)> 0 and continuous, whereas, along the inter-val,E2(t)

=

O.

Take'tt

<

t2

<

't2.

Multiplication of the left and right hand sides of the first inequality by a2and of the second inequality bya1and adding yields

(a2- a l) (eIl(ti)p(ti)-eIl(tl)P(tl))~

0,

implyingp (ti) >p (tt).

Just addition of the inequalities yield

(a2 - at)

(ell

ti) r (ti) -

ell

(tt)r(tt»~0, implying

r

(t i)

>

r

(tt).

Therefore(r (ti),p (ti) > (r (tt) P (tt) and(r (tl),P (tl))

e

W, contradicting Lemma

A7.

(]

Et(tt)~ 0,E 2(tl) > 0 ,(I2(tt)

=O. Hence

(II~ eIl(tt) (p(tt) - atr (tl)),

92=

ell

(tl) (p (tl) - aff (tl».

,

Ad

iil.

The first part ofii) follows immediately from i) since 11j (Cj) is bounded. The asymptotic growth rate of

Cj is{pj-r(oo»/11i (0).This proves the secondpart. (]

Proof. This is immediate from the fact that

r

decreases and /;" < O. Theorem 11.

Suppose(r(O),p(0» ~(rJf). Yj (t)>0 implies Xj(t) > O. Theorem 10.

i) There existsT~

0

such that,

C

(t)<

0,

i =

1,2,

for all, t >T

ii) Ci (t) ~ 0

as

t ~ 00 and

Ct/(C

I

+

Ci) ~ 0

as

t ~00 if and only if

(P2-r (00»/1'12(0)

>

(Pt-r(oo»/11t (0).

Proof. Ad

n.

Itfollows from (3.9) and (3.10) that

CJCi

=

(Pi - r)/11dCj ) •

11dC

j)

<

O.Ifr (t)

=

r

then pj

>

r

for bothi. Ifr (0)

<

r

thenr (t) ~0 as t ~00.

It will turn outto be crucial in the proof of Theorem

12

that

ell

(t) K (t) ~0 as t ~ 00. This holds if

Lemma B2.

(r(O),p(0» :I;(rJJ)

=>

limCl>(t)K(t) = O.

t-+oo

~ Fort large enough exploitation and production are specialized. Therefore indices i are omitted here.

DefineZ by

K

=

P -ar Z .

r

.

«j

-af)r

-r

(p

-ar) p -ar .

K= 2 Z+ Z

r r

=

~

K - (C1

+

C~

(from (3.6».

a+x

It follows from (3.11) and (3.12) that

(jJ -af)=r(p -ar).

Furthermore / (x)

=

xr

+

p

(from the homogeneity ofF)and . _ r (p-ar)x

p - x +a '

using

r

=

/"i ,

Ii

=

-xiI".

Then it is easily shown that

. r

Z

=- - -

(C1+ C~ .

p-ar

So Z is monotonically decreasing. Denote by t* the instant of time after which there is complete spe-cialization andr (t*)by r*.Then

00 00

K

J

E (t) dt

=

J -

dt

,.

,.

a

+x

must converge. Hence

00 0

J

-L

dt

=

J

p -ar

--L .!!.

!!E. dr ,. a +x ,.. r a +x dp dr

o

=

J

p -ar

--L

_-;;.;..a_-,;,;,x_ dr ,.. r a +x rep -ar)

,.. Z

=

J

-;:dr<oo.

o

r We also have dr I a -dZ/dr

=

dZ

.!!.

!!E. = _ _r_ (C₁

+

C~

dp!!E.

dt dp dr p -ar rep -ar) dr

C1+C

_z

= 2 (a +x)>

0 .

(p -ar)

Take e

<

r* and consider

,.. , . . , . . dZ

J

.l..

dr

=

_1.

z

1£ -

J -

1. -

dr

=_

Z(r*) + Z(e) +

Jr-

(C1+C:z)(a;x) dr.

r*

e

E rCp-ar)

It follows that

S

r-

(Cj +C:z)(a +x)2 dr

o

rCp-ar)

converges and limZ(r)lr exists. It equals zero for if Zlr ~A

>

0 for some A then

r~

would diverge. Finally

9K cjlK

=- - =

9ZIr . p-ar Hencecjl(t)K (t) ~

0

as

t ~00. Theorem 12. (K(t) ,u(t)} is PE.

Proof. Consider a program thatis feasible, Le. fulfils (3.2) - (3.8). Denote it by upper bars.

T

J {

_ae"""91'_U r:l """92 """91' - r:l """92' - _\} 1(C1)+ ...e U2(C:z)-ae U1(C1) - ...e U 2(C2/ dt o T

~

J

{ae"""91'

U;

(C₁₎

(CI-Cl)+~e

"""92'

U;

(C:z) (C₂-C:z)}dt

o

T =

J

cjl(CI+ C2

-C

1

-CV

dt

o

T T

=

J

cjl(Yl+Y2-Y1-Yvdt -

J

cjl(K-K)dt

o

0

(if

Kl

>

0 thenFiK =r and FiR = p.Therefore we continue) T

~

J

cjl{r(KI+K!-KI- K D+p(R_{1+R 2-R1}-R:z»)dt o T T --4>(K -K)

1

0

+

J

(K

-K)~dt

o

T

~

J

cjl[r(KI+K!-KI-K~-K+K)+p(_1

K f + -I

K~

o

al a2

_..!.

Ki _ _I Ki)} dt - cjl(T)(K (T)-K (T» al a2 T

=

J

{(91 - al)(E1 - £1)+ (92 - a:z)(E2 - E:z)} dt - cjl(T)(K(T) - K (T»

o

~

--4>

(T) (K (T) - K (T» ~- cjl(T) K (T).

cjl(T) K (T) ~ 0asT ~ 00(LemmaB2).

Here we prove one additional theorem. Theorem BI.

Let(a, ~)given with a

+

~= 1. For allK0there existsM (K0)such that

r-J

(Zlr~dr

o

o

J

{ae-1>1' U₁(C₁₎ +

P

e-1>z' Uz(C~} dt ~ M _{(K o).}

o

~Take (r.p) E (int VI) n (inlV~.Then

F; (KI,RJ~

r

KI

+

P Rj,

i

=1,2.

Hence K ~ r (K{

+

Kn

+

p (R1

+

R~ ~rK+pE-C, whereE

=

E1+E z,C

=

C1+C z.Therefore

,

J

e-r8

C

(s)ds

~

_{p (SlO+Szo)+Ko.}

o

There are positiveAloA2,BloandB2such that

Ul(Cl)~Al+BICI ; U2(C~~A2+BzC2 So ae-1>1'U1(C1)

+

P

e-1>z' U2(C

~ ~

e-1>'(aA1

+

aB1C1

+

pA2

+

P

B2C

~ ~

~ e-1>t(aAl+pA2+(aBl+pB~C), where

P

=

min

(Ph

p~. Hence

ae-1>1'U₁(C₁₎

+

pe-1>z'UiC~~ e-1>t (A +B C)

~

e-

rr(A+B C) for 0<

r

<P . It follows that

GO

o

Appendix C

This appendix derives(SI0,S20,KlO,K20)(v), used in Section 5.

Consider the quadruple v:= (r'"Iro, a. 9₁₎ with 0 < r'" <

r.,

0 <r'" < ro <

r.

0 < a < 1 and 9₁> 0

where

r.

is defined byr =a2P. (r,p)E

oW.

Define p'" -a2r'"

92_{= p -aIr'" '" 9}1 ,

wherep'"

=

p (r"') such that(r'" •p"') E

oW.

Letr (I, v) be the solution of

. p(r)-alr

r

=-

r, r(O)

=

roo r'" <r ~ ro.

al +x(r)

. p(r)-a2r

r=- r r~r"',

a2+ x (r) ,

wherep(r) is such that(r ,p)E

oW

andx =

_!!£.

on

oW.

Remark that these differential equations

fol-dr

Let1(r ,v) bethe inverse function ofr(I,v).Hence ro al +x(s) 1(r, v)

=

J

(p ( ) ) ds , r*

~

r

~

ro, r S S -als r* a2+

x

(s) 1(r , v)

=

1(r* , v )

+

J

(p ( ) ) ds, 0<r

~

r* . r S s -a'lfl DefineCl(r, v)andC2(r. v) by U'1(C (1r •v

» _

- -1

e

p1t(r,v)

a

1 r

*

~ r ~ ro

a.

P-air

U~

(Cl(r, v»

=

.1

eP1t(r,v)

a

2 0<r

~

r*

a.

P-a2

r

U' (C

(»

1 p2t(r,v)

a

1

*

2 2r, v

=

- e

r ~ r ~ ro

I-a.

p-alr U;(C_{2(r , v»}

=

_1_ e

P2t_(r,v)

a

2

₀

_<

_r

~

_{r* .}

I-a.

p-a2r

For given r(I) and v the amounts extracted from the resources

can

be calculated

as

follows (see also the proof of Lemma B2):

ro

S ( ) -

10 V -

J

Z+(r , v)2 d

r ,

r*

r

r* S20(V)

=

J

Z-(r;v) dr ,

o

r

where

a

2

Sr*

a2+

x

Sr*

al+

x

Z+(r , v)

=

-a

2 C(s , v) ds - 2 C(s , v) ds , 1 0 (p -a'lfl) r (p -als ) _ r a2+

_x

Z (r, v)

=

S

2 C(s , v) ds ,

o

(p -a'lfl) with C(s •v)

=

C1(s , v)

+

C2(S , v). Finally define

a

2

Sr*

a2+ x }

+ -

2 C1(s,v)ds - SlO(V)

a

1 0 s(p -a2s)

9

z

Sr*

az+x

9Z}

+ -

z C2(S.v)ds -

-9

Szo(v) ,

9

1 0 s(p -aZs) 1

Straightforward calculations yield

PO-al'O

Ko(v)

=

KIO(v)

+

Kzo(v)

=

Z+(,o.v) ,

'0

We conclude that ifthere exists

v

such that SIO(V)=SlOt Szo(v)=Szo

KIO(v)=K IO • Kzo(v)=Kzo

there exists a general equilibrium characterized by

P

-ai'

r=-

'.

,(0)='0. O~t<t(,,,,.v)

al+x

. p

-az,

'th . ' ( " ' ) ( " ' ) , = - , •WI ,continuous mt , •V tt ~ t , •v .

az+x

Hence the remaining problem is whether or not such a v exists. The functions (SI0.Szo,K10,Kzo) (v)

are continuous inv and continuously differentiable inv unless of course'0and,'" are located in points where

aw

is not differentiable. These properties however are not sufficient to have a solution. Unfor-tunately the formulae derived above do not allow for much analytical work.