A general equilibrium model of international trade withexhaustible natural resource commoditiesCitation for published version (APA):Geldrop, van, J. H., & Withagen, C. A. A. M. (1986). A general equilibrium model of international trade withexhaustible natural resource commodities. (Memorandum COSOR; Vol. 8612). Technische UniversiteitEindhoven.

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Faculty of Mathematics and Computing Science

Memorandum COSOR 86-12

A general equilibrium model of international trade with exhaustible

natural resource commodities

by

Jan van Geldrop and Cees Withagen

September 1986

Jan van Geldrop

Faculty of Mathematics and Computing Science

Cees Withagen

Faculty of Philosophy and Social Sciences

Eindhoven University of Technology

P.O. Box 513

5600 MB Eindhoven

The Netherlands

-2-

1. Introcluct.lou.

In the last. dec&de economic theory hu been enric:hed by an abundant lit.er&twe on natural

exhaustible reso~. It is (:omJJlonly agreed upon that the origins of this (since Hotelling

(1931) renewed attention are rooted in the 1973 oU crisill and Forrester's (1971) book on

World Dynamic:s and the subiJequent 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 luas

been developed on both aspects. We refer to Petenon and Fisher (1977) and Withagen

(1981) for surveys and to Dasgupta and Heal (1979) fot a standard introduction. Here we

shall be dealing with trade in exhaustible resource (:Ommodit1es.

The existing lit.er&twe can be divided into two broad cate&ories : one branch follows a par­

tial equilibrium approach. the other is of a ,enerat equilibrium nature. In the partial

equilibrium literatwe 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 interestlna (:Ontributions were made bya.o.

Vousden (1974), Kemp and Suzuki (1975), Aarrestad (197&) and Kemp and Long (1979.

1980 a.b) for the competitive cue. by Dasgupta, Eastwood and Heal (1978). who deal

with monopoly. by Lewis and Schma1ensee (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 hu been paid to general equilibrium

models of international trade in raw material8 from ahaustible resources. A. to our

knowledge. exhaustive survey can be found in Witha,en (1985). Kemp and Long (1980 c)

present a two economy model. One of the economiel Is resource-rich. the other is

resource-poor but has the disposal of a technology to (:Onvert the raw material into a (:On­

sumer good. Each economy then aims at the maximbration of its social welfare function

under the condition that equilibrium on ita current ac:count prevails. Kemp and Long

analyse general equilibrium under several UBumptioDi 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 production

(which takes place according to a Cobb-Douglas function) and. second. he allows also for

lending and borrowing between the countries involved. Elbers and Withagen (1984) drop

the dichotomy between the economies by postulatine each (:Ountry to possess an exhausti­

ble resource. The withdrawal from the resource is (:Ostly. A discU88ion of the results of

this literature is postponed to the final section of th.is paper.

The purpose of the present study is to generalize and to add several new aspects to this

general equilibrium approach. It is straiJbtforward to .. that there is a fair number of

good reasons to do so. First of all it eoes without .. yine that a eenerai equilibrium

analysis of trade should be preferred to a partial equilibrium treatment. if only from a

, methodological point of view. Furthermore. it may enable UI to explain the time path of a

crucial variable in the theory of exhaustible 1'eIIOurce8. namely the internationally ruling

-3-

interest rate. In the partial equilibrium approach this variable is always assumed a fixed

constant. which will tum out below to be justified only in a very special case. Second. the

concise summing 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 own­

ership of a natural resource can be dropped and the assumption of a Cobb-Douglas tech­

nology. describing non-resource production possibilities. seems rather restrictive. Further­

more. 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 aftirmative. how can these prices and the corresponding commodity allocation over

time for the economies participating in trade be characterimd 7 It turns out that the litera­

ture 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 turn 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-efticient (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 com­

parative dynamics. Finally Section 6 summarizes and concludes. The formal proofs are

given in Appendix A. B and C.

-4-

2. The modeL

We consider two economies which can be described IS follows. Economy i's (i -1,2) social

w~lfare functional is given by

00

I' (C, ) == f.-PI' u, (C,(t» dt • o

(2.1)

where t denotes time. p, is the rate of time preference (p, > 0). C,(t) is the rate of con­

sumption at time t and U, is the instantaneous utility function. It is assumed that U, is

increasing. strictly concave and provides a strong incentive to consume:

(P.t) U;(C,) > 0 .u;(C,) < 0 for all CJ > 0 .u;(0) .. 00

U;C, 'I). (C, ) : = -u:- = bounded.

Each economy has the disposal of an exhaustible resource of which the initial (at t - 0)

size is denoted by SOl •

The resources are not replenishabIe. Let E, (t) be the rate of exploitation of resource i. Then it is required that

.... f H, (t ) dt tii; SOl' • == 1.1.

o (2.2)

(2.3)

Exploitation is not costless. It is assumed that. in order to exploit. one has to use capital

(which is perfectly malleabIe with the consumer good) IS an input. Following Heal (1978).

Kay and Mirrlees (1975) and Kemp and Long (1980 b) we postulate an extraction technol­

ogy of the fixed proportions type :

Kf(t ) = 4, Hi (t) • (2.4)

where 4i is a positive constant and xt(t ) is the amount of capital used at t by economy i •

Economy 1 is cheaper in exploitation than economy 2.

Capital can also. together with the (homogeneous) resource commodity. be allocated to

non-resource production. Let R, (t ). Xl(t ). Y. (t) and F, denote the rate of use of the

resource good. the use of capital. the rate of non-resource production and the technology. respectively.

Then

(2.5)

-5-

About Fi 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 furthermore assumed that each input is necessary for produc­

tion. Finally. the e1astic:ity of substitution (1£.) 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 so but 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 terms of capital-intensity. We shall make

such assumptions as well by using the concept of factor-price frontier. This concept will

be intuitively clear but a rigorous statement can be found in appendix A.

CP4) The fpC's have. in the strictly positive orthant. a finite number of points in com­mon.

CPS) At such points p ;C air .

These assumptions merely state that the non-resource technologies essentially differ among economies.

Define

Z, (t) = {C, (t), Y, (t). Xl (t ).R, (t )Et (t ).E. (t) ) .

Let Xia be economy ,'s given initial non-resource wealth and Xi (t) its wealth at time t. We next define a general competitive equilibrium.

(X 1 (t), X 2 (t), Zl (t), Z2 (t). pCt). ret)} with p(t) > 0 and r(t) > 0 constitutes a general competitive equilibrium if

1) for i = 1.2: Zj (t) maximizes J' (CO subject to (2.2) - (2.5) and

GO

of 1J{t){ p (I) R, (t) + r (t )(xt (t) + Xl (t » + C, (t ) } dt ~

GO

f w(t){ p (t ) E, (t) + Y, (t) } dt + Xi 0 • o

(2.6)

where

, -I" h') ,IT

1J{t) := ., .

2) no excess demand :

R 1 (t ) + R 2 (t ) ~ E 1 (t ) + E2 (t ) • (2.7)

Xl (t)+X~ (t)+X! (t)+x~ (t)~ X t (t)+X2 (t). (2.8)

-6-

(2.9)

3) p(t) OIl: 0 when (2.7) holds with strict inequality; dt) = 0 when (2.8) holds with

strict inequality.

Condition (2.6) needs some clarific:ation. It is assumed that there exists a perfect world

market for resource commodities which have spot price p(t) and a perfect world market

for capital services with spot price r(t). Since capitalls also a store of value this implies

the existence of a perfect world market for "1inanciat c:apltal where the rate of interest is

r (t ). Hence condition (2.6) requires each economy to make plans such that the discounted

value of total sales exceeds the discounted value of total expenditures.

Bewley (1970 and 1972). Hart e.a. (1974) and Jones (1983) have studied the problem of

existence of general equilibrium for the case of infinitely many (dated) commodities.

Unfortunately their approach is not suited for the problem at hand bec:ause they work in

discrete time. require an a priori bounded production possibility set and/or do not allow

for additively separable preference schedules etc. This observation gives rise to the desira­

bilityl necessity of making a detour. which is based on BODle very well-known results in

general equilibrium theory and which has proved its usefulness before (see Elbers and

Withagen (1984) and Takayama (1974». In the sequel attention will be concentrated on

Pareto efficient alloc:ations (PE alloc:ations). Such alloc:ations will be characteriad and

existence of PE alloc:ations is c:arefully studied. Next the analysis will be directed towards

the application of the modified second law of classical welfare economics.

Without proof we state :

Theorem 1

Let {K l(t ).x 2(t )2 l(t )22(t ).p (t)". (t ») be a general equilibrium. Then

(K l(t ).x2(t )21(t ).Z2(t)} is PE. D

-7-

3 Pareto etJici.eDcy

Consider the problem of maximizing

co

J(C 1.C2)= J (ae-Plt U 1(Cl(t» + fJ.-P'/ U2(C2(t»)dt

o

subject to

... J Hi(t )dt ~ SIO' i" 1.2 •

o

HI (t) ;;, O. i = 1.2 •

xf{t) = 4j HI (t). i = 1.2 •

Yi (t) = F, (Kl (t ) • R/ (t». i = 1.2 •

it (t) = Y 1 (t ) + Y 2 (t) - C 1 (t ) - C 2 (t) •

R 1 (t ) + R 2 (t) ~ H 1 (t ) + H 2 (t ) •

K (t );;, Ki (t) + K~ (e) + Kl (t) + X~ (t) •

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.S)

where X (0) is given and equals X 10 + X 20• a and fJ are given positive ~onstants and (PI), (P2) and (P3) are satisfied. Clearly an allocation is PE if and only if it solves the

problem posed above. The intention of this section is to characterize PE allocations. The

formal proofs of the theorems presented here ~ be found in appendices A and B. Suppose there exist

- continuous X (t) > 0 and fJ (t) > 0,

- nonnegative constants 6 1 and 62•

- piece-wise continuous u (t) : = (C 1 (t ). C2 (t ). Xi (t), Ki (e ). Xl (t). X~ (t). R 1 (t ).

R 2 (t). HI (t), H2(t»;;' 0 and y (t): = (r (t). p (t). 0'1 (t). 0'2(t» > 0 where y (t) is

continuous except possibly at points of discontinuity of u (t ),

fulfilling (3.2) - (3.8) and such that for all t ;;, 0

CIte -PI' U ~ (C 1 (t) ) = fJ (e) •

fJe -Pal U ~ (C 2 (t » = "(t).

" (t) (p (t) - air (t » + 0' j (t) = 6,. ,= 1.2 •

-(p (t )'" (t) = r (t) •

Fi (Xl (t ) • R/ (t» - r (t) Xl (t ) - p (t) RI (t ) >

(3.9)

(3.10)

(3.11)

(3.12)

FjCKI,R,)-r(t)Kl-p(t)R, forall(KI.R,)~ O. i" 1.2. (3.13)

... 91 (S,O - J HI (e)de) = O. i" 1.2. (3.14)

o

-8-

CT. (t) E. (t) = O. i = 1.2.

r (t) (K (t ) - KI (t) - Ki (t ) - Ki (t) - K~ (t»... 0 •

p (t ) (E 1 (t ) + E 2 (t) - R 1 (t) - R 2 (t » - 0 .

(3.15)

(3.16)

(3.17)

This system strongly resembles the familiar Pontryagin necessary conditions for the prob­

lem stated above. although. for practical purposes. it is not equivalent to these conditions

(in particular (3.13) is a modification).

The equations can be given a nice economic interpretation when t/>. 6;. r and p 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 and p will here­

after 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.

i p

Ficure 3.1

In view of the homogeneity of F,. the left hand side of (3.13) equals zero for both i. Let i

be fixed for the moment. 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 (fpr). It is negatively sloped and hits

(one of) the axes and/or has (one or) the axes as an asymptote. There cannot be positive

-9-

asymptotes because of the necessity of both inputs. Examples of fpt's 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 = p la, is the locus where shadow­

profits of exploitation from resource, are nil.

Let us now list some properties of the solution of the system given by (3.2) - (3.17). To

each of the following theorems we add a description of the 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 proofs are invited to go carefully through the

appendices.

Theorem 2. The real shadow-prices move along that fpf which. given r • has maximal p.

This is clear from the fact that otherwise maximal shadow-profits of one of the economies

would not be zero.

Theorem. 3.

p (t) > 0 for all t ;, O.

If the theorem would not hold. there would be no exploitation (CT I > 0 from (3.11». nor

non-resource production. But then capital is useless. its price zero and (3.13) is violated.

Theorem 4.

r (t) > 0 for all t > o.

Theorem. 5, p (t) and r (t) are continuous for all t ;, o.

The intuition behind Theorem 5 is simple. Inspection of equation (3.11) learns that it

should hold along intervals of time where CT, is zero for some i for then a jump of one of

the shadow-prices is accompanied by a jump of the other in the same direction which con­

tradicts Theorem 2. As long as the shadow-price of capital services is positive there is sup­

ply of such services and hence non-resource production takes place. which necessarily

requires exploitation. So then the condition with respect to CT, holds. What one wants to

exclude therefore is the possibility of r (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 elastici­

ties of substitution. This type of argument has also been followed by Dasgupta and Heal (1914).

Lst (f' .j) be defined as the point of intersection of the line r = pIa 1 and the fpf for

which in that point r is maximal. See figure 3.1. Evidently real factor shadow-prices

- 10-

should allow for non-negative shadow profita from ezploitation of the eheapest resource.

Hencer (t)' F.

Theorem 6. For the real shadow-price trajectories there are two possibilities.

1. (r (I).p (I» = (F .j) for all t > 0;

2. r (t) < F • p (t) > j for all t > O.

If possibility 2 oc:eurs then r (t) monotonically decreases towards zero and p (t) mono­

tonically increases to infinity or a given constant. namely where a fpf is tangent to the p­

axis.

If the real shadow-prices equal (F.p) for an instant of time. then it follows from (3.11)

that they will have these values forever. In this cue the second economy will never

exploit (otherwise 92 < 0). The first half of the second statement then immediately fol­

lows. The time-path of the real shadow-prices follows from (3.11). (3.12) and the fact

that 6 i 's are constanta. The result is in fact a kind of Hotelling rule.

Theorem 7.

If the real shadow-prices are (F.j) then min (Pl' pi) > r. Since the capital-resource input ratio is constant in this cue and the stock of the resource

.... is finite f Ie (t )dt < co. This implies Ie (t) - 0 IS t - co. It follows from (3.9).

o (3.10) and (3.12) that Cl is decreasing only if Pi > r. This is necessary to prevent Ie (t) from becoming negative (see (3.6».

We now turn to the description of commodity trajectories. Some of these immediately fol­

low from the shadow-price paths.

Theorem 8.

Suppose that the fpf's do not coincide for r -= pial and that if they intersect for positive

real shadow-prices. the number of points of intersection is finite. Then non-resource pro­

duction is always specialized.

This follows from Theorem 2.

Theorem 9.

Exploitation is always specialized. Moreover. the second resource is only taken into exploi­

tation after exhaustion of the first resource.

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

tation r,p and PtP would be constanta (see 3.U) and real shadow-prices would both be

increasing. contradicting Theorem 6. The second part is less straightforward but rests on

the idea that cheaper resources should be exhausted first (see also Solow and Wan (1977».

-11-

The order of non-resoun:e produc:tion can easily be traced by looking at figure 3.1 as an

example. One could start at point I where economy 1 is producing. Over time point S is

reached. Afterwards economy 2 takes over ad infinitum. The resource of economy 1 gets

exhausted. This happens after point U has been passed. Hence. eventually the second econ­

omy carries out both production and exploitation.

Theorem 10.

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

The theorem follows from (3.9) and (3.10). where it should be recalled that -~/tf> .... 0 as t .... 00.

Theorem 11.

Production gets more capital-intensive over time.

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

Theorem 12. {K (t Lu (t)} is PE.

The proof of this theorem concentrates on asserting that tf> (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 concavity of the functions involved.

In view of the strict concavity of utility and production functions PE allocations are

unique (for given Q! and fJ) so that the system (3.1) - (3.17) is necessary and sWlicient for

Pareto efficiency. This turns out to be important when examining existence.

-12-

4 General equilibrium

This section addresses the relation between PB-allocations as discussed in the preceding

section and general equilibrium.

Let {K 1 (t).K2 (t).Zl(t),Z2(t),P (t),r (t)) constitute a general equilibrium. Then,

according to Theorem 1. (K 1 (t ).K,(t). Zl (t ). Z2 (t)J is Pareto-eflicient. Since conditions

(3.2) - (3.11) are necessary and sufftcient for Pareto-e1li.ciency it follows that all the

results of the previous section. which characterize PE allocations. hold. Furthermore a gen­eral equilibrium is unique. This is summarized in

Theorem. 13.

Let {K 1 (t ), K 2 (t ). Z 1 (t ).Z 2 (t ). p (t ). r (t)J be a general equilibrium. Then there exists

no other general equilibrium and Theorems 2 - 11 hold. IJ

This result needs no further comment except possibly for the following. In partial equili­

brium models involving ahausible resources it is customary to use constant interest rates.

In view of Theorem 1 (and Theorem 15 of the next IeCtion) it can be doubted if this is in general justUied.

- 13-

S Existence of general equilibrium

Hereto nothing has been said about the existence of a general equilibrium. This is the aim

of the present section. In view of the 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 (r). Subsequently

we shall go into the existence problem when the initial conditions do not allow for a con­

stant interest rate.

The analysis will be conducted along the following lines. If there exists a general equili­

brium of some type it is Pareto-efficient. This implies that there exist 01 and 13 (13 == 1- 0)

such that the equilibrium commodity trajectories and the equilibrium. price trajectories

solve (3.2) - (3.17). where the additional variables cr, and 9 i 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. 6 J. equals zero. This implies that the bal­

ance of payments condition (2.6) can be written as

"" J e-Ft C, (t )dt ~ X,O. j == 1.2. o

In equilibrium an equality will prevail since the instantaneous utility functions are

increasing. These observations give rise to the following.

Calculate C 1 and C 2 from

(5.1)

u' (C ) =.!MEl (Pa - F")t 2 2 1 e • -01

(5.2)

where 4>(0) and 01 are fixed positive constants for the moment and 01 < 1. If PI > F and

pz > r there clearly exist ~ (0) > 0 and 0 < ci < 1 such that

"'" J e-Ft Cl(~(O).ci.t)dt = X 10 o

"'" J e-Ft C 2 (4)(0),ci,t)dt = K ao , o

(5.3)

(5.4)

where eland C z are the solutions of (5.1) and (5.2). ~ (0) and ci are unique. Let X (t) be the solution of

i: (t) == FX (t)- C 1 (~(o),ci,t) - C2(4) (O).ci.t), K (0) = X 10 + Xao .

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

l<;:f;hnology at r. Define x by r = FIX (x .1). Consider

S 10 - S t(t) : = j ! (s) ds . o x + at

(5.5)

- 14-

The right hand side of (5.5) is total extraction of the first economy's resource before t.

along the proposed program. For

and

Now if S 1 (t ) ~ S 10 for all t • all the conditions for a general equilibrium are satisfied. The

desired condition is therefore that S 10 is sufficiently large relative to K 10 + K 20' Hence we

state

Theorem 14.

Suppose PI > rand P2 > r. If S 10 is large relative to K 10 + K 20 there exists a general

equilibrium with (r (t ). p (t » = (r. j). 0 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 fj = 1- a). r (0).91 and 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 positive and r* < r (0). In addition. exploitation of the

second resource must be profitable: p * - G zr * ~ O. 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 10. K 20. S 10. S 20 can be calculated which

would induce a general equilibrium. More formally. define v = (ro.r*.a.9 1).

Theorem 15.

Suppose

then there exists a general equilibrium characterized by r (0) = roo r (t) < O. E 1 (t) > 0

as long as r (t) > r*. E2 (I) > 0 for all t such that r (t) < r*. D

This theorem does not solve the existence problem. The functions S.o (v ) and K,o (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 con­

structive. Second. it can easily be generalized for an arbitrary number of economies partici­

pating in trade.

Finally we sketch an alternative approach of the existence problem. which has been used in

Elbers and Withagen (1984). Consider the problem of finding PE allocations (3.1 - 3.8)

- 15-

with ~ = (1- ex). Suppose that for all 0 < ex < 1 there exists an optimum. Parametrize

the solutions with respect to ex. This yields £; (er}.9 i(er) etc. Form

co

v,(er) : = tjJlc,) K jo + f tjJ(er){y,clll) - Cj(o.) + p(lII) (E,(OI) - R,(er»

°

CO>

= tjJ&l~,>KIO+ 6,(III)S,O- f tjJ(er)c,(er)dt

° Straightforward calculations. taking into account that 4;(,,) K (t'> .... 0 as t .... co. show

that

Furthermore

V 1°) > O. V JO) < O. V 11) < O. V Jl) > 0 .

In order to apply the intermediate value theorem we need the continuity of vi(er) in ex. It is

easily seen that a sufficient condition for this is that C fill) is piece-wise continuous in ex

(by the dominated convergence theorem). So if cia ) is piece-wise continuous in ex there exists Ii such that vf~) = vl~) ==: O. Then (Kl~) .Kl;) .zl;) .zl;) .p(;).r(~} constitutes a

general competitive eqUilibrium.

This approach is economically very appealing. It should be stressed however that it

presupposes the existence of PE allocations and the piece-wise continuity of the rates of

consumption in the weighting factor ex. Although intuitively correct. these assumptions are

not easily confirmed by formal inspection of the model.

- 16-

6 ConclusiODL

In this paper we have presented a general equilibrium. model of trade in natural exhausti­

ble 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 com­

pared with other general equilibrium. approaches. Existence of an equilibrium. has been

examined. departing from the characterization of Pareto-e8icient 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 pro­

duction. Second. the simplicity of the description of the extraction technology. Third. one

could introduce (manufactured) substitutes for the resources. Finally there is the assump­

tion of perfect mobility of capital goods which allows for instantaneous switches in pro­

ductive activities from one economy to the other. Nonetheless 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.

- 17-

AppendlxA

In this appendix some duality results are derived which are frequently used in the main

text and appendix B. The following notation will be adopted. For a vector y = (y 1.Y2) we

write

Y ~ 0 if Yi ~ 0 for all i •

y~o if y~O andy;ltO.

y > 0 if Y. > 0 for all i .

O>nsider a concave and homogeneous production function F (K. R) with the following

properties

Define

F (0. R) := F (K. 0) = 0 •

Fit; : := 8F 18K > 0 for all (K. R) > 0 •

Fr : = 8F loR > 0 for all (K. R) > 0 .

V : = {(r.p) I F (K. R) - rK - pR , 0 for all (K. R);iiII o} .

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

the boundary of V.

Lemma At.

BV = {(r. p) ~ 0 I (r .p) e V and there exists (if. i);iiII 0 such that F (if.;:)- rif - pi = oJ.

Proof.

(r. p) E BV ~ (r. p) E V and there exists a sequence (r,.. p,.) f V with

(r,..PA)'" (r.p).

(r" • Pn. ) t V ~ there exists (K,. • R,. ) such that

F (Kn.. R,.) - r,.K,. - PAR,. > O.

(K,.. R,,) can be chosen on the unit circle. converging to (if. R) ~ 0 with

F (K. R) - rK - pR ~ O.

Since (r • p) E V this expression equals zero. Conversely. suppose (r • p) E int V and there exists Cif. if) ~ 0 such that

F (K. R) - rK - pi = O.

But then there exists (; • ;) E V. close to (r . p) such that

- 18 -

F (it. R) - rK - pR > 0 .

a contradiction. []

LemmaA2.

Suppose (r1,P1) E BV • (r2,P2) E BV and P2 > Pl' Then

j) r2 ~ r l'

ii) if r 2 = r 1 then r 1 = 0 and P 1 > O.

Proof. The proof follows immediately from the previous lemma. []

Lemma A3.

Suppose (r1,P1) E BV • (r2'PZ) E BV and rz > rl' Then

i) P2~PI

ii) if P 2 = P 1 then PI = 0 and r 1 > O.

~ This follows directly from Lemma 1. []

Lemma A4.

(r .p) E B V 1\ (r.p) > (;. p) => (;. p) f V.

Proof. This is clear from the definition of V and Lemma A1. []

Consequently V and BV can take the shapes as depicted below.

r r r

V

P P P P

- 19-

Remarks.

1. It cannot be that the curves displayed have an;' > 0 or a p > 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. However. 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 F 1 and F 2 with respective

inputs X and R indexed by 1 and 2. V 1 and V 2 are defined analogously to V. Define

furthermore

Lemma AS.

8W={(r.p)~OI(r.p)eW and there exists (il.jD~o such that

Fi (X. i) - ril - pi = 0 for some i.).

Proof. This is evident.

Without proof we state

Lemma A6.

Lemmata A2. A3 and A4 hold with V replaced by W.

Now suppose

there exists (r • p) • (X 1. R 1) and (X 2. R 2) such that

F 1 (K 1. R 1) - rX 1 - pR 1 ~ F 1 (X. i) - rX - pi for all (X. i) ~ 0 •

F2(X2.R2)-rX2-pR2~ F 2(il.i)-rX -piiforalI(K.ii)~ o.

This is equivalent to (3.13).

Lemma A7.

(3.13) implies that (r . p) E W.

Lemma AS.

(Xj.Rj ) > o for some i implies (r.p) e 8W.

Clearly Lemma A 7 proves theorem 2. Lemma A8 will turn out useful in Appendix B.

o

o

o

o

- 20-

Appendix B

In this appendix the theorems of Section 3 are proved. For conveniency the assumptions of

the model are restated here.

U;Ci • (PO U; > O. Ui" < 0, U; (0) = 00 ,1Ji (Cj ): = -,- IS bounded.

Ui

(P2) a 1 < a2'

(P3) F j (Kl.Rj ) is concave and homogeneous and satisfies

Fi (O.R j ) = Fi (Kf,O) = O.

FiK : = a Fi/O K! > 0 for all (K!.Rj ) > O.

Fi/? : = a Fi /a R; > 0 for all (K!.Rj ) > O.

The elasticity of substitution </JI is bounded.

(P4) The set l(r.p) I (r.p) > 0 and (r.p) E BV 1 n BV 2} is finite.

(PS) (r .p) E B V I n 8 V 2 => P ;C aIr.

Assumption (N) says that essentially the Fl's differ. whereas (PS) excludes the possibil­

ity of intersection at a specific point. In the sequel the argument t is omitted when there is

no danger of confusion. The following notation will be used.

It is easily seen that

FiK = fi'. FiR = fi - xif,', fi" < 0, fj (0) = 0,

<Pi = - f/(fi - Xj fj')/fi"xdi .

Finally

y (t +): = lim y (t + h). y (t -): = lim y (t + h) . IIlO lito

Theorem 2 has been proved in Appendix A.

We first present a lemma that is frequently used in the sequel.

Lemma B1.

r (t) > 0 => Y 1 (t) > 0 V Y 2 (t) > O.

Proof. Suppose that there exists t 1 ~ 0 such that r (t 1) > 0 and Y 1 (t 1) = Y 2 (t 1) = O.

Then Kl (t 1) = K2 (t 1) = O. otherwise (r.p) f Wand Lemma A7 is violated. If

p (ll) = 0 then CT dt 1) > 0 for both i (since 9. = </J (t)(p (I) - air (t»

+ CT. (t) ~ 0) and El (t 1) = E 2 (t 1) = RI(tl) = R2(t 1) = 0 (3.11.3.15 and 3.7). If

p (t 1) > 0 then also R 1 (t 1) = R2 (t 1) = 0 since (r.p) E W. Therefore in both cases

- 21 -

Kt (t 1) :: O. i = 1.2. But K (t 1) > O. It follows from 3.17 that r (t J):: O. a tontradie-[] tion.

Theorem 3,

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

~ Suppose there exists t 1 such that p (t 1) :: O. Since (0.0) f Vi. i :: 1.2. r (t 1) > o. 6, ~ 0 for both i. hence CT, (tl) > 0 for both i (from (3.11» implying

E 1 (t 1) :: E2 (t 1) = R 1 (t 1) = R2 (t 1) = O. Therefore Y. (t 1) :: O. j = 1.2 and Lemma B1

is violated. []

Theorem 4.

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

Proof. The proof will be given in several steps.

1) Suppose that there exist t 1 ~ 0 and;' such that r (t 1) :: 0 and R, (t 1) > O. Then

Kl (t 1) > O. otherwise (r (t 1) • P (t 1» f VI in view of the fact that p (t 1) > O.

Hence (r (t 1) • P (t 1» E 3V. (Lemma At). But F fJ{ > 0 and this implies

(r (t 1) • P (t 1» f 3Vj • Hence. r (t 1) == 0 implies R, (t 1) • O. i:: 1.2.

2) Suppose that there exists t 1 > 0 such that r (t I) > O. whereas r (0) :: O. Then

tP (t 1) ~ tP CO) in view of (3.12). Since r (t 1) > O. Y. (t 1) > 0 for some i and there­

fore R, (t 1) > 0 for this i. This implies that there exists j such that EJ (t 1) > 0 and

(} J = tP {t I)(P (t 1) - aJ r {t 1» = <I> (0) (p (0) - a) r CO»~ + CT J (0) .

So p (t 1) > P (0) and (r (t 1) • P (t 1» > (r (0) • P (0». But then

(r (0) • p (0» f W. contradicting Lemma A7. Hence r (0) = 0 implies r (t) :: 0 for

all t ~ O.

3) r (0) > O. This is so since, if r (0) = O. r (t) = 0 for all t (2) implying R. (t) = 0

for all t and both i; hence E, (t) == 0 for all t and both i (3.17) contradicting

(3.14).

4) Suppose there exists t 1 > 0 such that r (t 1) :: 0 and r (t) > 0 for all 0 < t < t 1 •

a) Suppose that r (t 1 - ) > O.

In view of the piece-wise continuity of r there exists 1.. < t 1. close enough to t 1. such

that r CD > O. Hence Yj CD > 0 for some;' (Lemma B1). Then also P CU ~ P (t 1)

otherwise (r (t 1) • P (t 1» f V, (Lemma A4). Since tP (t ) is continuous by assump­

tion we then have (J' J <V > CT J (t 1) ~ 0 (j:: 1.2). Therefore E 1 CU :: E2 CU = R 1 CD :: R 2 W = O. contradicting Y, W > 0 for some i. So

- 22-

b) r (t 1 - ) :II: O.

Assumption P4 implies that there exists an interval (1' , t 1) such that YI (t) > 0 for

all t E (1' , t 1) and just one I. Since E, (t) is piece-wise continuous for both i the

interval (1' • t 1) can be partioned in a finite number of subintervals such that

Ei (t) Ci = 1.2) is continuous along each subinterval. Observe first that there is no _fc'

subinterval with E 1 > 0 and E:2 > 0 a10n, a non-depnerated subinterval of it. For

suppose that there exist1. and i with l' , 1. < i < '1 such that E 1 (I) > 0 and

E z > 0 for all t E k.. il. This implies that t/> (P - air) and t/> (P - a.zr) are con­

stants for 1...' t , i. But ~ (I) < 0 for 1...' t , i. Since al pi!: a2 t/>p and t/>r are constants. implyine that (r (t) . p (t» > (r (tJ • p (tJ). Therefore

(r (tJ • p (tJ) f. W. wh.ic:h is not allowed by Lemma A7. So for all partitions

E 1 (t ) = 0 .... E:2 (t) > O. Hence there exists if < t 1 such that along the interval

(if .tl) E 1(I) = OandE2 (t) > O.orE1(t) > o and E2(t) = O. Assume, without

loss of generality. that Y 1 (t) > 0 and El (t) > 0 for all t E (if • t 1). Strai,htfor­

ward calculations yield

where it should be recalled that I-' 1 is the elasticity of substitution. If r (t ) ~ 0 for somet E (f' • t 1) then % 1 (t ) , 0 because ,. (% 1) < O. Therefore

%1/%1' 1-'1 il (% 1)/%1 .

Since r (t ) .... 0 • % 1 must become arbitrarily large. But

lim il (%1)/%1 = 0 x 1""00

so that % 1 is bounded on any finite interval.

Theorem ~.

o r (t + ) = r (t - ).

ii) p (t + ) = p (t - ).

~ 61 is constant. Hence

IJ

CT I (t - ) - CT I (t + ) = t/> (t ) (a.l (r (t - ) - r (t + » - (P (t - ) - p (t + »}. j = 1.2.

a) Suppose that r (t 1 - ) > r (t 1 + ) for some t 1 > O. Then there exist 1.. and i. close

enough to t l' with.!. < t 1 < i such that r (tJ > r (t). Since r (tJ > 0 • Yl (tJ > 0

for some i (Lemma B1). Then also p (tJ , p (t) otherwise (r (t) • p (i» f VI

(Lemma A4). Therefore CT) (tJ > CTJ (i)~ 0 (j = 1.2) and E1(tJ = E2 (tJ

= R 1 (V = R2 W = 0 contradicting that Y j W > 0 for some i.

- 23-

b) Suppose that r (t 1 - ) < r (t 1 + ) for some t 1 > O. In this case the proof is analo­

gous to the proof under a. This shows the validity of part i) of the theQrem. The

proof of part ii) is similar and will not be given here. 0

Let (F.p) be defined by p = atF and (F.p) E 8w. See figure 81.

r

fig. 81

In view of the assumptions made (F.i) exists. In fact F is the maximal feasible r.

Theorem 6.

o (r (t 1) • P (t 1» = (F.p) for some tl ~ 0::::;:. (r (t) • p (t» = (r.p) for all t ~ o. ii) Suppose r (0) < r. Then

a) r (t 1) > r (t 2) and p (t 1) < p (t 2) for all t 2 > t 1 ~ O.

b) lim r (t) = O.

Proof.

Ad 0.

t .... 00

1) r (t) ~ r for all t. For suppose that for some tl ~ 0 r (t 1) > r. Then Yj (t 1) > 0

for some i (Lemma B1). p (t 1) ~ P otherwise (F.p) ~ 8W > 0 (Lemmata A6 and

A4). Since (I J ~ 0 for both j it follows that CT J (t 1) > 0 for both j. which implies

that EJ (t 1) = RJ (t 1) = 0 for both j contradicting Yj (t 1) > 0 for some i.

- 24-

2) Suppose (r (t 1) • P (t 1» = (r .ji) for some t 1 ~ O. Then E2 (t 1) = 0 because

ji - a2 r < 0 and 92 ~ O. But. since r (t 1) > o. R 1 (t 1) + R2 (t 1) > 0 (Lemma B1).

implying that E 1 (t 1) > O. Therefore 9 1 = 0 and p (t) - air (I) ~ 0 for all t. If.

for some t 2 ~ O. P (t 2) - air (t 2) < 0 then E 1 (t 2) = 0 (00 1 (t 2) > 0). A fortiori

E 2(t 2) = O. Hence Yj (t2) = 0 for both i. contradicting Lemma B1. So

p (t) = air (t) for all t ~ O. This proves the first part of the theorem.

Ad ii).

t) Suppose r (t 1) ~ r (t 2)' r (t) > 0 for all t (Theorem 4). hence Y. (t 1) > 0 for some

i and Yj (t 2) > 0 for some i (Lemma 81). Therefore (r (t 1)' P (t 1» e 8Vj for some

and (r (t 2).P (t 2» e 8 Vi for some i. So (r (t l)'P (t 1» E 8 Wand

(r (t 2).P (t 2» e B W. (Lemma AS). Using Lemmata A6 and A2 we find

P (t 1) ~ P (t 2)' By the same argument r (t 1) = r (t 2) if and only if P (t 1) = P (t 2)'

2) Now suppose r (t 1) = r (t 2).

9 j = ~ (t Hp (t) - aj r (t » + 00) (t) for all t and both j.

Because ~ is continuously decreasing and p (t 1) = P (t 2)' CT J (t 2) > 0 for both j.

So R 1 (t 2) + R 2 (t 2) = 0 and Y 1 (t 2) = O. contradicting Lemma 81. We can therefore

restrict ourselves to the following case.

3) Suppose r (t 1) < r (t 2)' Then p (t 1) > p (t 2) in view of 1). Then a fortiori

Y 1 (t 2) = Y 2 (t 2) = O. contradicting Lemma 81. This proves ii) a).

4) Suppose

lim r (t ) = r > 0 . t .... 00

Then ti> (t )/~ (t ) .... -? as t .... 00 (3.12). So ~ (t) .... 0 as t -+ 00. Furthermore

p (t) .... p > 0 as t -+ 00. In the case at hand (}) > 0 for both j. Hence

6 j /~ (t ) .... 00 as t -+ 00. implying that 00 J (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 one i. I]

Th~rem 7.

er (t ).p (t» = (r.ji) => PI > r. P2 > r.

Proof. 0 = p (t ) - aIr (t) > p (t ) - a 2 r (t). Hence E 2 (t ) = 0 for all t ~ O. Since

~/~ = -r it follows that ~ (t) = ~ (0) e-Fr • Hence

. ( ) !I!..S!!l (Pei'lt . U· (C ) =!I!..S!!l (P2- r )t U 1 C 1 =a e . 2 2 IJ e

Furthermore

K = Ki + Ki + at (R t + R2 )

K / R = KlI R + KV R + a 1 •

whereR : = Rl + R 2•

- 25-

If Yj (t 1) > 0 then Kl (t 1)/ R j (t 1) is constant. If Yj (t 1) = 0 then Kl (t 1)/ R; (t 1) = o. So

there are constants b 1 and h 2• such that

Hence

co

J K (t ) dt < 00 .

o

In view of the homogeneity of Fi

Therefore

and

t

K(t)= Koe N - J e;(t-s)C(s)ds. o

CIO

where C = C 1 + C 2• It follows from K ~ rK. K ~ 0 and J K (t)dt < 00 that o

K (t) -+ O. For suppose that this does not hold. Then

00

V,. VT 3, >T K (t ) < .!. . otherwise J K (t ) dt diverges n 0

300 VT 3, >T K (t) > E • by assumption.

Hence there exist E > 0 and sequences l' /l and tn such that for n > 1 E

Furthermore the sequences can be chosen such that K (t)' , for 'I'll , t , tn. See figure B2 below.

1

K

E

1 n

n + 1

Hence

- 26-

Fig. B2

In

J K (t ) dt ~ 1. (t -.,. ) ~ ..L (1 __ 1_> n" II nr En 1'n

and

co

of K (t )dt

diverges.

Now. if PI ~ r or P2 ~ r then C (t) ~ C > 0 for some C and for t large enough K (t> becomes negative. which is not allowed. IJ

As a corollary we mention

Corollary Bl.

er Ct).p (t)) = (P .p) :::;.. 1> (t) K et) - 0 as t - 00.

Theorem S.

Take t 2 ~ t l' Suppose Y 1 (t ) > 0 and Y 2 (t ) > 0 for all t E [t 1.t 21. Then t 1 = t 2'

Proof. Suppose there exist t 1 and t 2 with t 2 > t 1 such that Y 1 (t) > 0 and Y 2 (t ) > 0 for

all t € [t It 2]. If (r (t ).p (t » = (F.p) for all t ~ 0 then assumption P5 is violated. If

(r (O).p (0)) ;I'! (P.p) then r is monotonically decreasing. Hence P4 is violated.

- 27-

Theorem 9.

i) Take t2 ~ tl' Suppose E 1 (t) > 0 and E2 (t) > 0 for all t E {t It2)' Then t 1 = t2' ,

ii) E 2 (t) > 0 => J El (t )dt = S10' o

Ad n.

D

The argument haS already been given in the proof of theorem 4 but will be repeated here

for convenience. Suppose there exist t 1 and t 2 with t 2 > t 1 such that E 1 (t) > 0 and

E2 (t) > 0 for all t E [t It 2]' Then. from (3.11) and (3.15)

t/J (p - aJr) = OJ. j = 1.2. t E {ttt2].

Since a 1 ;C a2. t/Jp and t/Jr are constants along [t Ih]. But t/J (t 2) < t/J (t 1) and therefore

(r (t 2)'P (t 2» > (r (t 1)'P (t 1»' So (r (t 1)'P (t 1» f W (Lemma A4). which is not allowed (Lemma A 7).

Ad in,

Suppose there exists t 1 such that E',2 (t 1) > 0 and

tl

J E 1 (s ) ds < S 10 • o

00

a) J E 1 (s ) ds < S 10 • o

In this case 61 = O. implying (r (t).p (t» = (F.p) for all t ~ O. Therefore 0"2(t) > 0

for all t as well as E',2 (t) = 0 for all t • contradicting E2 (t 1) > O.

co

b) J E 1 (s)ds = SIO. o

There exists an interval [1"1.1"2]. tl lEt 1"1 < 1"2. with El (t) > 0 and continuous. whereas.

along the interval. E 2 (t ) = O. Take 'T 1 < t 2 < 'T 2.

E 1 (t 1) ~ 0, E2 (t 1) > 0 , 0'"2(t 1) = O. Hence

91 ~ ¢ (t 1) (p (t 1) - a lr (t 1»'

92 = ¢ (t 1) (p (t 1) - Q2r (t 1»'

E \ (t ,,) > o. E 2 (t 2) = o. 0'" 1 (t 2) = O. Hence

So

6 1 = rp (t 2) (p (t 2) - aIr (t 2»'

62 ~ rp (t 2) (p (t 2) - a~ (t2»'

- 28-

~ (t 2) (p (t 2) - aIr (t 2» ~ ~ (t 1) (p (t 1) - aIr (t 1»'

~ (t 1) (p (t 1) - a zr (t 1» ~ rp (t 2) (p (t 2) - a 2r (t 2»'

Multiplication of the left and right hand sides of the first inequality by a2 and of the

second inequality by a 1 and adding yields

(a 2 - a 1) (rp (t 2) P (t 2) - ~ (t 1) P (t 1» ~ 0,

implying P (t 2) > P (t 1)'

Just addition of the inequalities yield

(a 2 - a 1) (~t 2) r (t 2) - rp (t 1) r (t 1» ~ 0,

implying r (t 2) > r (t 1)'

Therefore (r (t 2), P (t 2» > (r (t 1) P (t 1) and (r (t 1). P (t I»! W, contradicting Lemma A7. I]

Theorem 10.

n There exists T such that for all t > T 6; (t) < 0 i = 1.2.

H) e; (t ) -+ 0 as t -+ co and e I/(e 1 + e 2) -+ 0 as t -+ co if and only if

P2/1'J2(0) > Pl/1'Jl (0).

Proof.

Ad n. It follows from (3.9) and (3.10) that

Ci lei = (Pi - r )/1'J; ee,) .

1'Ji (e, ) < O. If r (t) = r then Pi > r for both i. If r (0) < r then r (t) -+ 0 as t -+ co.

Ad iD. The first part of ii) follows immediately from i) since 1'Jj (e, ) is bounded. The asymptotic

growth rate of e j is p;l1)j (0). This proves the second part. D

Suppose (r (O).p (0» ¢ (r .]i). Y/ (t) > 0 implies XI (t) > O.

Proor. This is immediate from the fact that r decreases and /;" < O. I]

- 29-

It will turn out to be crucial in the proof of Theorem 12 that tI> (t ) K (t ) ... 0 as t .... co.

This holds if (r (O).p (0» = (F.p) (Corollary B1). For r (0) < F it is proved in the fol­

lowing lemma.

Lemma B2.

(r (O).p (0» ¢ (F.p)::;, lim tI> (t) K (t) = O. , ......

Proof. For t large enough exploitation and production are specialized. Therefore indices t are omitted here.

Define Z by

K=p-ar Z r

K = (p - ai )r -/ (p - ar) Z + I - ar i r r

(from (3.6» .

It follows from (3.11) and (3.12) that

(p -ai)= r(p -ar).

Furthermore I (x) = xr + p (from the homogeneity of F) and

. _ r(l-ar)x p - x +a .

using r = I"x . P = -xii". Then it is easily shown that

So Z is monotonically decreasing. Denote by t* the instant of time after which there is

complete specialization and r (t*) by r*. Then

00 00

t.f E (t) dt = ,.1 a! x dt

must converge. Hence

QO 0 1 -L dt = 1 p - ar _Z_ !!!... !!.P. dr ,. a + x ,.. r a + x dp dr

o = 1 p - ar _Z_ -a - x dr ,.. r a + x rep - ar)

,.. Z = 1 -r dr < co.

o r

- 30-

We also have

dr l-a-

dZ/dr = dZ ~!!£. = _ r (e 1 + e 2 ) dp!!£. dt dp dr p - or dp - ar) dr

Take E < 1"* and consider

r* Z 1 I r* _ r* 1 dZ J - dr = - -Z. J - - - dr £ r2 r c r dr

= _ Z (1"*) + Z eE) + f (e t +e 2)(a +x) 1"* E c r (p - or )2 dr.

It follows that

converges and lim Z (r )/r exists. It equals zero for if Z /r .... A > 0 for some A then r-O

r*

J (Z /r 2) dr would diverge. Finally o

OK </>K = = 9Z/r .

p-or

Hence </> (t ) K (t ) -+ 0 as t -+ 00.

TheQrem 12.

{K (t ) . u (t )} is PE.

[]

Proof. Consider a program that is feasible. i.e. fulfils (3.2) - (3.8). Denote it by upper bars.

T J {as -Pit U 1 (C 1) + (3e -Pit U 2 (C 2) - ae -Pit U 1 (C 1) - /3e -P2' U 2 (e 2)}dt o

T

~ J {ae-Pit U~ (Cl)(Cl-Cl)+(3e-Pr U~ (e 2)(C2-CZ)}dt o

r = J </> (C 1 + C:2 - C 1 - C 2) dt

o

T T

= J </>(Yl+ Y2- Yl- Y2)dt - J c!>or-K)dt o 0

(if Kl > 0 then FiK = r and FiR = p. Therefore we continue)

- 31 -

T

~ J cf>Ir (XI +X~ -K\ -K~) + p (R 1 + R 2 -R1 -R2 )ldt o

T T

-cf> (X - K) 10 + J (X - K).j,dt o

T

~ J cf> {r (Xi + x~ - Xl - x~ - X + K) + p (..Lxi + _1 K~ o al a2

1 1 ---Xi - -X~ )}dt - cf> (T)(X (T)- X (T» a1 a2

T

= J{(81-Ul)(E1-E1) + (82-U2) (E2- E 2)}dt - cf>(T)(X (T)- K(T» o

~ -cf> (T) (K (T)- K (T» .

cf> (T) X (T) -+ 0 as T .... OQ (Lemma 82). []

Here we prove one additional theorem.

Theorem Bt.

There exists M > 0 such that

00

oj {ae -Pl' U 1 (C 1) + pe -Pi U 2 (C 2)} dt ~ M .

Proof. Define: E: = El + E 2 • C: = C 1 + C 2• F (X .E): = Fl (K.E) + F 2 {K .E).

u (C): = aU1 {C) + PU2 (C),p: = min(Pl.P2).

Take r > p and (r .p) E (int VI) n (int V 2)' Then

Hence

K ~ F (K .E) - C ~ r K + pE - C .

t I

e-rt X - X (0) ~ J pe-r. E (s )ds - J e-r• C (s )ds

t

o 0

• ~ P(SI0+S20)- Je-rsC(s)ds.

o

J e-r6 C (s)ds ~ P (S10 + S20) + X (0). o

Furthermore

- 32-

for some positive constants A and B. Therefore

r J {ae -Pi' U t (C I ) + fje- Pr U 2 (C 2 )!ds , .!! + Bp (5 10 + 5 20) + BK (0) . [1 o p

- 33-

AppendixC

This appendix derives (S 10.S2C).K 10. K 20) (v) • used in Section S.

Consider the quadruple v: = (r*. r o. a. 6 1) with 0 < r* <!:. 0 < r* < r 0 < r. o < a < I and 61 > 0 where!: is defined by r = a2P. (r.p) E 3W. Define

where 1'* = p (r* ) such that (r* .1'*) E a w . Let r (t • v ) be the solution of

p(r)- air r == + () r. dO) = ro. r* < r E.; ro

al ;)I; r

r ==

where p{r) is such that (r.p) E aw and ;)I; == -~ on aw. Remark that these

differential equations follow from (3.11) - (3.13) and the constancy of the 9.·s.

Let t (r , v ) be the inverse function of r (t • v ). Hence

"0 (

f al+;)I; s)

t(r,v)= «) )ds ,. s p s -als

fr" a2+;)I;(s)

t (r , v ) = t (r* • v ) + ( ( ) ) ds , 0 < r E.; r* . ,. s p s -a~

Define C l(r • v ) and C 2(r , v ) by

U· (C ( » - 1 Plt(,. ... ) 61 1 1 r,v - -e .

a P -air

u~ (C 2(r, v» = _1_.P2t (,. ... ) 81 , r* E.; r E.; ro 1- a P -alr

U~(C2(r,v» = _1_ eP2f(,., .. ) 82 , 0 < r :t:;; r* .

1- a P -a2r

For given r (t) and v the amounts extracted from the resources can be calculated as fol­

(OWS (see also the proof of Lemma B1):

- 34-

S () Jro Z +(r . v) d 10 V = :2 r .

,.. r

,.. S () J Z-(r. v) d

20V= 2 r. 4) r

where

62 J'" (Z2+ X J'" (Zl+ X ( ) Z +(r • v ) = T" (p )2 C (s • v ) ds - ( )2 C s. v ds • 111 0 - (Z zS ,. P - IZ IS

- Jr az+x Z (r. v ) = ( )z C (s . v ) ds •

4) P - IZZS

Finally define

Straightforward calculations yield

We conclude that if there exist v such that

there exists a general equilibrium characterized by

- 35-

p-ar r = + 2 r. with rcontinuous in t (r*. v) . 112 x

Hence the remaining problem is whether or not such a v exists. The functions

(S 10. S 20. K 10. K 20) (v ) are continuous in v and continuously differentiable in v unless of

course ro and r* are located in points where 8W is not differentiable. These properties

however are not sufficient to have a solution. Unfortunately the formulae derived above do

not allow for much analytical work.

- 36 -

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