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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.
Document status and date:Published: 01/01/1986
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Download date: 28. Jul. 2020
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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