Technological renewal of natural resource stocks

Journal of Economic Dynamics and Control 4 (1982) l-36. North-Holland

TECHNOLOGICAL RENEWAL OF NATURAL RESOURCE STOCKS* C. FOURGEAUD CEPREMAP,

UniversitP

de Paris

I, Paris,

France

de PEconomie,

Paris,

B. LENCLUD Direction

de

la P&vision, Minis&e

France

Ph. MICHEL Universird de Paris

i, Paris,

France

Received November 1980 This paper presents a macroeconomic model of optimal growth with stocks of an exhaustible resource. These stocks can be renewed by technological processes, which imply some specific investment costs. The dates, levels and the order of the processes are endogeneous decisions. The main results are: (1) the present value of the exhaustible resource follows a step function in time falling whenever a new stock becomes available and related to the physical return of the process; and (2) shadow prices allow to decentralize the allocation of resources to stock renewal, but they do not permit to decentralize the choice of optimal timing. This would justify the existence of a coordinating agency.

1. Introduction The problem of utilization and pricing of exhaustible resources has stirred new interest, for evident reasons and provoked a considerable literature [Peterson and Fischer (1977), Dasgupta and Heal (197911. The seminal contribution to the field appears to be that of Hotelling (1931) according to ‘which the optimal pricing of an exhaustible stock must be constant in terms of present value, i.e., its current price must rise at .a rate equal to’ the rate of actualization. This price is ‘similar to a scarcity rent, which is in this case, a ‘mining’ rent. In fact, as Heal (1976) has pointed out, the notion of exhaustible resource is dependent upon a certain state of society and its *A fast version of this paper was presented at the European Meeting of the l%onometric Society, Geneva, 1978. We are indebted to R. Guesnerie for his suggestions on earlier draft. We are also grateful to the anonymous referees of this Journal for valuable comments. Of course, we alone are responsible for any remaining errors.

0165-1889/82/0000-0000/$02.75

0 1982 North-Holland

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technology. At a given moment, the amount of resources which are either known or accessible at relatively low costs, is limited; nevertheless if we take the example of energy or of raw minerals necessary for industrial production, their quantities are essentially unlimited, but their extraction implies investments or production costs which are prohibitive when compared at market prices to those of techniques currently in use. This fact has led Nordhaus (1973) to develop the notion of a ‘backstop technology’. Heal, utilizing this notion, assumes the existence of a continuous set of techniques of increasing cost. In particular, he deduces that the price of energy, defined as the sum of extraction costs and rent, evolves in such a way that the latter tends to zero as we approach the backstop technology. This is not a surprising result, since at that point, the previously limited resource becomes infinitely abundant. This property is evidently contradictory with those of the preceeding approach: the rise in rent with the interest rate. Furthermore, Heal shows that when the backstop technology becomes available the price of energy is not discontinuous. Optimal choice between alternative substitutes has been the object of numerous studies [Hanson (1978), de Montbrial (1975), Solow and Wan (1976), among others]. The problem of developing and renewing existing resources can be approached in a different manner by explicitly introducing the technologies which would allow a growth in the initial stock. Let us suppose the existence of a set of known techniques which implemented during a fixed period of time provide for a given new stock. An example of this case would be that of oil fields where new drilling techniques permit to extract greater quantities of oil thus increasing the amount of proven reserves. Another example of a similar case would be the construction of a nuclear plant to which a virtual stock of fixed energy could be assigned, given the norms of production and the life expectancy of the equipment. In fact, the partial renewing of the stock of exhaustible resources places them in an intermediate case between strictly limited resources and those which are renewed by nature (forest, fish, etc.). For the latter case, with constant technology, their present value is decreasing. Intuition suggests that the case of partially renewable resources would fall between these two extremes. Several questions arise from this approach. First, one could examine the validity of the classical results concerning the price of a scarce but renewable stock of a resource which is itself non-renewable. Must the price of energy rise indefinitely? Will it stay constant? Secondly, what will be the price effect of the introduction of a renewing technique at the instant when it increases the available stock. The preceeding authors have either postulated (Nordhaus) or proved under particular assumptions (Heal) that at that moment, the price would not present discontinuities. A different sort of question relates to the possibility of decentralizing the

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decisions relative to new techniques. Is knowledge of future prices sufkient for decentralized profit maximizing agents to make optimal decisions on the dates in which the different techniques should be put into operation? Or, on the contrary, is it necessary to create a coordinating organization to guarantee the utilization of resources according to the general interest? This paper intends to shed some light on these issues. We consider an initial stock of energy or raw materials which can be renewed through the implementation of a certain number of technologies, where each one, when utilized, will reconstitute a stock of variable quantity dependent on the intensity of utilization. This new stock will be available at a date chosen by the user. The production system utilizes this energy together with productive capital and its evolution is studied for a finite period of time. At the end, we impose the existence of a remaining stock of given size. At every instant, production can be assigned either to increasing productive capital, to the implementation of energy renewing technologies or to consumption. Among all possible evolutions of the system, .the choice is made in the classic manner, by maximizing a discounted utility function dependent only on consumption. Thus, our model is one of growth which is classical in many respects but presents new aspects and difficulties. The latter are due to the fact that the evolution equations of the state variables can take different forms depending on the nature of the renewing technologies put to use, the intensity of their utilization and the date at which the new stock will be available. Due to these peculiarities, Pontryagin’s classical principle cannot be applied. A new formulation specially designed to treat this problem, was deemed necessary. However, when taking as given the dates, the order and intensity of use of the renewing technologies, one obtains the classical results, thus contributing little to the current state of the art pasgupta and Heal (1974), Stiglitz (1974)]. But, when applied to the general case, this approach permits, through the optimization of these new elements, to provide new answers to the questions posed beforehand, under the assumption that an optimal solution of the problem exists. A particular result is that the present value of energy varies stepwise in discontinous fashion. The discontinuities appear at the moment when a new stock becomes available and the steps are strictly decreasing. This result implies that the current price of a stock grows with the interest rate up to the point when the stock in question is exhausted. Then, at the moment in which a new stock becomes available, there is a sudden reduction in the price of energy and then a progressive rise with the rate of interest until the stock is exhausted; the process repeats itself. This property may be related to the steep rent decrease given a discontinuous increase in extraction costs as established by Herfmdahl and Kneese (1974) and reiterated by Fisher (1979). The reason here is different as will be shown later.

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This property makes partially renewable resources appear as an intermediary case between the definitively limited resources and those renewed by natural means. We will see that the interpretation of the fall in the price is related to the returns of the renewing investment; it expresses the fact that the operation yields a positive energy return. With respect to the possibilities for decentralization, the answer is more cqmplex. The model used, as we -will see, is not of the classical ArrowDebreu type and knowledge of the price and output of energy does not constitute sufficient information to allow the managers of the different resources to make decisions according to the general interest. More precisely, we will see that although the intensity of utilization of the different technologies can be decentralized, the same does not apply with respect to the choice of optimal dates. This would justify the existence of a coordinating organism whose mission would be not only to announce future prices, but also to announce the dates at which the different techniques should appear, and may be even to provide the necessary financial stimulations. The model considered in this article remains still very partial..To simplify the study we have assumed that the costs of extraction are negligibie’when compared to the investment costs on which this study is based. On another ground, and this is certainly an important omission in our approach, uncertainty -over the costs or results of investment were not taken into account. However, they have been included in simpler models [Arrow and Chang (1978), Crabbt (1977), Deshmukh and Pliska (1980)], but technical difficulties in our model have made us limit the analysis to the deterministic case. In order to make the properties of the model more intuitive, we will present, in the first section, a simplified example with two time periods and one renewing technology where the resulting properties are easily interpreted. The second section will describe the full model while the third part will deal with the solution and the interpretation of its results. The relevant version of Pontryagin’s principle is considered in the appendix together with other mathematical properties of the model. 2. Introductory

example

2.1. A simple management model of an exhaustible resource can be written down in the following form: maxjlogC(t)e-dcdt, 0

C(t)=Ae@E(t),

bE(t)dtJS,-S,,

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where C(t) is the consumption p.roduced by the energy consumed E(t) with a rate of technical progress y; 6 is the discount rate. The availability quantity of energy for the period [0, ZJ is the difference between the initial and fmal stock of energy S, - Sr both of which are given. Taking the residual stock at time t to be expressed by S,+Q(t), the utilization of stock -Q(t)20 is defined by (j(t)=

-E(t),

Q(O)=O.

The global constraint on the stock of energy can be written as

Applying Pontryagin’s maximum co-state variables 4 and IT:

principle gives the following relations, with 1 -e-d=

C(t)=6Ae(Y-d)t(So-ST) 1 se+

’ q=d(s,-s,)’ 1 -e-dT

E(t)=se-d’&-w l-e-dT

’

e-dt

n(t)=6A(So-S,)e-y’=C(t).

The optimal value of utility from consumption AtqS,-S,) 1 -esdT

is

y-6 -(T)[(T+i)e-“-i],

where q can be interpreted as the implicit present value of energy and lT as that of output; q is therefore constant and IT is decreasing (if y > 0). 2.2. Let us introduce a renewing operation requiring a constant investment RON over the interval [a - 0, a] to produce a new available stock N at date tr. The corresponding program can be written as maxje-dtlogC(t)dt, 0

under the following constraints:

(1)

eo= -Eo,

Qo(O)=O,

&=

Ql(O)=O;

-EJ,,..,

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QlV)+NZO, Q&‘-)+&IO, s,+N+Q,(T)+Q,(T)-S,~O;

(2)

O$aSlT; (4)

NZO, E,(t)zO.

E,(t)20,

W)lO,

E, is the consumption of energy levied upon the initial stock, while E, is that taken from the new stock. -Q. and -Q1 are, respectively, the quantities of energy consumed from the initial and the new stock. lL=,B, is the characteristic function of the interval [a, /3].

We will now compare the solution obtained for N =0 (studied above) with that corresponding to a positive value for N. (i) The optimality conditions implicit price q(t) such that

for the case of N >O lead to a definition

of

t ECa,Tl,

cl(t)=419

and checking that: q(t)=A e(Y-d)r/C(t). We will determine q0 and q1 using the property (which will be demonstrated for the general case) according to which the initial stock is exhausted at the moment when the second one becomes available and also by using the preceeding results. (ii) Over the interval [a, T],

where E 1 (r)_e.-d’

3 41

IE,(t)dt=I$

dt=-((e-d”-e-dT)=N-ST, q16 1

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with e-d.-e-dT qi=

6(N-ST)

’

and E,(t)=O. (iii) Over the interval analogous fashion, C(t)=A

[O,a[

the calculation

in

rE[O,a-O[,

eY’E,(t),

C(t)=AeY’Eo(t)-R,N,

We derive, by calling B(u)=fi-

t~[a-@,a[.

e(Roe-“/k)dt=(Ro/Ar)(e-Y(O-

@-e-Y$

1 -e-da

s =l-ewd” 0

of q0 is determined

-+BN, &lo

qO=b(So-NB)’

B is the consumption of energy necessary for the production of a unit of future energy available at time a. B is therefore an energy input coefficient and q,B is the present value of the cost of development.

We have E,(t)=O. (iv) Let us determine now the level of optimal renewing N. The objective function, ilog Ce -dcdr

with

C=s,

is written log (l/qo)~e-dcdr+log(l/q,)~e-d’dr+constant. 0 (I Its derivative with respect to N will be zero, -&,+a

=Q

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which means the marginal benefit of the new stock is equal to the marginal cost of its production. We deduce N_BS,(1-e-d”)+S,(e-d”-e-6’) (1 -emdT)B

.

(v) To determine the optimal date, we must still optimize the objective function with respect to a by taking into account the fact that OSt
C(t)_6Ae(Y-d)‘(S,-BST) l-emdT

a
C(t) =

’

A e(y-b)f(So- BS,) B(1 -e-“‘,

&a)=[logC(t)e-“dt -daee-dT

1 -e-aT

=~log(S,-BS,)-e 6 The derivative of the utility renewing date is

6 of consumption

log B + ct.

function

with respect to the

Calling B(T) the value of B(a) when a= T we obtain the following situations: If B(T)> 1, the technical coefficient B of the energy renewal process is greater than one for all renewal dates since J3 is a decreasing function of that date. The optimal solution in that case is simply not to renew the stock. Effectively, the difference in the consumption utilities of cases with or without renewing,

=-~(e-~'-e-6T)log~-~(l-e-dr)log~'O~~~

, 0

is negative for B> 1.

T

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GA(So-ST1 , -e-6T

t

t

A

0

0

t

a-9

a

t

a

t

1

, -,-6T

q

GAISo-ST1 I I

I\ I 0

Fig. 1. Case N=O.

n

n

e t

0

I

.

a

t

Fig. 2. Case N > 0.

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If B(T)=l, the difference in the consumption utilities of cases with or without renewing is negative for all renewing dates smaller than T; it is zero for a renewal at the final date TT:Here, there is an indifference, i.e. identity of utilities, between non-renewal and renewal at the final date: the yield of the operation is zero. If B(T)< 1, the optimal solution is renewal: there exist dates in which the technical renewal coefficient is less than unity. The optimal solution will be to renew at a date which can be: (a) L?=@, and then we must have 4’(O) j0; the renewal process is started at the first instant in the time period; and (b) a*= T and then we must have @(T)zO; the renewal process is started at the latest possible time (in T-O), and then the newly constituted stock will coincide with the final desired stock ST; (c) 0
2.3. Conclusions. Renewal takes place when the technical coefficient B is less than unity. It follows that the present ualue of energy falls at the renewal date when B < 1. It is this result which we plan to study in the general case. Beforehand, we will point out the discontinuities in the consumption of goods and energy as well as thbse of implicit prices for the case of renewal (N>O) and compare the evolution of these variables with those obtained for the case of non-renewal (N = 0). The results are shown in figs. 1 and 2.

3. The model and its solution 3.1. Data and hypotheses Variables

C: consumption, K: capital (with depreciation investment, and Y: production.

rate m), E: energy, I: gross

The initial value of the stocks of energy K, and S, are given. Also, we impose on these variables minimum final values K,, S.,. which constitute the model’s horizon. Production function

The following hypotheses are made on the F function:

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Hypotheses

(la)

For K>O, E>O and tE[O,TJ, continuous.

(lb)

ForK>O

(lc)

ForK>O,E>Oand

F(K,E,t),Fk(K,E,t)

and Fi(K,E,t)

are

and tE[O,r[l,F(K,O,t)=O. tE[O,T],Fk(K,E,t)>OandFk(K,E,t)>O.

Energy stock renewal We have at our disposal L different processes in order of energy. Each process k (15 k SL) is characterized duration of investment and by an investment function corresponding to quantity N, of new energy stock. The the new stock is labeled a, and its consumption is labeled

to renew the stock by the necessary R,(t,N,), te [O, 8 J availability date of E,(t), for t >a,.

Hypotheses concerning R,

(2a) Ok< T. (2b) R,(t,N,)=O

for each t, iffNI,=O.

(2~) Rk(t,Nk) is continuously differentiable, hypothesis simplifies the mathematical essential conclusions). (2d) For IV, >O, (8R,.JN,)(t,iV,)

and zero for ~$10, S,[ (this study without changing the

20, and not identically zero.

Objective function For a consumption

trajectory

ie-“U(C(t))dt,

C(t), t E [0, 7’J the corresponding

utility is

6>0.

Hypotheses concerning U

(3a) U(C) is continuous for C 2 0 and differentiable (3b) U’(C)>0 (3c) lim,,,

for C > 0.

for C>O. U’(C)=

3.2. Optimization

+ co.

problem

We consider a continuous parameters and constraints.

time program

with state and control variables,

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State variables For capital K,

g(t)= -d(t)+z(t),

K(o)=&,

.K(T)zK,.

Energy consumption - Qb, - Qr, . . ., -QL levied on the initial stock So and on the new stocks N,, . . ., N,,

where lt,,,Jt)

is the characteristic function for the interval [a,, T].

Control variables and parameters C 20: consumption, Z LO: gross investment, E 2 0: total energy consumed, E,zO: energy consumed from stock N,, N,>,O: new energy stock obtained from the kth process, a,, 0, $ a, S T: availability date for the energy obtained from process k. Constraints Resources -

uses of production,

C(t)+Z(t)+

i

R,.(t-ak+Ok,Nk)5F(K(t),E(t),

k=l

Resources -

uses of energy,

Qk(T)+Nkzo

QoU-I+,+

for

15 ksL,

i (Qk(T)+Nk)~& k=l

Problem (P) max [e-dTJ(C(t))dt,

t)

a.e.

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under the following constraints :

Ii= -mK+z,

i &l~or,~,, QdO)=Q

t&,=-E+

(1)

K(O)=K,,

k=l

(2)

&I+ c N,+&(T)+

c &(T)--&zO;

k=l

k=l

(3

F(K(r),E(t),t)lC(t)+z(t)+

(4)

c(t)zo,

i

Rk(t-ak+@k,Nk)

a-e.;

k=l

z(t)zo,

E(t)ho,

&(t)zo,

i E,(t)~E(t). k=l

One of the peculiarities of this system is that there are control variables (Nk and ak) which are also parameters and thus the system can evolve differently depending on their value. Consequently, we must utilize a new form of Pontryagin’s principle in order to obtain the necessary optima&y conditions. This new version appears in appendix 1. 3.3. Hypotheses concerning the optimal solution Let R(t), Qk (OSkSL),

IL)

e(t), r(t), E(t), &k(t) (lsksL),

tik and Nk (1s;

be an optimal solution. We make the following hypotheses:

Hypotheses

(4a) There exists consumption.

a realizable

(4b) Gross production i.e., Vt,

trajectory

is never completely

F@?(t)&(t),+

i k=l

with

a non-identically

zero

utilized for energy investment,

&(t-tik+@,,&).

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(4c) The function

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U(t)=e-“U’(C(t))

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is decreasing with respect to t in

CO,Tl. Remarks

(i) Hypothesis 4b implies C(t)>O, Vt (appendix 2, property 1). (ii) In order to satisfy hypothesis 4c, it is sufficient that Vt,

Fh(K(t),E(t),

t)-m>O

(positive net marginal productivity

of capital).

Proof

For C, et,, the decrease of C(t) by a quantity s>O for [t,,r, +A[ permits the increase of C(t) over [r2,r2 +A[ by a quantity E’ such that in the first order E’=Eexp~(FIK(~(t),B(t),t)-m)dt>s. ‘1 (This results from the theorem of differentiability with respect to the initial conditions of a solution of a differential equation.) Therefore, the corresponding change of Jiemd’U(C(t))dt, which is nonpositive, is equal to the first order at -Aee-dffU’(C(t

1

))+h’e-

d’ZU’(e(t2))>~&[n(t,)-n(t,)].

Hence n(t,)
hypothesis 4c is equivalent

to 6 - (d/dt) log U’(C(c)) > 0. (This means that all along the optimal growth path, the discount rate in the economy is positive.)

3.4. The necessary optimality

conditions and the associated multipliers

---_ Let {K, QkrC,I,E, si,, m,} be an optimal there exist p(t), qk(t), ck, such that

solution to the problem.

Then

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$=pm-e-“U’(C)F;, O$ksL

&=Q

lik s

l&-e,

aRk e-W(C)-(t-iik+

aNk

0 k,iV,)dtzq, =qk

if

m,>o.

And we show (see appendix 2) that: (1) the optimal values of a,, such that tik < T and fl, > 0, are different: ii~l
ci+tscl;elk, =wg vmm,t),~k,w=m), (2)

Qo(~k,)+SO=‘o,Qk,(~k,+,)+~,,=o,

(3)

qO’qk,‘qk2’**->qk;

lsiss-1,

4. Economic interpretation 4.1. Prices and the continuity

of variables

(a) We can interpret n(t)=e-“U’(C(t)) as the implicit present value of the consumption good, .and p(t) as that of a capital good. The two prices are

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equal when investment T(t) is positive, ll(t)$p(t)

and

~(t)>O*l7(t)=p(t).

When T(t)>O, we find the usual relation defining the discount rate i(t), i(t)=6-$logU’(Qt))=

fi(t) -t=,,

--m

-m=

P(t)

(b) We can equally interpret q(t)=L’(t)FH and we find [if r(t) > 0]

K

.

as the present value of energy,

i(t)+m=$logF;. On the other grounds, for a positive renewal of stock (N, > 0), qk is equal to the marginal cost of development of source k, when renewing dates are optimal, qk=

y n(t$$(t-i,+o,,N,)dt. 4-8, k

(c) The continuity of p(t) implies that of C(t) when investment is positive on both sides of the instant of utilization of a technique (4 - 0,) and of that of its availability (~7~);however, when investment vanishes, consumption is discontinuous in general as illustrated by the example where, in the absence of capital, investment is always zero by definition. The discontinuity of the utilization of energy l?(t) results from that of its implicit price q(t). In fact, the latter is a piecewise constant function and strictly decreasing at each renewal date. 4.2. The decrease in the present value of energy

As was illustrated in the introductory example, the renewing process consists of a transformation of energy currently available into energy which will be available at the end of the process. The evolution of present values is directly related to the energy input coefficient B of the operation by the relation

and we found that renewing was effective for B< 1 which implies the decrease in present values, q1
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In the more general case where productive capital intervenes, we can give a similar interpretation to the strictly decline of the prices at the instant of renewal (4) by considering the two relations

qk=

y

n(t)$dr

where R, means Rk(t - Cr,+ Ok, N,).

4-8,

k

Let B,=

7

4-8, -c

$ k

FL dt. I>

B, is called the marginal energy input coefficient for process k.

We deduce

qkzqk-I

where

eqUdity

7

OCCUrs

F;, dt=q,-,B,,

2

ik-e, -c if iik -

k I>

Ok 2 L& _ 1,

i.e., if the renewing process started in

the previous period 4, during the period of utilization of the stock (k- 1). Property qkV1 >qk implies that B,< 1, i.e., that the process’ energy input coefficient is less than unity and hence, that the physical returns are positive. Also, we can understand the decrease in the present value of energy by comparing the renewal process to two extreme cases. For a non-renewable stock the implicit present value is constant. For a naturally renewable resource (e.g. a forest), the implicit present value is descreasing. Our case corresponds to an intermediary situation as illustrated in fig. 3.

Fig.

3

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4.3. Optimal renewal dates

Let us consider the total expenditure of energy 9 defined as the sum of direct expenditures in consumption and of expenditures in stock renewal, 9=.j’q,,B(t)dt+‘~ 0

‘r”q,$(t)dt+ i=l

ok

t

j q,/%)dr Ok*

i=s

+ 1 7 irlOk-q. I

Il(t)R,,(t-a,,+Oi,N,,)dt.

We show (see appendix 2, properties 14-16) that

~(~ki-0)=(4ri-~-4k,)E(~ki-O)-ck,, k, ~(a,,+0)=(4k,-,-ei)E(4,)-Ck,, ki

and that g(tik,-O)
and

~(ti+o. ki

Optimal dates achieve a local minimum in the total expenditure of energy, and thus, 9 appears as the social cost of energy consumption. If we now consider the investment expenditure corresponding to process k13 Iii

D/c,= j n(t)R,i(t-a,,+O,,,N,,)dt. -8, 111 Using property 15 of the second appendix, we obtain

and we know (appendix 2, property 7) that cki > 0. By analogy with the interpretation of qki in terms of marginal costs of development, we can interpret cki as the ‘marginal cost of the availability date’. Noting that ax, -. T; Rkidn, ck,=o 5, n(t)?dr= It k, kt I,

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ck, represents the decrease in the cost when investment is postponed (since l7 is decreasing). 4.4. Decentralization

Let us consider the present balance of a manager of resource k, 2&=qkNk-

y n(t)R,(t-&+&,N,Jdt. 4-8,

Since R, is convex with respect to N,, Bk achieves a maximum in mk for the fixed optimal dates (appendix 2, property 17). The level of activity of the process k, N,, is therefore decentralizable with respect to prices, when the availability date tik is fixed. The same does not apply to this date. Effectively, the receipts of the manager of resource I( is constant and equal to qkN, for any date, in the interval CC&,, Q+ i [, where the new stock is available; but the discounted investment costs decrease as the availability date is postponed as we saw in the previous paragraph. Knowledge of present values is not sufficient to allow decentralization of both the’level of activity and the dates of activation of the different processes. Yet, decentralization remains possible if we introduce a ‘price for the dates’ which could be interpreted as a penalty for lateness. Under these conditions, if we consider the balance @ modified by the introduction of this supplementary cost,

we notice that &8,Jaak, the marginal balance, is zero for the optimal date tik and given the ‘appropriate’ hypotheses, it attains a maximum for this date. This peculiarity shows that the model considered here is not directly of the Arrow-Debreu type. Available energy at two different dates in the same interval cannot be considered to be the same good, in spite of the fact it has the same price. The good must be defined by the process which allows to obtain it and by the date of its availability; and this is due to the irreversibility of time.’ It would be necessary to consider a coordinating organism which, besides prices, would also impose investment dates, or equivalent, would have the power to impose delay penalties. In any case, even with the introduction of these dates as additional goods, it is difficult to imagine that their value could be revealed by the market.

‘This interpretation

was suggested to us by R. Guesnerie.

C. Fourgeaud

20

et al., Technical

renewal

of natural

resource

stocks

4.5. The ordering of processes

Let us characterize the process ki available at &, by the interval [a,,,- I, ii,(a[ in which the investment starting date a,- Oi is included (by noting ai=Ck, and Ri=R,,(t-ci,,+Oi,N,i). From the formulas qi=

y n(t)aRi &t, I ai-ei

n(t)F;=q(t),

we obtain

qi’qi-l “s’dRidt+qi-Zaijl !!k dt a,-2aNi Fk *,-,aNi Fk

or by noting ri-l=

1 aRi dt ai- 1aNi Fk ’

rim2=

a’-~aR. dt j 27, 0,--2aNi FE

. . ..

a*(‘) a& dt r&i)-I=

j

-Ts

O,-e,aNi FE

we obtain

The terms riej represent the energy quantity levied on the stock (i-j) available at the instant ai- j needed to obtain a unit of stock i available at a,. They are partial energy input coefficients and the expression C:idl(i)+l qimjrjsj represents the (marginal) discounted balance. Additionally, we have seen (cf. section 4.2) that i-d(i)+

Bi=

C

1

rjejcl.

j=l

If we consider now the coefficient Bi defined by qi = l?,q,- 1, we cm also write

C. Foutgeaud

et al., Technical

If-, Bi=rj-1

ano

+Bi_,

renewal

of natural

resource

stocks

21

r&o - I +“‘+Bi-l,...,Bd(i,’

find the following property concerning the sequence of processes: B”i
41

cl2

q”=~>Y&>“‘>~.

%

In fact, &AL
~=qi~l>qi-2’~. i

I

1

and hence, B”i appears as an energy input coefficient valid for the period [ai- Oi, Ui[. Of course, if d(i)=i- 1, i.e., if the process has been started in the interval CC&..-,,&,,[. Bi is equal t0 ri-l,

This property shows that processes are ordered in a decreasing order in relation to qi/Bi which indicates their profitability.

Appendix 1: Reformulation

of Pontryagin’s principle2

We consider the following problem:

m=So(x(?:h)), under the constraints

~=F,(w~>w~+

i l,,,,,F,(t,x(t),h,u(t))

I=1

a.e. in [0, Q,.

%e

also Michel

(1981).

22

C. Fourgeoud

et al., Technical

1[.I.T]=o

if

t-=a,,

=l

if

qIt$‘7;

renewal

of natural

resource

stocks

The controls u(t) are piecewise continuous functions over [O, T] and take their values in a topological space U. The vector x(t) = (x’(t), . . .,x”(t)) is an element of the Euclidian space R” and C is a convex of R”. Functions F, (OSISL), $, f, (Osps Q), and g, (15 15~5) are continuously differentiable with respect to x and h. We adopt the following notation: a=(a,,...,%.), F(t,x,h,u,a)=F,(t,x,h,u)+

i l[.,.T,W,X,h,U), I=1

F-(t,x,h,u,a)=F(t-O,x,h,u,a) =F,(t,x,h,u)+ The function F is continuous the left of t for all t. For GEE,,, let

i l~nr,~]F~(4X,h,U). 1=1

at the right of t, function F- is continuous

at

C. Fourgeaud

et

al., Technical

renewal

23

of nahual resource stocks

Let (z?,u*,ci,i) be an optimal solution. Without loss of generality, we suppose that the function C(t) is continuous to the right of t for t E[O, T[, and continuous to the left of T The constraints’ qualijkation

conditions

The following linear system admits the unique solution [b,=,O (O$pSQ) and cl=0 (l~~IL.L)]: for

b,zO

OSpSP,

b,f,(x^(T),h)=O

for

$cbr~(zZ(T),h)=O

for

lgpSP,

l$isn,

i$l(hi-&)(p=o 5 b‘ahi af, for all

hEC.

Theorem. If the constraints’ qualtfication conditions are satisfied and if (12,I?,h, a*) is an optimal solution, then there exist real numbers b, (OS&Q) and c, (1515 L), not all zero and a non-zero continuous function q(t) such that

(a) for OspSP,

b,zO, andfor

lSpSP,

b,f,(z?(T),h)=O;

lb) q(t) obeys hi(t) -= dt

BH t

-ax-(t,q(t),i(t),Ci,Q(t),a*),

lsisn,

(c) for all t E [O, T[ (respectively t~]0,7+J), the function H(t,q(t), Z(t), I;, u,ci) (respectively u+H-) attains its maximum (respectively M-(t)) over the space U at u*(t) (respectively t?(t-0));

(d) jar each h E C,

u+ M(t)

24

C. Foutgeaud

et al., Technical

renewal

of natural

resource

stocks

(e) for each 1 (1 S 15 L),

Appendix 2: Mathematical An equivalent formulation

study of the model of the problem

Property 1. Under the hypotheses 3 and 4, the optimal solution problem (P) obeys the condition C(r)>Ofor all r in the interval [0, T]. Proof.

to the

Let us fix the optimal values &, i?, w, and &, and label G(K,r)=F(K,E(r),t)-

i

Rk(r-~k+Ok,N’k).

k=l

The functions R(t) following problem:

and S(t)=T(t)/G(l?(r),t)

maxje-6’U((1

are optimal

solutions

to the

-s)G(K,t))dr,

0

subject to &=sG(K,t)-mK,

K(O)=K,,

K(T)LK,,

SEp, 11.

From Pontryagin’s maximum principle, there exists a couple (po,‘p(r)) nonidentically zero such that p. 20, p(T)>,0 and such that the Hamiltonian, H=p,e-“U((1

-s)G(~(t),t))+p(r)(sG(~(r),r)-d(t)),

reaches its maximum in the interval [0, 11 in S(r).

C. Fourgeaud

et al., Technical

renewal

Let us show that p0 is not zero. If homogeneous equation

$0) = -p(t)W)Gt(~(t),

of natural

resource

25

stocks

were zero, p(t), the solution of the

p.

t) - ml,

does not become zero and hence proves that p(t)>O, Vt. This implies that S(t)= 1, V t, since G@(r), t) is positive (hypothesis 4b). We obtain then that C(t)=O, Vt, and C(t) will not be optimal (hypothesis 4a). Consequently, we have p0 > 0 and, with hypothesis 3c, lim (&Y/as) = - 03. 5-l

H being maximum in s(t), we have S(t)cl, Property 2.

4

Vt.

The problem (P) is equivalent to the problem (P’) defined as (P’) maximize Z(T),

under the following four conditions:

zc=-mK+z,K(O)=K,, i=e-“U[F(K,E,t)-I-

f R,(t-h,+O,,N,)], k=l

(1) Qo=

-E+

ok=

-Ekl,ak,,>

i k=l

Qo(O)=Q

Ekl~cr,..T,r

Qk(O)=o,

K(T)-K,ZO,

Qk(T)+Nkzo,

l$kSL; OSkSL,

(2) k$o (Qk(T)+Nd-&-?o; (3)

UK&t)-I-

5 &(t-h,+@,,N,)zO

a.‘?.;

k=l

ak=hk, h,>=@,, osUk61: IlkSL,

(4)

N,=S,,

N,zo,

lSkSL,

z>=o,

E,zO,

lSkSL,

EZ i k=l

E,.

Z(O)=O,

C. Fourgeaud

26

et al., Technical

renewal

of natural

resource

stocks

The problem (P’) is obtained by making Z(t)=ie-“U(C(s))ds 0

and eliminating C(t). Property 3.

n

The optimal solution to the problem (P) define with

z(t)=ie-‘YJ(C(s))ds, 0

locally an optimal solution to the problem (P”) obtained from problem (P’) by eliminating constrainr (3).

This is a consequence of property 1: constraint (3) is never binding. Application of the necessary optimality

conditions

Property 4.

The conditions of qualification

are verified.

The following:

system which

qualification

Proof.

alO,

b’?O,

b’(J%‘-F&)=0,

defines the

b,ZO,

n

(OSkSL),

conditions

is the

b20, OSk=
b,&&(T)+NJ=O,

b i @k(T)+W-ST =Q k=O

>

b’=O,

a=O,

ckzo

if

&=?:

ck(hk-h&O, (&-‘&)(bk+b)sO,

b,+b=O,

OS;kSL,

ck=O

if

Vh,z@,, VNk.0,

tik<‘T;

lS:kSL,

lSksL, 1SkSL.

The relations of this system give b’=O, a =0, and b =0= b, (OS k S L), since we have b 20, b, 20 and b + b, =O. For k such that ‘ik < ?: we have directly ck= 0. And for k such that ii, = T, we have Zik> Ok (hypothesis 2a); in

C. Fourgeoud

et al., Rchnical

renewal

of natural

resource

21

stocks

this case, the inequality ck(hk-&)ZO,

Vh,BO,,

implies’

The system which define the conditions trivial solution 0. n

c,=O.

of qualification

admits only the

Property 5. The necessary optima&y conditions for the problem (P”) are: there exist real numbers a, b’, bk (OS ks L), b, c, (1 S k6 L), and functions p(t), r(t), qk(t) (05 kSL) not all zero, which obey relations (5~(13):

(5)

~120,

b’B0,

bk.O

b’@(T)-K,)=O,

(OSksL),

bz0;

bk(Qk(T)+&.)=O,

OSksL,

$1 b

i

(&(7-)+&)-S,

k=O

=O. >

Let

Ho=P(--++I)

+re-“U

F(K,E,t)--I-

i

&(b-hk+@&.)

k=l

fh=ho--qk)Ekr

H=HO+

i k=l

Hkl,,,,T,.

Then p= -aH/X=pm-re-6W’F;,

(7)

(8)

i='-aH/dZ=O, (ik= -aH/aQ,=o,

OSkSL;

pV)=b’,

qk(T)=b,+b,

r(T)=@.,

The maximum of the Hamiltonian

(9)

OSksL.

with respect to 1 gives

p-re-W(C(t))SO

=o if @)>O.

1

-q&,

C.

28

Let E = E, + z,

Fourgeaud et al., Technical renewal of natural

1 I&; H is at a maximum

resource

stocks

for

all E, 2 0 (0 5 k 2 L). We obtain

if

Eo(ct)>O,

aH/aE,=re-6’U’FH-q,~0 =o (10)

aH/aEk = r e-“U’F’,

- q0 60,

=re-“U’F;-q,

50 =o

t<&, tzi&+lY if

E,(t)=-0,

All the h, and Nk parameters are independent; iIt. follows then that, for 15 k dk

dts;;c,

5 dk-e,

=C

k

=b,+b if

cT,=‘T:

(&,-qk)Ekk(a,)&k,

if

cs,<‘7;

($,-qk)Ekk(Ek)&kr

M(iik)-M-(iik)~C~Cl;~,=iik},

where

-q$(&---0)+

c {I; a,
(qO-q,)~,(ik-O)*

Consequences Property 6.

The multipliers obey the following r=cr>O,

(14)

iik>@,;

dt>=b,+b

(12)

(13)

l$

p(t)>O,

qk=bk+b, VI
qk>O,

OSkSL, OSkgL.

relations:

ij- iif,>O;

C. Foutgeaud et al., l&h&al

renewal of natural resource stocks

29

Proof

(i) r and & are constant [relation (7)] and respectively equal to a and (S)]. (ii) I is not zero. In the opposite cast we would have p(t)sO, Vt [relation (9)], p(T)= b’20 [relations (8) and (5)], therefore p(t)=O, Vt, since p(t) is the solution of 6 = mp if r is zero. We would also have that & = bk+ b 20 [relation (5)] and SO [relation (1211, i.e., qk =O. Hence, all multipliers p, r and qk would be zero, which is not possible. (iii) If p(t) takes on a value p(&,)SO, then &)-CO [relation (7)] and o(t) < 0 for all t g t,, which implies that p(T) < 0, which implies a contradiction. (iv) For Nk =0, 4 is arbitrary and can be chosen to be smaller than x which implies that qk > 0 [relation (lo)]. And for Nk > 0, relation (12) which hypothesis 2d imply that qk > 0. 1 bk+ b [relation

Notation.

Due to the preceeding property, one can choose a = r = 1. Let

(15)

n(t)=e-W’(C(t)).

*Relation (9) can be written (16)

P(t)sm) =U(t)

Property Proof.

if

f(t)>O.

7. For all k such that nk >O, the multiplier

ck obeys ck>O.

Relation (11) can be written

kntegrating by parts, we get

And for nk>O, we get Rk(t,mk)hO not identically zero. Hence, hypothesis 4c gives as conclusion that ck>O. n Property 8. At the availpbility date, all preceeding stocks are exhausted. The optimal dates 4 < T corresponding to a stock renewal efictive when &>O JEDC-

B

30

C. Fourgeaud

obey the following

et al., Technical

renewal

of natural

resource

stocks

relations:

(17)

~&-+<...
(18)

n(t)~(~(t),E(t),t)=q,i,

2ik,st m=&,,(t), llils-1, -Z&ST;

m=Jqw,

(19)

Qo(~kl)+SO=QQk,(q+,)+&=Q

llils-1. --

ProoJ: Suppose that at date E,+, all preceeding stocks have not been exhausted. We would then be able to choose E,..(&,)=O without modifying E(t), C(t) and r(t), since we can utilize the previously existing stock. We obtain therefore an optimal solution which obeys the necessary conditions (5)-(13) and, in particular, ck5 0 which results from relation (13). This contradicts property 7. The exhaustion of the preceeding stocks at a,( implies relation (19), and also that the preceeding availability date &..~ti~, are strictly inferior to &..; therefore we get relation (17). Additionally, for all t in the interval [c&&,+~[, we have &,(t)=E(t), and relations (10) imply relations (18).

Property 9. Let us consider the following

sets:

L, ={k;Nk=O}, L,={k;N,>O,ci,
,..., k,},

L,={k;N,>O,&=T}.

We have the following (20)

properties:

n($$I@),E(t),t)j

1. For k E L, and t E CO,, T], 1 L’(s)$(s-t+O,,O)ds. t-9, k

C. Fourgeaud

et al., Technical

renewal

of natural

resource

stocks

31

2. For kELzvL3, (21)

3. For kEL,, (22)

qk=b,

VlSl$L.

qksql,

Proof: 1. For kE L,, 15, is zero and 4 can be chosen arbitrarily in [O,, 7’l without modifying the optimal solution. We then obtain, through relations (10) and (12),

j n(s), aRk (8-t+@,,O)ds. n(t)~(~(t),B(t),t)~q,~ t-0, k

2. For Rk > 0, relation (12) implies relation (21). 3. For k E L,, we have 4 = T and wk >O, which implies that, with &(T) =Gk(0)=O, &(T)+fik>O. Therefore b,=O, qk=b and q,sb+b,=q, for all 1IIIL. n -Property 10.

(23)

We have the following

b>O

and

i

relation:

(f&(T)+&.)-&=O.

k=O

Proof If the last renewed stock constituted before time ‘1 k= k, is not exhausted by ?; we have G,‘(T)+fik, > 0, then bkS is zero and b= bk, iS positive (property 6). If the stock w,, is exhausted at T (Zk,(T)+Nk,=O, then the relation

implies that at least an additional stock must be available at ‘I: Hence, there exist k E L, and we have b = qk > 0. Finally, b > 0 implies

Property 11.

(24)

“j’ 6,-e,,

For 15 iss, we have n(t)~(t-o,,+Qk,,N,,)dt~(qk,-,-B,l~(4,).

C. Fourgeaud

32

et al., Technical

renewal

ofnatural

resource

stocks

Proof. Let us consider the optimality problem obtained by imposing that the renewing dates be distinct from each other and that the energy consumption over [~,,,a,,+,[ be extracted from stock Nki,

This problem admits the same optimal solution since the latter obeys those conditions. For this problem, the necessary optimality conditions imply the following relations, with the multipliers cLi and qLi:

hr 3Rk, -4$(&J s n(t)--tSc;,~h;,-, at b,-%( Lik,

f

q;,=

lSl(t)$dt. ki

ik,-ek,

The last equality implies that qii = qk,. Consequently, (24).

the multipliers qki obey

Property 12. The series of implicit present values of energy of available stocks before IT:qO, qkl,. . ., qk,, are strictly decreasing. Proof

Integrating

by parts the first member of inequality

(24), we get, for

llils --, h-l-qk,)@k,)~

-

y’ q-e,,

Rkdn>o,

since we have R, dnS0 and non-zero (hypotheses 2b and 4c). Consequently n E(Gk,) and qki-, - qk, are both POSitiVe. Property 13. (25) :Proof

cki ’

For 15 iss, we have (qk+,

-qk,)‘%ki

-Oh

We have M(a,i)=H,(~(a,,),~(((T,,),a,,)+

(q,,

-qki)Ek,(cki).

C. Fourgeaud

et al., Technical

Since the Hamiltonian

renewal

of natural

resource

stocks

33

is at a maximum with respect to I and &,, we have

~(~k,)z?h(wk,-

o),~~~k,,-o)~~k,,)+(q,-qk,)~(4,-0).

This inequality is strict: if H attained its maximum and I =T(&,, - 0), we would have mcr,, - waJwk,),

&k,-o>,

at tikl for E =I?(&., -0)

akJ = qkp

and relation (18) implies n(rs,,-O)F~(~(rS,,),E(cS,,-OO),~kT,,)=qk,-,, which is excluded by property 12 (qkl #qkiWl). In addition, we have

Relation (13) implies therefore n

Ck,~~(cSk,)-M-(tikr)>(qki_r-qk,)~(Lik,-oO)’

Property 14. (26)

Total expenditure of energy consumed,

‘y’q,,B(t)dt+

9=‘[‘q,,E(t)dt+s~ i=l

ak

T qk,E(t)dt, ‘k

I

I

‘admits with respect to a,, a derivative at the left of (Sk,,

(27)

~(4,-O)=(qk,_,-qk,)~(4,-O), ki

and another derivative at the right of &,,

(23) Proof:

~(a,,+O)=(qk~-,-qk,)~(4,). ki

For a piece-wise continuous function Jl(t), the expression

admits the limit I/@ +0) [respectively

$(b-O)]

when h tends to zero from

C. Fourgeaud

et al., Technical

renewal

o/natural

resource

stocks

35

Remark. For L&..> Okl, we have ck= I(a~@&.)&)l, the multiplier c, is the rate of interest with respect to the time of expenditure of the necessary investments available at & of the stock R,.

Property 16 can be interpreted as an extremum condition in or, of the sum of energy consumption ‘and investment expenditures: properties #‘(ak - 0) s 0 and $‘(ak+O)zO are the necessary conditions for a function- which is differentiable at left and right, be minimum at Cr,. Property 17. If the investment function Nkr then the discounted profit of a process

Rk(t,Nk) of type

is convex with respect to k,

(31) is positive or zero and maximum at Nk=nk. Proof.

Relation (12) implies

=o if if+-0; hence if Rk(t, *) is convex, we obtain

References Arrow, K.J. and S. Chang, 1978, Optimal pricing, use and exploration of. uncertain natural resource stocks, Technical report no. 31 (Harvard University, Cambridge, MA). Crabbb, P., 1977, L’exploration des ressources extractives non-renouvelables: ThCorie konomique, processus stochastique et vbrification, L’Actualitb Economique 4,559-586. Dasgupta, P. and G. Heal, 1974, The optimal depletion of exhaustible resources, Review of Economic Studies Symposium, 3-28. Dasgupta, P. and G. Heal, 1979, Economic theory and exhaustible resourcw (Cambridge University Press, Cambridge, MA). Deshmukh, S. and S. Pliska, 1980, Optimal consumption and exploration of nonrenewable resources under uncertainty, Econometrica 48, no. 1, 177-200. Fisher, A.C., 1979, On measures of natural resource scarcity, in: V.K. Smith, ed., Scarcity and growth reconsidered (Johns Hopkins Press, Baltimore, MA). Hanson, D., 1978, Efficient transitions from a resource to a substitute technology in an economic growth context, Journal of Economic Theory 17, no. 1,99-113. Heal, G., 1976, The relationship between price and extraction cost for a resource with a backstop technology, Bell Journal of Economics 7, no. 2, 371-378.

36

C. Fourgeaud

et

al.,

Technical

renewal

of natural

resource

stocks

Heriindahl, 0. and A. Kneese, 1974, Economic theory of natural resources (Merrill, Columbus, OH). Hotelling, H., 1931, The economics of exhaustible resources, Journal of Political Economy. Michel, P., 1981, Choice of projects and their starting dates: An extension of Pontryagin’s maximum principle to a case which allows choice among different possible evolution equations, Journal of Economic Dynamics and Control 3, no. 1,97-118. de Montbrial, T., 1975, La tarilication des ressources naturelles non-renouvelables, World Congress of the Econometric Society, Toronto. Nordhaus, W., 1973, The allocation of energy resources, Brookings Papers on Economic Activity. Peterson, F.M. and A.C. Fisher, 1977, The exploitation of extractive resources: A survey, The Economic Journal 87,681-721. SOlOW, R.M. and F.Y. Wan, 1976, Extraction costs in the theory of exhaustible resources, Bell Journal of Economics 7, no. 2, 359-370. Stightz, J., 1974, Growth with exhaustible natural resources: Efficient and optimal growth paths, Review of Economic Studies Symposium, 123-138.