Price shocks and fringe supply of natural exhaustible resource commodities

commodities

Citation for published version (APA):

Withagen, C. A. A. M. (1984). Price shocks and fringe supply of natural exhaustible resource commodities. (Memorandum COSOR; Vol. 8410). Technische Hogeschool Eindhoven.

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

~emorandum COSOR 84 - 10

Price shocks and fringe supply of natural exhaustible resource commodities

by Cees \Vithagen

Eindhoven, The Netherlands September 1984

Cees Withagen

Faculty of Philosophy and Social Sciences Eindhoven University of Technology

P.O. Box 513 5600 ME Eindhoven the Netherlands

Summary

In the present paper we investigate the impact of an anticipated shock in the world market price of an exhaustible resource commodity on the planning of optimal resource extraction in a small economy. It is found that the supply of this economy is not necessarily dis-continuous and that, if a discontinuity occurs, the direction of the discontinuity depends on the value of the elasticity of marginal utility.

1. Introduction

In the past decade the world "suffered" from two shocks in the world market prices of oil. It is commonly agreed upon that they originated from political circumstances and that the possibility of sharp price hykes occurring was due to a coherent cartelization on the supply side. In the eighties this situation has changed drastically. First, Opec cannot be considered anymore as a coherent entity and second, possibly implying the former reason, demand for oil (or energy sources in general) is nowadays met by a number of non-opec members as well. Therefore, the market structure should be described by a cartel-versus-fringe model, on which Newbery (1981), and Ulph and Folie (1980) have very elegantly elaborated. In view of a.o. the lack of satisfying models of von Stackelberg rational expectations equilibrium, it seems however premature to predict that price shocks will never occur again. It is not the purpose of the present paper to give such a model but our far more moderate aim is to describe the impact of an anticipated price shock on the supply schedule of fringe members. Consequently, we construct a simple model of such a supplier, who is expecting a sudden price increase at a given instant of time (section 2). In section 3 the outcomes of the excercise are presented. Section 4, finally, summarizes and concludes. If not already obvious from the foregoing, it should be stressed that our analysis is of a partial equilibrium nature. But given the very origins of the oil crises, this seems justified.

2

-2. The model and some preliminary results

The economy under consideration has the disposal of an exhaustible non-replenishable, resource, whose initial size is known with certainty and denoted by SO. The reserve remaining at instant of time t is denoted by Set). Due to, for example, geographical circum-stances, the rate of exploitation (E) is bounded from above by some positive constant E. The first constraints the economy faces, are then:

set) = -E(t) So given (1)

S (t) ~ 0 (2)

o

~ E(t} ~ E (3)

For the resource commodity there exists a world market. Our economy, being small, takes the price coming about on this market as given. At the outset of the planning period (time 0) the economyts (firm) expectations are described by:

pet) _POe Yl t _{, Y 1} > 0 I for 0 ~ t ~ T

,

(4a)

pet)

_ y

2t

Y₂ > 0 for t > T

pe

,

,

(4b)

where T is some given instant of time. It is assumed that p jumps upwards:

p(T-) lim pet) < p{T+) := lim pet) (4c)

If there would exist a perfect world market for lending and borrowing, the problem would not be very interesting. The best extraction policy for the economy is to sell so as to maximize present discounted profits from exploitation. So, therefore, we consider the case where, on a priori grounds, the economy wishes or is forced to establish permanent equilibrium on its current account. In order to simplify matters further, it will be assumed that exploitation is the only source of income of the economy. From the revenues consumer goods (e), serving as the numeraire are bought. It follows that

e(t}

=

p(t)E(t) • (5)

The economy's objective is to maximize

00

J :=

J

e -pt u(e(t»dt (6)

o

where pC> 0) is the constant rate of time preference and U denotes the instantaneous utility function, assumed to be increasing, strictly concave and such that U' (0)

=

00.

This is the model. Before giving the Lagrangean of the problem, we remark that, in view of the assumption U' (0)

=

00, the rate consumption

is positive along an optimal program, implying from (5) that the rate of extraction is positive as well and that, therefore,constraint (2) will never be binding. Then the Lagrangea~ reads

4

-Let {S(t},C(t},E(t}} be the solution of the problem. Then according to the Pontryagin maximum principle, there exist a constant

A

and Set) 0

-and ~(t), continuous, except possibly at points of discontinuity of

E(t) and C(t) such that:

(7)

-

-dL/dE

=

0 : ~(t}p(t) - Set)

=

A (8)

set}

(E -

E(t» = 0 • (9)

In order to facilitate the analysis we assume that U is of the constant elasticity of marginal utility type:

U(C) = 1 c1+n

1 + n (n < 0 , n -f: -1) •

Then (7) and (8) are reduced to (omitting t where there is no danger of confusion)

-pt-n 1+n _ Q ,

e E p ~ = 1\ (10)

In the sequel the following convention is adopted. For a variable x(t) we define

x(t +) = lim x(t + h) , h+O

x(t-)

=

lim x(t + h) • htO

Next we establish some lemmata. The first asserts that, at T, the optimal rate of extraction is discontinuous if it is interior.

Lemma 1.

Proof

Suppose the contrary. Then

B

is continuous at T and equals O. Since p jumps at T and 1 + n

#

0, this contradicts (10).

The following lemma says that, in order to have an optimum, the rate of time preference should be large enough. We define:

i = 1,2 (11)

Lemma 2.

If w

2 ~ 0, then no optimal program exists. Proof

Suppose the contrary. It is convenient to distinguish between the cases w

2 > 0 and w2 = O. Consider the former case first. Suppose that, during some interval following T,

E

=

E.

Along this interval

S

is continuous and it follows from (10) that

B

is increasing. The phase with

E

=

E

cannot last forever in view of the limited availability of the re-source, hence, at some point of time, a transition to interior

exploi

-tat ion must be made. At that point of time,

B

jumps downwards. But then E jumps upwards in view of (10), which is ruled out by (3). There-fore for all t > T the rate of exploitation is interior. But then

E/E = w

2 > 0, contradicting (2).

The same argument can be used to show that, if w

2 = 0, the rate of extraction is constant for t > T. But there does not exist a constant extraction rate, not exhausting the resource within finite time. This proves the lemma.

o

6

-Henceforth it will be assumed that w

2 < O. Then the optimal program for t > T is easily characterized. Consider the equation

-E/W₂

=

S(T) -

E(T -

T) , T

-

~ T. (12)

-The right hand side of (12) is the amount of the resource left at T, if from T on the resource is exploited at a maximal level. The left hand

-

-side of (12) is the reserve of the resource needed when from Ton, E

-is interior and E(T)

=

E. This can be verified as follows. Differentiation

-

-of (10) with respect to time and keeping in mind that

a

=

0 for t ~ T

yields

E(t) for t ~ T

The left hand side of (12) then equals

00

f

-w₂ (t - T)

_ Ee dt •

T

Now consider figure 1

-E/w 2 T* -T fig. 1.

-T-T

-

*

If SeT) ~ -E/w

2, (12) has a solution for T, denoted by T • The optimal exploitation program is then to exploit at a maximal level for

*

*

T ~ t ~ T and to have 0 < E(t) < E for t > T • If SeT) < -E/W

2, then it is optimal to have interior exploitation for all t > T. The

optimality of these policies is easily seen in view of the concavity of U. This proves our third lemma.

Lemma 3. Suppose w

2 < O. Then, for t > T, the optimal exploitation policy is a~ follows:

-E/W_{2 '}

-

-

-I f SeT) < then

o

< E(t) < E, E(T)

=

-W

2S(T).

*

-

-If SeT) f;; _-E/w

2, then there exists T f;; T such that E(t) == E for

*

*

T ~ t ~ T and 0 < E(t) < E for t f;; T

.

3. The optimal program

In this section we derive the characteristics of an optimal program and perform a comparative dynamic analysis on some of the parameters.

(1) p < y 1 (1 + n) I P > y 2 (1 + 11) •

Since i t has been assumed that y. > 0 and

~ p > 0, 1 + 11 > 0 in this

case. We remark that, before and after T, the rate of exploitation is continuous. As far as the period after T is concerned, this

follows from lemma 3. Let us then consider the period of time before T and suppose that at, say t _i, with 0 < ti < T, the rate of exploitation is discontinuous. It follows immediately that it is impossible to

-

-

-have 0 < E(t

i-) < E and 0 < E(ti+) < E, since S(ti) is 0, contra-dicting, from (10},the discontinuity of E. Furthermore, an upward

jump of

E

to

E

is ruled out, because that would require 8 to jump

8

-to a negative value. A downward jump of E (from E) requires

S

to jump to a positive value, contradicting

S(E -

E)

= O. We conclude that along the optimal program the only discontinuity possibly occurs at T.

Let us suppose next that the optimal rate of exploitation is interior. throughout. It then follows from (10) that

1 - (l+n) B(t) = A Tl

Po

11

a

~ t ~ T , Since 1 - (1+Tl) t w 2 E(t}

i

Tlp(T+) n e e

f

E(t)dt = So '

o

we find upon substitution:

1 - (l+n)

• 11 n So =

A

Po

n

_{t > T .}

The solution proposed is indeed optimal for So not too large. To see this, remark that w

1 > 0 and w2 < 0 and that a "small" So therefore implies a "large"

A,

which on its turn gives "small" rates of

extraction from (10), ceteris paribus of course. Hence, the optimal program can be depicted as in figure 2 below.

E

- - - r - - - -

E fig. 2

I

I I T t fig. 3

The upward discontinuity in E is caused by the price jump.

What happens, then, if So is increased? A, the shadow price of the resource, decreases and, for So sufficiently large, the solution proposed is not optimal anymore. Then the optimal program becomes as in figure 3. Since E(T₂) =

E

we have

(14)

and A and T2 can be solved from (13) and (14).

Enlarging So further, could have the effect of just increasing the

A

interval [T,T

2

J.

This is not the case. Increasing T2 lowers A (from (14», which has its impact on the rate of exploitation before T. Hence the following two figures depict optimal exploitation schedules for "large" and "very large" So respectively.

E E T1 fig. 4 - 10 -T t E

-E fig. 5

We can, therefore, state the following theorem.

Theorem 1.

Suppose p < Y1 (1 + n), P > y

2(1 + n). Then the optimal exploitation program is characterized as follows.

There exist Tl and T2 with 0 ~ Tl ~ T and T ~ T2 such that

-

_~

_E

:.

a) for O~t~Tl' 0 < E(t) E(t) > 0

,

b) _{for Tl} ~ t ~ T2

,

E(t} E

-c) for t > T

2, o < E (t) < E

.

If T1

=

T2

=

T, then E(t} shows an upward jump at T. The smaller is SO' the larger is Tl and the smaller is T

2•

The above analysis has been carried out by varying SO' When So is kept fixed, we can do another interesting excercise by varying the price jump at T. A look at equation (13) reveals that, for given SO' it

is satisfied for some p(T+). Decreasing p(T+) implies decreasing A which yields larger E for the period preceding T. Then, after T, the rate of exploitation should be smaller on average.

(2) (1 + T) >

a .

In this case, obviously, there occurs an upward jump in E, if E is interior around T. In the previous case it was clear that there would be no maximal exploitation before T, unless there was maximal

ex-ploitation after T. This result does not hold true here. What actually will happen depends on the rate of time preference and the price jump. Consider the following equation

(15)

Suppose the rate of extraction is interior throughout the planning period. Then if

1 + T) 1 + n

(e T) P (T+) Tj

is smaller than unity (occuring for example when p{T+) is "large" and/

-

-or p is "small"), then E(O} is smaller than E(T}. In that case,

in

-creasing So will yield maximal E after T, before the initial E is maximal. We therefore state

Theorem 2.

Suppose p > Y1 (1 + Tj), P > Y2(1 + Tj), (1 +n ) > O. Then the optimal exploitation program is characterized as follows.

12

-a) for

a ;;;

t ;;; T1 E(t) = E

-b) _{for T1 ;;; t} ;;a T

,

0 < E (t) < E

,

c) for T ;;; _{t ;;a T2 E(t) E}

-d) for t > T2

,

a

< E(t) < E

.

-If T1 = T2

=

T, then Eet) shows an upward jump at T. The smaller is SO' the smaller is T1 and the smaller is T2 • If

-pT 1 +

n

1 +

n

e

n

_p(T+) n _fpO n

is l~rger than unity, then T1 >

a

implies T2 > T. o

This case cannot occur since p >

a

and Y1 > O.

,

(4) (1 + n) <

o.

If, around T, the rate of extraction is interior, then at T there will

-now occur an upward jump in E, since (1 + n) < O. using the same

arguments as under 1), it is easily seen that the following holds.

Theorem 3. Suppose p > Y

1 (1 + n), P > y2(1 + n), (1 + n) < O. Then the optimal exploitation program is characterized as follows.

a) for 0 ~ t ~ T, E(t)

₌

E

-b) _{for T1} ~ t ~ T

,

0 < E(t) < E A c) for T ~ t ~ T 2, E(t)

=

E d) for T2 ~ t

,

o

< E(t) < E

.

If Tl

=

T2

=

T, then E(t) jumps downwards at T. The smaller is So the smaller is Tl and the smaller is T

2•

4. Conclusions

In this paper we have investigated the impact of an expected future price hike on the optimal exploitation policy for an exhaustible natural resource. It has been found that it is not necessarily true that the price hike is accompagnied by a discontinuity in the supply of the resource good. In fact, when the initial size of the resource is sufficiently large, no discontinuity will occur at all. (Except of course in the economy's rate of consumption.) If there occurs a dis-continuity in the rate of extraction, which is the case for a relatively small initial value of the natural resource, then the direction of

this discontinuity depends on the value of the elasticity of marginal utility (n). If the utility function allows for satiation in

con-sumption (n < -1), then, as a consequence of the price shock, the rate of extraction jumps downwards. Otherwise it jumps upwards.

When we, tentatively, extend the analysis to a more general equilibrium-like approach, then from the point of view of the cartel, a drastic reduction of supply will create a large price shock(assuming a smooth demand schedule) if the fringe's initial reserves are large and (or if

- 14

-its maximal capacity output (E) is small.

On the other hand, if the price shock induces an upward discontinuity of fringe's planned supply, then the existence of the fringe mili-gates the effect of the cartel's action.

References

[1] Newbery, ,D. (1981): "Oil prices, cartels and the problem of dynamic inconsistency", Economic Journal, vol. 91, pp. 617-646. [2] Ulph, A and Folie, G. (1980): "Exhaustible resources and cartels:

an intertemporal Nash-Cournot model", Canadian Journal of Economics, vol. 13, pp. 645-658.