There have been several formulations of models for crude oil production which tried to identify the elements of user cost and show their effect on production and investment decisions. In this chapter, previous results are extended by incorporating the uncertainty regarding the date of arrival of the backstop technology in the model. This uncertainty adds a new element to the user cost identified previously and is shown to affect the production and investment decisions

Al-Adhadh, Naim H.

Caltech Authors - Main

DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES 
CALIFORNIA INSTITUTE OF TECHNOLOGY 
PASADENA, CALIFORNIA 91125 
THE EFFECT OF A RANDOM PLANNING HORIZON ON PRODUCTION AND INVESTMENT 
FOR PETROLEUM RESERVOIR -- A NOTE ON KULLER'S AND CUMMING'S MODEL
Naim H. Al-Adhadh 
""'s1\lUTE OF \'\"' ,.� �'to � � � 
§ 
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� ltj 
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'Pt,-/; 4..� 
Jt SHALL t.\�"'"� 
SOCIAL SCIENCE WORKING PAPER 221  
June 1918 
ABSTRACT 
There have been several formulations of models for crude oil 
production which tried to identify the elements of user cost and show 
their effect on production and investment decisions. In this chapter, 
previous results are extended by incorporating the uncertainty regard-
ing the date of arrival of the backstop technology in the model. This 
uncertainty adds a new element to the user cost identified previously 
and is shown to affect the production and investment decisions. 
THE EFFECT OF A RANDOM PLANNING HORIZON ON PRODUCTION AND �NV�STMENT 
FOR PETROLEUM RESERVOIR -- A NOTE ON KULLER'S AND CUMMING'S I MODEL 
Introduction 
2 
Since the classical paper by Davidson [lJ, there haje 
role of user's costs in oil produ 
een many 
models which illustrate the 
However, a paper by Kuller and Cunnning offers the most compre 
treatment of user costs by introducing the following assum-
ptions: 
1. Total recovery, as well as annual production rates from 
natural drive, depends not only on cumulative proquction, 
but also on the rate at which production has tare� place. 2. The recoverable stock, as well as the productior �ate, 
·depends on the time path of investment as well as Ion 
c=ulativo inve•tment (i.e., the oapital otuck)
In their model, n firms are exploiting a given petrrleum 
reservoir under centralized management which maximizes the exoected . I profit function,Il,over a known planning horizon T, subject lo Icon-
:::::::: ::::e::::gc::: :::::n::: a::��t:::: :::t::n::::::r:�::�r They 
costs, user costs of capital consumption, and production uslrlcosts. 
Their policy prescriptions are simple: 1) produce at-a rabelwhich 
equates marginal net income to firm J and the user cost association 
with firm J's production, and 2) equate the marginal cost of inv�stment 
(to firm J for capital-type k) with the'marginal present value of the 
reservoir�wide benefits associated with such investment. The latter 
includes not only direct impacts on the marginal productivity of.J's 
capital.and J's future variable and boundary costs, but also external 
impacts on other firms' variable and boundary costs as well as on the 
recoverable stock [2]. 
This note will extend the results of Kuller and Cumming by 
introducing an additional source of randomness in the planning model, 
that which pertains to' the planning period. 
The Effect of Random Planning Horizon 
One element in the decision matrix of the oil producer is 
uncertainty about the arrival date, T*, of the "backstop" technology 
that will replace hydrocarbon fuels as the principal source of energy. 
This uncertainty introduces another element into user cost and 
modifies the production decision of the producer. Assume that the 
·central management of a field believes that T* is randomly distributed 
on the range [O,T] .. To facilitate comparison of these results with 
those obtained by Kuller and Cumming, assume further that their T 
corresponds to the expected value of T* in this framework. 
Let Kt 
be the capital stock at period t; Rt= Cr1, 
r
2, • • • , 
r
t
) 
is the history of production; V = (v
1
, v , 
•
• •  , v )is the history of 
t 2 t 
investment; C
t 
is the generalized cost function of period t. Then, let 
3 
c 
t ct (
R
t
, vt
,Kt) 
ac ac t > 0 t > 0 rr- ·av'"' • 
ac t < 
-il -
-
act < -�
- 0 0, ilK 'L t v, t 
'[ "' t 
r = the volume of petroleum extracted by firm J, J = 
Jt 
R 
t 
VJkt 
Vt 
KJkt 
KJt 
DJkt 
x 
FJt 
CJt 
during period t 
annual production rate by all firms during all 
i. e. , R 
t (
r , r , • • .  , r , 
• • •  , r , 
11 21 .ill ln-1 
rlt' • • •  , rnt) 
gross investment by firm J in capital component k, 
k = 1, . • .  , q, during period t 
gross investment for all capital components by all 
during the periods 1, . • •  , t 
firms J's stock of capital components k at the 
of period t 
(KJlt' • • •  , KJqt) 
net depreciation of firm J's stock of capital compqnelnt 
during period t 
the recoverable stock 
an upper (physical) bound on firm J's capacity to 
petroleum during period t 
firm J's cost function during period t 
4 
k 
1\ a discount factor, (1 + r)-
t where r is the appropriate 
discount rate 
pt 
unit price of petroleum during period t 
wherel 
an an an Jkt < 0 _lli > 0 _lli > 0 avJkt - ' arJT 
- ' aKJkt 
-
aFJt aF < 0 _:I!_ ariT - , a viT 
aF 
> 0 Jt > 0 - · � -Jt 
ax < 0 � > 0 ariT - ' avi,-
T = l, • • •  , t ; 
i, J 
• 1, 2, • . .  , n; 
k = 1, 
•
•
•
 ' q 
1:=t:='r. 
Chance Constrained Formulation 
The problem will be formulated as a chance constrained opti-
mizing decision [4]. In particular, the constraint relating to the 
total recoverable stock becomes of the form 
Probability � x(�*' 
T n 
VT*) - I: I: T=l .J=l 
r > JT - 0 f 1 • 
1This "all eir nothing" situation for the lifetime of the oil 
industry is unrealistic, since it is known that oil will command a 
positive price long after the emergence of the backstop technology. 
5 
And the problem is then: 
T* n 
Max E { I: I: [PtrJt - CJt(Rt,Vt,KJt)]St} t=l ;J=l 
subject to T* n 
P {- I: I: rJT + x(�*'VT*) � O} = 1 T=l J=l 
KJk,t+l = KJkt - DJkt(
rJt' vJkt'KJkt) 
rJt � FJt(Rt,vt,KJt) 
r Jt 2 0, v Jkt � 0 V 
J, k and t, I 0 t < T*. 
* 
Let T obey a probability mass functiol\ Yt•defined on fOJT
l such 
that 
and 
T 
I: 
t=O 
yt > 0 for 0 $ t $ T, yt 0 t ¢ [O, T] 
yt = 1.
Define the probability that the "backstop" technology 
emerge in the period 0 to t by $t' i.e., the 
is in the range t to T is 
T 
$ = I: yt t T=t 
Let R be the production plan for the entire period 0 tolT 
T 
6 
Thus, 
T t n 
E {n(R)} = E Yt E E [PTrJT - CJT(RT,VT,KJT)] ST tocl T=l J=l 
or, changing the order of summation: 
Let 
n T 
E {n(R)} = E E $t St [PtrJt - CJt (R ,Vt,KJt)]. J=l t=l t 
* 
n T * 
S(T )= .x(R *'V *)- E E rJt T T J=l t=l 
Then the problem becomes: 
n 'f 
Max E E � S [P rJt - CJ (R ,V ,KJ )]J=l t=l t t t t t t t 
p(S(Th� 0) = 1 
KJk,t+l = KJkt - DJkt (+Jt' vJkt'KJkt) 
rJt.:::. FJt (Rt,vt,KJt) 
r > O, v k > 0 VJ, k and t . Jt - J t -
* 
But p(S(T ) � O) = 1, under the assumption that yt > 0 for 
0 < t < T,is equivalent [3] (up to a set of yt-measure zero) to 
S(t) 2'._ 0 for all t. Thus, the Lan grangian for the problem is: 
n 'f 
L = E E $tSt [PtrJt - cJt (Rt,Vt,KJt)]J=l t=l 
T n q 
- I: E l I: LIJk t+l St+l {KJk t+l - KJkt + DJkt (r Jt' vJkt'KJkt)} t=l J=l k=l ' • 
7 
- ipJtst {rJt - FJt (Rt,vt
,KJt)} 
- (AS) t 
t n 
{ I: I: rJT - x(RT,VT)} 
T=l J=l 
n q 
+ l;Jt st UJt + I: I: 0JktStvJkt l. J=l k=l 
Characteristics of Optimum 
Production Rates 
or, 
_£_J.: 
arJt 
From the Langrangian expression: 
T n ilc 
(Pt$tSt - I: I: __!! ST $t) T=t i=l ilr Jt 
q 
- I: LI S 
k=l 
Jk, t+l t+l 
ilDJkt 
ilrJt 
T n 
s + I: I: - ipJt t 
�l i=l 
T (l - E (),f3) (1 - _x_) + l; S = 0 
t=l t a 
Jt t rJt 
(le 
(P - _.:!!. ) � 13 t a t t 
ax 13 (Al3)T (l - --) + ipJt t arJt rJt 
T n F q 
+ L l: ljliT !_ElT=t i=l rJt 
13T + I: LIJk t+ll3 +lk=l , t 
T (lcJt + E 
T=t+l arJt 
T-1 
T n (le 
S $+ I: r-1..113 T i T =t i=l ilr T 
i-/J 
Jt 
�T 
+ E (Al3) (1 - � ) 
T=l 
T ilr 
Vf. J.. = 11 • • •  , n; 
iT i < t < T 
8 
Elements of User Costs 
As in Kuller and Cumming, the following user cost components 
can be identified. 
Stock User Costs for Firm J 
X measures the increase in net incomes from the reservoir 
associated with an incremental change in the endogenously determined 
stock; the stock user cost for firm J in period t is given by 
�). x13r {l - arJt 
Bounda+}' User Costs 
9 
Since �iT measures_ 
the increase in net incomes which would 
result from �n increinental relaxation of the restriction, the boundary 
user cost is given by 
T n 1 aF 
I � 13 - E E �  __!!.13 Jt t T=t i=l iT arJt T • 
User Costs of Capital Consumption 
The multiplier AJk 1 associated with the capital equation ,t+ 
measures the marginal productivity of capital type k used by firm J in 
all future periods t+l, t+2, • • • , T; the user costs of capital consum-
ption is given by 
q 
E 6 13 an 
k=l 
Jk, t+l t+l __J}:
t 
arJt-
Production User Costs 
These user costs reflect the stock value of oil and gas to the 
firm, contributing to output as natural forces of productiqn,I and are 
given by 
T acii T n aci, E ar:-13 + E E -- 13 
T=t+l rJt T
 T=t i=l arJt T 
i,J=l, • • ;,n; 1 5 t � T 
However, a new user cost element is now introduced by the ranldomness 
of the planning horizon. This element will be termed "thelbqundary-
time cost." It is equal to 
Y-1 ax 
E {X13,){l - --ar--> 
T=l iT 
The Effect of the Optimal 
Production Rates 
Comparing these first order conditions with those r· flKuller and 
Cumming, the following can be noticed: 
1) The net marginal benefit of producing one extr[ unit is 
decreased by a factor �t (<l). This decrease causes the net 
marginal benefit curve-to-�hift downward. 
2) The effect of time-horizon uncertainty on marg�nal cost is 
 indeterminate, and depends on the relative maghltudes:of 
 
changes of opposite directions in the .terms ofl t:lhe first· 
order conditions· equation.· ·In comparison with! the corres-
ponding terms in Kuller and Cwmning, the term 
T n 
. E E ip 
T=l i=l i
T I 
aFirl 13t arJt 
is greater, because of the additional uncerta:llnt, •• 
10 
ax The terms (Al3)-T (1 - -" ) are smaller and the terms arJt 
T (le T 
E _-1'.E. R � and E 
T=t+l (lrJt 
T T c=t 
n oc., 
E --1...- 13 � 
i=lor3
t T T 
i1'J 
may increase or decrease 
depending on whether the extra terms-in the summation which correspond 
to T = T*, T* + 1, . . •  T balance the reduction in each term of the 
summation caused by the weighting factor �,. 
T-1 ox On the whole, if E (Al3) (1 - -"�-),the boundary time user cost, 
c=l T ari 
is sufficiently large, then the marginal user cost increases in 
comparison with that obtained from Kuller's and Cumming's formulation. 
This means, ·that a reduction in marginal benefit causes a reduction in 
production rate. In other cases, the effect on the production rate 
is ambiguous, since it depends on the shape and relative shifts in 
the marginal cost and marginal benefit curves.2 
Marginal Cost 
or 
Marginal 
Benefit 
MC 
r2 �rl 
Figure 1 
MB 
Production Rate 
r 
11 
2.r.n comparing the effect of .the introduction of the boundary 
time user cost on the production decision with that obtained from Kul­
ler 1 s and Cumming's formulation, it is here assumed that their T 
corresponds to the expected value of T* in this formulation. Thus,T > T. 
Characteristics of Optimum 
Investment Rates 
From the Langrangian expression: 
oD oCJt 
avJkt 
13 � = - A 13 
Jkt 
t t Jk, t +l t + l  av;rkt 
ox + Al3T OVJkt 
T n oF n ClC + l l $iT 13, ___!!_ - l __if. �tl3t av av c"' t i = 1 .:fkt i=l Jkt 
T 
I 
T=t+l 
n oCi I _T 
i=l avJkt
i1'J 
T-1 
�. 13, + I 
c=l 
i,;J = 1,- - - n; 
k = 1,- - - q 
1 < t < T. 
ox 
Al3T Clv Jkt
These first order conditions state that the optimal level of 
12 
irm ,J's 
- I investment in capital-type k during any t, 1 < t < T is give� by - -
I equating the present value of the marginal costs of such inv 
adjust for the uncertainty of the planning horizon, 
benefits of the reservoir associated with such investment. 
tion of the terms in the above expression follows closely t 
11 I by Ku er and Cummings's [2]. Comparing with their results 
I that the discounted marginal cost of the investment is reduc 
I factor of � < 1 and that the aggregate benefit to the reserv 
he interpreta-
:ote 
d by a 
ir as 
a whole has a new term as a result of the inclusion of uncertainty 
in the planning horizon. However, even if D, F, x and C are the same 
functions as those considered by Kuller and Cunnnings, the effect 
on the aggregate benefit of the reservoir is ambiguous. Only, if 
T-1 
l 
T"'l 
ax Al'.\ 
avJkt
is large enough to swamp all the changes in the 
other terms on the right hand side of the first order conditions that 
the aggregate benefit increases at all levels of investments for 
all capital components. In ·this case the optimal investment level 
increases unambiguously. 
See Figure 2. 
.13 
Discounted Marginal Cost 
This note captures the effect of only one aspect 
uncertainty, that which is related to the time of the eme�g 
f 
nee 
of the backstop technology. Other sources of uncertaintyl rtain 
unexamined, such as uncertainty related to the price pat� a d 
particularly the uncertainty regarding the prevailing pr�cel of 
the emerging alternatives. Moreover, a more realistic tjeatment 
should deal with the situation where: 
a) the oil commands a positive price after the em�rgence 
of the backstop technology; 
b) the strategic aspects provide the oil producers 
I strategy of delaying the emergence of the altern 
technologies. 
The preceding analysis demonstEates that the theo� o 
oil production is affected by incorporating the type of uncer 
ith a 
tive 
crude 
ainty 
Marginal 
' ' ' '- / considered in this note. 
Values 
1 
Aggregate benefit 
vl v2 Investment rate 
Figure 2 
14 
REFERENCES 
1. Davidson, P. "Public Policy Problems of the Domestic Crude Oil 
Industry" American Economic Review 63 (1963): 85-108. 
15 
2. Kuller and Cumming. "An Economic Model of Production and Invest­
ment for Petroleum Reservoip" American Economic Review 64 
(March 1974): 66. 
3. Yaari, M. "Uncertain Lifetime, Life Insurance, and the Theory 
of the Consumer" Review of Economic Studies (1965): 138-150. 
4. Charnes, A. and W. Cooper. "Chance Constrained Programming" 
Management Sciences 6 (1959). 


The Effect of a Random Planning Horizon on Production and Investment for Petroleum Reservoir -- A Note on Kuller's and Cumming's Model

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The Effect of a Random Planning Horizon on Production and Investment for Petroleum Reservoir -- A Note on Kuller's and Cumming's Model

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