Adjustment Cost, Uncertainty, and the Proved Reserves of Crude Oil John Boyce * Xiaoli Zheng † Abstract This paper discusses an important issue for the oil industry: why do firms hold large amounts of proved reserves relative to crude oil productions? To answer this questions, we first clarify the definition of proved reserves and then build a simple deter- ministic and a stochastic model. In the deterministic model, we find a delayed response is necessary for the firm to hold proved reserves while adjustment costs govern the periods of delay. In the stochastic model, this result still holds. We find uncer- tainties from demand and exploration lower the firm’s optimal productions and thus resources would be exhausted within a longer period. The stochastic model predicts that firms would hold larger proved reserves than productions either if the de- mand volatility is much smaller than the interest rate or if the exploration volatility is not noticeably larger than the inter- est rate. By using U.S. data, we verify that this prediction is empirically relevant. The existence of these volatilities also ex- plains why the ratio of proved reserves to productions declines slowly or even remains stable. keywords: Adjustment cost; delayed response; uncertainty; proved reserves. JEL classications: Q410; G170; D210. * Department of Economics, University of Calgary, 2500 University Dr. NW Calgary, AB, T2N 1N4 Canada. Email: [email protected]† Department of Economics, University of Calgary, 2500 University Dr. NW Calgary, AB, T2N 1N4 Canada. Corresponding author email: [email protected]
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Adjustment Cost, Uncertainty, and the Proved
Reserves of Crude Oil
John Boyce∗ Xiaoli Zheng †
Abstract
This paper discusses an important issue for the oil industry:
why do firms hold large amounts of proved reserves relative to
crude oil productions? To answer this questions, we first clarify
the definition of proved reserves and then build a simple deter-
ministic and a stochastic model. In the deterministic model, we
find a delayed response is necessary for the firm to hold proved
reserves while adjustment costs govern the periods of delay. In
the stochastic model, this result still holds. We find uncer-
tainties from demand and exploration lower the firm’s optimal
productions and thus resources would be exhausted within a
longer period. The stochastic model predicts that firms would
hold larger proved reserves than productions either if the de-
mand volatility is much smaller than the interest rate or if the
exploration volatility is not noticeably larger than the inter-
est rate. By using U.S. data, we verify that this prediction is
empirically relevant. The existence of these volatilities also ex-
plains why the ratio of proved reserves to productions declines
∗Department of Economics, University of Calgary, 2500 University Dr. NWCalgary, AB, T2N 1N4 Canada. Email: [email protected]†Department of Economics, University of Calgary, 2500 University Dr. NW
U.S. and Canada’s proved reserves of crude oil share the common
trend with crude oil production for the past century. The ratio of
proved reserves to production, which measures how many years at
current production before current proved reserves are exhausted, is
around 8-10 years in U.S and 8-15 years in Canada.1 Moreover,
these numbers have declined slowly since the 1940’s in U.S. and the
1970’s in Canada (Figure 1). They also prevail on firm-level data.
Figure 2 below is a histogram that illustrates the the the ratio of
proved reserves to productions for 20 listed U.S. and Canada’s firms
from 1999 to 2013. More than 55 percent of the observations are
centered in the interval of 9-16 years, implying these firms hold 9-
16 times more proved reserves than current production. This leads
to the research questions on this paper; What are the forces that
drive a firm’s decisions on proved reserve accumulation; Why do
they accumulate large amounts of proved reserves? This paper is
interested in examining how economics factors, such as crude oil
prices, price volatility and exploration costs, affect a firm’s optimal
proved reserve accumulation.
1This ratio is very large compared to manufacturing sectors, where the in-ventory ratio ranged from 1.21 to 1.39 from 2000 to 2010 according to the U.S.Census Bureau.
1
Figure 1: The Ratio of Crude Oil Reserve to Production in U.S. and Canada(Source: Energy Information Administration and Canadian Association of
Petroleum Producers)
Figure 2: The Ratio of Crude Oil Reserve to Production in U.S. and CanadianFirms (Source: COMPUSTAT)
In order to answer these questions, we properly define what crude
oil reserves are and how they can be measured. Starting from Hotelling’s
2
seminal paper [7] on exhaustible resource extraction, studies on ex-
haustible resource focus on the time path of production and prices.
Many theoretical studies (e.g., [24] and [25]) confirm Hotelling’s pre-
diction that production monotonically decreases and price increases
over time. On the other hand, some studies (e.g., [22] and [6]) find
that prices should follow a U-Shaped path. In all these models, nev-
ertheless, reserves are treated as either a known fixed stock which de-
pletes over time, or in models of exploration, as the outcome of sub-
tracting cumulative production from cumulative discoveries. How-
ever, proved reserves are more than geological definitions. Proved
reserves are defined by the Energy Information Administration and
Natural Resource Canada as “recoverable crude oil with reasonable
certainty under existing economic and technological conditions”[3].
Under this definition, reserves are “proved” so that they may become
crude oil production in the near future. As such, proved reserves are
surely affected by economic factors such as oil prices, price volatil-
ity and discovery costs.2 Therefore, it is even bizarre why firms
hold such a great amount of reserve in proved status. Early studies
(e.g., [21] and [5]) explain this fact by specifying production costs as
reserve-dependent; that is, a firm accumulates reserves just because
production costs are lower if the reserve is larger. Although this may
be true for a single oil field, it is hard to believe finding a new field
lowers the costs of other existing fields [11].
This paper offers a new answer using the concept of adjustment
cost and delayed response in a simple determinist model, in which
a representative firm has constant marginal costs of production and
discovery. Therefore, unlike previous studies, we do not rely upon
reserve-dependent costs. Due to the delay between resource discov-
ery and production, a firm makes decisions on how fast the resources
are transferred to the proved and then production stage when it finds
a new discovery from the oil-in-place stock. We assume there are ad-
justment costs associated with a fact that moving discoveries quickly
to the proved and then production stage is more costly than doing
2Some empirical studies (e.g., [4] and [15]) verify crude oil prices have signifi-cant impacts on oil explorations and reserve accumulation in some countries.
3
it slowly [19]. We show that adjustment cost and delayed response
in transit are necessary for the firm to hold a positive amount of
proved reserves. However, the deterministic model solely relies on
the periods of delay to explain the large amounts of proved reserves
relative to production. Moreover, it fails to account for why the
ratio of reserves to productions declines slowly or even remains sta-
ble. This leads us to consider a second model attempting to explain
how demand and exploration uncertainties affects the the crude oil
production and the accumulation of proved reserves. This model fol-
lows Pindyck’s [18] and Mason’s ([12], [13]) models that considers
stochastic models of exhaustible resource extractions. Although the
stochastic model follows Pindyck and Mason’s framework, it differs
in two aspects: the stochastic model is based on the specification of
deterministic model, thus the feature of delayed responses and ad-
justment costs is still kept; Additionally, the stochastic model explic-
itly solves the analytical solutions of expected productions, proved
reserves and the ratio of proved reserves to productions. The re-
sults of this stochastic model suggest that a firm would hold larger
proved reserves than current production if demand volatility is much
less than interest rate, or if the exploration volatility is not notice-
ably larger than the interest rate. The existence of these volatilities
also explains why the ratio of proved reserves to productions declines
slowly or even remains stable.
The paper is organized as follows: Section 2 presents the deter-
ministic model and section 3 describes the stochastic model, with
Section 4 offering the concluding remarks.
2 The Deterministic Model
We consider an environment in which a representative firm explores
for oil resources underground, and then make discoveries available
for production. There are two key features in the model. The first
is that there are three types of stocks: Su(t), which are total oil
in place; S1(t), which are resources that have been discovered and
proved, but are not yet ready to be produced; and S2(t) which are
4
proved reserves that ready to be produced. These three stocks rep-
resent three different stages of oil production, where Su denotes the
most undeveloped stage and S2 is the most developed stage. There
are three stocks because this is the minimum number necessary to
illustrate the importance of delayed responses and adjustment costs.
Adjustment costs enter the model because it costs a greater amount
to transfer resources directly from the most undeveloped stage to
the production stage than it does a two-step route where resource
are first turned from most undeveloped stage to the intermediate
stage, and then producible stocks. Thus, production only can occur
from stock S2.
The second feature of the model is that there exists a delay in
the time for resources transferred from previous stock to the current
stock. For simplicity, this delay is assumed to be identical, l units of
time periods, for all transfers.
In sum, to produce crude oil, the representative firm has two
routes by which it make discoveries for production as illustrated by
Figure 3.
Figure 3: Resource Discovery, Transfer, and Crude Oil Production
In one route, the firm could find and then transfer the resource
from the total oil-in-place stock, Su, directly to the stock ready for
production, S2. Alternatively, it could reach production stage by
two sequential steps: step one involves discovering and transferring
the resource from Su to an intermediate stock S1, and in the next
5
step transferring it to the S2. To sum up, the law of motion of crude
oil discoveries, transfers, and production can be summarized as the
differential equations as below,
dSudt
= −yu1(t)− yu2(t), (2.1)
dS1dt
= yu1(t− l)− y12(t), (2.2)
dS2dt
= yu2(t− l) + y12(t− l)− q(t). (2.3)
where yu1(t) is the quantity of reserves transferred from the total
oil-in-place stock Su(t) to intermediate stock S1(t+ l) at period t+ l;
y12(t) is the quantity transferred from stock S1(t) to S2(t + l) at
period t + l. So both yu1 and y12 denote the two-step transfers,
while yu2 is the quantity directly transferred from the stock Su(t) to
S2(t+ l), with l-period delay; so it denotes as one-step transfer.
Since the representative firm has two routes to reach production,
it faces different costs for each route. We assume the two-step trans-
fers, yu1 and y12, have a constant marginal cost, c1, while the one-
step transfer, yu2, has a constant marginal cost, c2. The marginal
and average cost of production from stock S2 is c0. The concept
of adjustment cost implies c2 > 2c1. Intuitively, this requires that
one-step route is much more costly than two-step route. This is be-
cause it involves more efforts to immediately reach production than
to transfer resources available for production step-by-step.3. Below
we derive the exact relationship on costs where the two-step route
dominates the one-step route, or vice versa
The deterministic model is an optimal control problem for a rep-
resentative and competitive firm as follows,
J = maxq,yu1,yu2,y12
∫ T
tπ(τ)dτ, (2.4)
where J is an objective function at present time. The profit func-
3This extra effort involves hiring more geologists, building drilling facilitiesand pipelines faster, extending average working hours, and compensating moredepreciation in drilling facilities.
6
tion π is defined as π = e−rtp(t)q(t)− c0q(t)− c1[yu1(t) + y12(t)]−c2yu2(t), where p(t) is the crude oil price at period t, and q(t) is the
production of crude oil at period t.
The behavioral assumption is that the firm’s maximizing equation
(2.4) subjected to equation (2.1)-(2.3), S1 > 0 and S2 > 0. The last
two constraints ensure that S1 and S2 are nonnegative over time.
The present-value Hamiltonian is:
H = e−rt[(p(t)q(t)− c0q(t)− c1[yu1(t) + y12(t)]− c2yu2(t)]
Repeating the procedure from (3.14) to (3.17) for Su and S2 sim-
ilarly yields,1
dtEtd[JSu(t)] = 0, (3.18)
1
dtEtd[JS2(t)] = −φ2(t). (3.19)
Similar to the deterministic case, the expected rate of changes in
shadow values of stocks are non-positive over time.
9Ito’s differential generator is analogous to the time derivatives in the deter-ministic case. For its mathematical discussion, see Chow [2].
15
The one-step or two-step route may or may not occur according
to the following proposition.
Proposition 2 Only two-step route occurs in equilibrium if
c2 > (1 + erl)c1, (3.20)
Also only one-step route occurs if
c2 < (1 + erl)c1, (3.21)
However, it is impossible for both one-step and two-step route to
occur simultaneously.
Proof Proof of Proposition 2 is provided in Appendix B.
Thus, even in a stochastic environment, adjustment costs and
delayed responses still determine whether or not the firm chooses
one-step or two-step route as the deterministic case.
Applying Ito’s differential generator to (3.10),
1
dtEtd[e−rt(p(t)− c0)] =
1
dtEtd[JS2(t)]. (3.22)
Expanding the left hand side of (3.32) yields,
1
dtEtd[JS2(t)] = −re−rt[p(t)− c0] + e−rt
1
dtEtd[p(t)]. (3.23)
Solving for 1dtEd[p(t)] as
1
dtEtd[p(t)] = r[p(t)− c0] + ert
1
dtEd[JS2(t)]. (3.24)
Suppose c2 > (1 + erl)c1, thus only two-step route occurs and
(3.11) and (3.12) hold as equalities. Use these condition to rewrite
Js2(t) as,
JS2(t) = e−r(t−l)c1
[1
θ(t− l)+
erl
θ(t− 2l)
]+ JSu(t− 2l). (3.25)
16
Using Ito’s lemma and then Ito’s differential operator on 1θ(t)
yields,
1
dtEtd
[1
θ(t)
]=
σ21θ(t)
. (3.26)
Applying Ito’s differential operator on (3.25) and using (3.26)
yields,
1
dtEtd[JS2(t)] = e−r(t−l)(σ21 − r)c1
[1
θ(t− l)+
erl
θ(t− 2l)
]. (3.27)
Substitute (3.27) into (3.24)
1
dtEtd[p(t)] =r[p(t)− c0] + erl(σ21 − r)c1[
1
θ(t− l)+
erl
θ(t− 2l)
].
(3.28)
Similarly, it can be shown the expected rate of changes in prices
follows the equation below if the one-step route occurs:
1
dtEtd[p(t)] = r[p(t)− c0] + erl(σ21 − r)
c2θ(t− l)
. (3.29)
Appendix C shows the expected rate of changes in production is,
Et[q(t)] = K − ηeσ2xt − βe(σ2
x+σ21)t − Zert. (3.30)
where K, η, Z, and β are positive constants defined as K = p+ σ2xqxr ,
η = rc0r−σ2
x, Z = K − q0 − η − β and
β =
(σ2
1−r)c1[e(r−σ
21)l+e2(r−σ
21)l
]σ2x+σ
21−r
, c2 > (1 + erl)c1
(σ21−r)c2e
(r−σ21)l
σ2x+σ
21−r
. c2 < (1 + erl)c1
(3.31)
(3.30) implies that there are three different forces that affect the
rate of expected productions over time: ert tends to lower the current
17
rate of production since Z > 0, which is consistent with the standard
Hotelling rule that high interest rates r leads to more resources being
left underground rather than turned into production. The other
two forces, eσ2t and e(σ
2x+σ
21)t, which are not taken into account in
the deterministic model, also reduce the current rate of production
given that η > 0 and β > 0. This result is consistent with real option
literature that predicts the higher uncertainty of resource prices yield
a firm’s stronger incentive to delay production([14], [10], [20], [8]).
Beyond this, the exploration uncertainty, σ21, also plays a role in
lowering the current rate of production. In this sense, with demand
and exploration uncertainties, resources must be exhausted within a
longer period of time than the deterministic model.
The proved reserves are defined similarly as the deterministic
case: if only two-step route occurs,
Et[R(t)] =
∫ t+2l
tEt[q(s)]ds
= 2lk − η
(e2σ
2xl − 1
σ2x
)eσ
2xt − β
(e2(σ
2x+σ
21)l − 1
σ2x + σ21
)e(σ
2x+σ
21)t
− Z(e2rl − 1
r
)ert.
(3.32)
On the other hand, when only one-step route occurs,
Et[R(t)] =
∫ t+l
tEt[q(s)]ds
= lk − η
(eσ
2xl − 1
σ2x
)eσ
2xt − β
(e(σ
2x+σ
21)l − 1
σ2x + σ21
)e(σ
2x+σ
21)t
− Z(erl − 1
r
)ert.
(3.33)
When (3.30) is combined with(3.32), the ratio of expected proved
reserve to expected production is greater than 1 if the inequality
below holds.
Ω(σ2x, σ21) < (2l − 1)K − Z
(e2lr − r − 1
r
)ert (3.34)
18
where Ω = η
(eσ
2xl−1−σ2
xσ2x
)eσ
2xt + β
(e(σ
2x+σ
21)l−1−σ2
1−σ2x
σ2x+σ
21
)e(σ
2x+σ
21)t is a
positive and increasing function of σ2x and σ21. The right hand side of
(3.34) is more likely to be positive given that K is large and l > 0.5.
We interpreted condition (3.34) in two separate cases.
In one case, only exploration uncertainty exists such that σ1 > 0
and σx = 0. So condition (3.34) implies,
Ω(σ21) < (2l − 1)K − Z(e2lr − r − 1
r
)ert. (3.35)
We obtain Ω(r) < Ω(σ21) by σ21 > r, so Ω(r) < Ω(σ21) < (2l −1)K − Z
(e2lr−r−1
r
)ert.
This implies r < σ21 < Ω−1[(2l − 1)K − Z
(e2lr−r−1
r
)ert], where
Ω−1 is the inverse function of Ω. This result suggests that in the ab-
sence of demand volatility, the magnitude of exploration volatility is
upper bounded. Therefore, a firm would hold larger proved reserves
than the current productions if the exploration volatility, σ21, is not
noticeably larger than the interest rate.
In the other case, only demand uncertainty exists such that σx >
0 and σ1 = 0. Thus, condition (3.34) reduces to,
Ω(σ2x) < (2l − 1)K − Z(e2lr − r − 1
r
)ert. (3.36)
Since Ω is an increasing function of σ2x, we obtain Ω(σ2x) < Ω(r)
by σ2x < r and it can be shown Ω(r) is less than the right hand
of (3.35). So in this case the condition (3.36) turns into Ω(σ2x) <
Ω(r) < (2l − 1)K − Z(e2lr−r−1
r
)ert, which reduces to σ2x < r <
Ω−1[(2l − 1)K − Z
(e2lr−r−1
r
)ert]. Therefore, in the absence of ex-
ploration uncertainty, a firm would hold larger proved reserves than
the current productions if the demand volatility, σ2x, is much less
than the interest rate.
To summarize, the above reasoning suggests a firm would hold
larger proved reserves than current production if either demand volatil-
ity is much less than interest rate or the exploration volatility is not
noticeably larger than the interest rate. It is interesting to examine
19
this prediction by the data observed from real world. We estimate σ2xby calculating the variance of (lnPt − lnPt−1) in the period of 1977
to 2013, where Pt is the annual WTI price obtained from EIA. The
estimated σ2x is 0.0478. In next step, we estimate σ21 by calculating
the variance of (lnyt − lnyt−1) in the period of 1986 to 2013, where
yt is the discovery of proved reserves at year t obtained from EIA10.
The estimated σ21 is 0.156. Finally, according to a 1995 survey by
the Society of Petroleum Evaluation Engineers (SPEE), the median
nominal discount rate applied by firms to cash flows is 0.125 [10]11.
Thus, r−σ2x = 0.077 and σ21− r = 0.031. The difference between the
interest rate and the demand volatility is two times larger than the
difference between the exploration volatility and the interest rate.
This indicates demand volatility is much less than interest rate and
the exploration volatility is not noticeably larger than the interest
rate. Thus, we verify our prediction from the stochastic model.
Divide (3.32) by (3.30), time differentiate it and rearrange to
yield an equation below,
d(Et[R(t)]Et[q(t)]
)dt
=1
q(t)2
−ηK(e2lσ2x − 2lσ2x − 1)eσ
2xt
−ZK(e2lr − 2lr − 1)ert
−βK(e2l(σ
2x+σ
21) − 2l(σ2x + σ21)− 1
)e(σ
2x+σ
21)t
+ηZ(r − σ2x)(e2lr−1r − e2lσ
2x−1σ2x
)e(σ
2x+r)t
+βησ21
(e2l(σ
2x+σ
21)−1
σ2x+σ
21− e2lσ
2x−1σ2x
)e(2σ
2x+σ
21)t
+βZ(σ2x + σ21 − r)(e2l(σ
2x+σ
21)−1
σ2x+σ
21− e2lr−1
r
)e(σ
2x+σ
21+r)t.
(3.37)
The first three components in (3.37) are negative, while the next
three terms are positive given that e2rv−1v is increasing in v. On one
10EIA reports three sources of discoveries proved reserves: extensions, newreservoir discoveries in old fields and new field discoveries. We aggregate thesethree sources to the discovery of proved reserves.
11A more recent SPEE survey (2008) showed that 71 percent of companiesused a cost-of-capital discount rate in the range 9 percent to 11 percent, with anaverage of 10.4 percent. See Moore [16], page 35
20
hand, the ratio of expected proved reserve to expected productions
is still decreasing over time since K is larger than any other param-
eters; on the other hand, unlike the deterministic model, three pos-
itive components introduced by volatilities in (3.37) provide forces
that lower the rate of decrease in this ratio. Therefore, the ratio
of expected proved reserves to expected production tends to decline
much more slowly or even remains stable over some periods.
4 Conclusion
This paper answers important questions in the oil industry: what are
the forces that drive firms’ decisions on proved reserve accumulations
and why do they accumulate large amounts of proved reserves? We
consider a representative firm that maximizes the lifetime profits by
choosing an optimal plan of oil discoveries and production over time.
We recognize that there are important delays and adjustment costs
in transferring resources from the oil-in-place stock to the production
stage. We found the larger the delayed periods are, the more likely
the firm will reach production quickly. This relationship is still true
in a stochastic model in which we allow the random shocks from
demand and exploration to play a role in determining the resource
transfers. The deterministic model explains why firms hold proved
reserves by the delayed responses and adjustment costs, but does not
account for why the ratio of proved reserves to productions declines
very slowly or even remains stable.
In the stochastic model, we found both demand and exploration
volatilities reduce current rate of production and thus the stock of oil
in place exhausts over a longer period. The results of the stochastic
model suggest that a firm would hold larger proved reserves than
current production if demand volatility is much less than interest
rate, or if the exploration volatility is not noticeably larger than the
interest rate. We examines this prediction using the U.S. data and
find it is consistent with empirical evidences. The intuition behind
the stochastic model suggests that proved reserves refer to expected
future productions within a fixed period of time and they differ from
21
the concept of oil in place. Therefore, although demand and ex-
ploration volatilities lower current rate of production and leave more
resources as oil in place, proved reserves may not necessarily increase.
Proved reserves are proven to be economically and technically recov-
erable under current conditions so that they are ready for expected
productions in the near future. Since uncertainties lower the rate
of current production, the expected productions in the near future
would also be lowered. In this sense, large uncertainties may dis-
courage a firm’s incentive to accumulate proved reserves. Thus, in
order to encourage a firm to hold a greater amount of proved reserves
than the current production, these volatilities cannot be very large.
Specifically, we show that the demand volatility is much smaller than
the interest rate, and exploration volatility is slightly larger than the
interest rate. The existence of these volatilities also yields the ratio
of reserves to production to decline slowly or even remains relatively
stable.
22
References
[1] Boyce, W.E., Diprima, R. “Elementary Differential Equations
and Boundary Value Problems”. John Wiley, U.S.A, 7 edition,
2003.
[2] Chow, G.C. “Optimal Control of Stochastic Differential Equa-
tion Systems”. Journal of Economics Dynamic and Control,
1(2):143–175, 1979.
[3] EIA. “U.S. Crude Oil and Natural Gas Proved Reserves”. Web-