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. NWCalgary, AB, T2N 1N4 Canada. Email: boyce@ucalgary.ca†Department of Economics, University of Calgary, 2500 University Dr. NW

Calgary, AB, T2N 1N4 Canada. Corresponding author email: xzhen@ucalgary.ca

1 Introduction

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)]

− λu(t)[yu1(t) + yu2(t)] + λ1(t)[yu1(t− l)− y12(t)]

+ λ2(t)[yu2(t− l) + y12(t− l)− q(t)] + φ1(t)S1(t) + φ2(t)S2(t).

(2.5)

The first-order conditions are4,

∂H

∂q(t)= e−rt (p(t)− c0)− λ2(t) = 0, (2.6)

∂H

∂yu1(t)= −e−rtc1 − λu(t) + λ1(t+ l) ≤ 0, (2.7)

∂H

∂y12(t)= −e−rtc1 − λ1(t) + λ2(t+ l) ≤ 0, (2.8)

∂H

∂yu2(t)= −e−rtc2 − λu(t) + λ2(t+ l) ≤ 0, (2.9)

λu(t) = 0, (2.10)

λ1(t) = −φ1(t), (2.11)

λ2(t) = −φ2(t). (2.12)

Note that by (2.11) and (2.12) the shadow value of S1 and S2

are weakly decreasing over time, because the Lagrangian multipliers

4Since there are lagged controls in the model, the technique of delayed responseis applied [9].

7

φ1(t) and φ2(t) are non-negative. (2.10) indicates the shadow value

of Su is constant and positive over time. Moreover, (2.6) holds as

equality because crude oil production occurs every period over the

history; while (2.7), (2.8) and (2.9) hold as inequality because either

one-step or two-step route is alternative to each other and thus may

or may not occur. These necessary conditions lead to a proposition

as shown below,

Proposition 1 Only two-step route occurs in equilibrium if

c2 > (1 + erl)c1, (2.13)

but only one-step route occurs if

c2 < (1 + erl)c1, (2.14)

However, one-step and two-step route cannot occur simultaneously .

Proof Proof of Proposition 1 is provided in Appendix A1.

Proposition 1 states the representative firm would follow two-step

route when the adjustment cost for one-step route c2 is sufficiently

large as given by (2.13). However, it does the one-step route when

the adjustment cost for one-step route c2 is less than the cost of two-

step route c1 discounted by time value erl. If l→ 0, (2.13) reduces to

our original concept of adjustment cost: c2 > 2c1. Therefore, even if

there is no delayed response, the firm is still prevented from doing the

one-step route given the adjustment cost is too high. On the other

hand, (2.14) is legitimate only for l > 0. Thus, delayed response

is necessary for the firm to switch to one-step route. In general,

a large lag of delay l causes the firm to choose one-step route and

reach production more quickly. This is because large lag of delay

raises the cost of two-step route, due to firm’s waiting more time for

production. Consequently, it lowers the opportunity cost of one-step

route.

Now let us consider a feasible solution of this model. First we

find that S1(t) = S2(t) = 0 along the equilibrium path 5. Next we

5The proof is provided in Appendix A2

8

consider a simple linear demand function as

p(t) = p− q, (2.15)

where p > 0 denotes as the cost of backstop technology.

Suppose only two-step route occurs. Lag (2.7) by l periods, and

substitute it into (2.8) to eliminate λ1(t). Then (2.8) becomes λ2(t+

l) = e−rtc1 + e−r(t−l)c1 + λu. Lagging this by l periods yields

λ2(t) = c1(1 + erl)e−r(t−l) + λu. (2.16)

Substitute (2.15) and (2.16) into (2.6) and solve for equilibrium

productions q∗(t) as,

q∗(t) = p− c0 − c1(erl + e2rl)− λuert. (2.17)

The time derivative of equilibrium production is,

q∗(t) = −rλuert < 0. (2.18)

Therefore, the equilibrium production decreases over time. In the

terminal period T , oil-in-place in stock Su are exhausted such that

Su(0) =∫ T0 q∗(s)ds. By using (2.17), this yields Su(0) = [p − c0 −

c1(erl + e2rl)]T + λu(1− erT ). On the other hand, the present-value

Hamiltonian H(T ) = 0. This implies λuerT = p− c0 − c1(erl + e2rl).

These two conditions allow us to pin down the value of T and λu.

Since S1(t) = S2(t) = 0, S1(t) = S2(t) = 0. Thus, if condition

(2.13) applies, (2.2) and (2.3) can be written as,

y12(t) = yu1(t− l) (2.19)

y12(t− l) = q(t) (2.20)

From (2.17), (2.19), and (2.20), the equilibrium two-step transfers

can be found as,

9

y∗12(t) = q(t+ l) = p− c0 − c1(erl + e2rl

)− λuer(t+l), (2.21)

and

y∗u1(t) = q(t+ 2l) = p− c0 − c1(erl + e2rl

)− λuer(t+2l). (2.22)

Thus, q∗(t) exactly equals the y∗12(t− l) and y∗u1(t− 2l) and equi-

librium production of crude oil comes from transferring resource from

Su by two steps and it takes 2l periods to complete. Then the proved

reserves are defined as those have been discovered and proved from

the oil-in-place stock, but which are not yet available for production

at time t:

R(t) =

∫ t

t−2lyu1(s)ds =

∫ t+2l

tq(s)ds, (2.23)

(2.23) implies proved reserves at t is cumulative productions be-

tween current period t and the future period t+2l. Substitute (2.17)

into the (2.23), the proved reserve can be found analytically:

R(t) = 2l[p− c0 − c1(erl + e2rl)]− λur

(e2rl − 1)ert (2.24)

Note that the firm will not hold any proved reserve if l = 0. This

is because all discoveries are immediately transferred to production

and there is nothing held as in situ proved reserves. The ratio of

proved reserves to production is then expressed as,

R(t)

q(t)=

2l[p− c0 − c1(erl + e2rl)]− e2rl−1r λue

rt

p− c0 − c1(erl + e2rl)− λuert(2.25)

The time derivative of this ratio is then computed as,

10

dR(t)q(t)

dt=λu[p− c0 − c1(erl + e2rl)](2rl − e2rl + 1)ert

[p− c0 − c1(erl + e2rl)− λuert]2(2.26)

(2.25) indicates proved reserves are larger than production if

(2l − 1)[p − c0 − c1(erl + e2rl)] − λue

rt( e2rl−1r − 1) > 0. This im-

plies that large lag l leads firms to hold more proved reserves than

current production6. This is true because proved reserves are cu-

mulative production between t and t + 2l. Additionally, because

(2rl − e2rl + 1) < 0 in (2.26), the ratio of proved reserves to pro-

duction is monotonically declining over time and it decreases more

rapidly if l is larger.

On the other hand, if the one-step route occurs, by (2.6) and (2.9),

it can be shown the ratio still monotonically decreases over time as

is the case in the two-step route. In sum, the deterministic model

presented above shows why the representative firm holds proved re-

serves: because transferring discoveries to production is delayed by

a time interval, either 2l or l, which is governed by the adjustment

costs. This leads to an amount of in situ proved reserve held by

the firm. Also, it predicts the ratio of proved reserves to production

decreases over time and its rate of declining mainly depend on the

magnitude of delayed periods l. However, this model dose not ex-

plain why this ratio declines so slowly or even remains stable over the

sample period. In that sense, we need account for something missed

in the deterministic model that affects a firm’s decisions on proved

reserve accumulations. This leads to the next section in which we

examine how uncertainties play a role in optimal productions and

proved reserve acclamations.

6The first component, (2l−1)[p−c0−c1(erl+e2rl)], is positive and increasing

in l given l > .5 and r ∈ [0, 1), but the second component, λuert( e

2rl−1r− 1), is

also positive given e2rl−1r− 1 > 0. Given p and l are sufficiently large, the first

component must outweigh the second one.

11

3 The Stochastic Model

The literature widely recognizes uncertainty as an important issue

for natural resource modelling. Pindyck [18] identifies two sources of

uncertainty on exhaustible resource extraction: one is the stochas-

tic demand that results in resource prices oscillate randomly; the

other is the fluctuation in explorations. Following Pindyck’s frame-

work, Mason ([12] and [13]) argues the stochastic demand leads to

stockpiles of short-run inventories of crude oil. Slade [23] shows the

“least-cost-first” principle in resource mining may not hold when the

demand is stochastic. Finally, Kellogg [10] provides empirical evi-

dence that firms fully take into account the expected price volatility

of crude oil when making drilling decisions. Here we consider random

shocks from demand and exploration following Pindyck and Mason’s

specifications.

The model below exhibits all the assumptions in deterministic

model, but it contains the new feature that realized prices depend

upon a stochastic variables x and realized discoveries rely on another

stochastic variable θ. With these stochastic shocks, we may write an

optimal control problem for the representative firm as follows.

J = maxq,yu1,yu2,y12

∫ T

0e−rτp(τ)q(τ)− c0q(τ)− c1[yu1(τ) + y12(τ)]

− c2yu2(τ)dτ(3.1)

subject to,

Su = −θ(t)yu1(t)− θ(t)yu2(t) (3.2)

S1 = θ(t− l)yu1(t− l)− θ(t)y12(t) (3.3)

S2 = θ(t− l)yu2(t− l) + θ(t− l)y12(t− l)− q(t) (3.4)

p = x(t)f(q(t)) (3.5)

12

dx(t)

x(t)= σxdzx(t) (3.6)

dθ(t)

θ(t)= σ1dz1(t) (3.7)

S1, S2 ≥ 0 (3.8)

where x(t) is a random shock that shifts the demand function f [q(t)]

up and down at the period t; θ(t) is a random shock that shifts

resource discoveries at the period t. Note that θ and x are multi-

plied by planned discoveries and demand. Thus, yu1(t), y12(t), and

yu2(t) are planned discoveries at the beginning of each period, while

θ(t)yu1(t), θ(t)y12(t), and θ(t)yu2(t) are realized discoveries at the

end of that period. The differentials of these random shocks are

specified by Geometric Brownian Motions, (3.6) and (3.7), respec-

tively. Note that both θ and x are thus ensured to be positive over

time and Et(θ) = Et(x) = 1 7. Moreover, the initial values of x and

θ equal 1 if the initial values of those drift terms, zx and z1, are zero.

This implies there is no random shock to demand and discovery at

the initial period.

Both demand and exploration shocks have constant volatilities,

σx and σ1, and we assume 8σ2x < r < σ21; zx and z is a Wiener process

which can be described as dz = ε√dt where ε is a serially uncorre-

lated normal random variable with zero mean and unit variance. At

the beginning of each period, the firm chooses the plan of produc-

tion, q, and discoveries, yu1, y12, and yu2, and pays the corresponding

costs, but at the end of each period the realized discoveries deviate

the planned transfers by the factor θ. Thus, we define a positive

shock if θ > 1, that is, the realized discoveries are greater than the

planned ones; but it is a negative shock if θ < 1. x behaves simi-

larly with θ. For simplicity, we assume θ and x are independent of

each other. The remaining setup keeps identical to the deterministic

model in Section 2.

7See Øksendal [17], page 64-65.8We later justify this assumption. See page 20-21.

13

The fundamental equation of stochastic optimality in this prob-

lem is,

0 = maxq(t),yu1(t),yu2(t),y12(t)

e−rtp(t)q(t)− c0q(t)− c1[yu1(t) + y12(t)]

− c2yu2(t)+ Jt(x)− JSu(t)θ(t)[yu1(t) + yu2(t)]

+ JS1(t)[θ(t− l)yu1(t− l)− θ(t)y12(t)]

+ JS2(t)[θ(t− l)yu2(t− l)) + θ(t− l)y12(t− l)− q(t)]

+σ212θ(t)2Jθθ(t) +

σ2x2x(t)2Jxx(t) + φ1(t)S1(t) + φ2(t)S2(t),

(3.9)

where JK is the derivative of J with respect to K where K =

t, Su, S1, S2, x, θ, and JKK is its second derivative.

The necessary conditions are,

∂J

∂q= e−rt(p(t)− c0)− JS2(t) = 0, (3.10)

∂J

∂yu1= −e−rtc1 − θ(t)JSu(t) + θ(t)JS1(t+ l) ≤ 0, (3.11)

∂J

∂y12= −e−rtc1 − θ(t)JS1(t) + θ(t)JS2(t+ l) ≤ 0, (3.12)

∂J

∂yu2= −e−rtc2 − θ(t)JSu(t) + θ(t)JS2(t+ l) ≤ 0. (3.13)

Differentiating (3.9) with respect to S1:

0 = JtS1 − JSuS1(t)θ(t)[yu1(t) + yu2(t)]

+ JS1S1(t)[θ(t− l)yu1(t− l)− y12(t)θ(t)]

+ JS1S2(t)[yu2(t− l)θ(t− l) + y12(t− l)θ(t− l)− q(t)]

+σ212θ(t)2JS1θθ1(t) +

σ2x2x(t)2JS1xx(t) + φ1(t).

(3.14)

14

Next, let us write down JS1 in terms of state variables and time

t, so we have JS1 = JS1(t, Su, S1, S2, θ, x) and apply Ito’s lemma to

it:

dJS1 = JtS1 − JSuS1(t)θ(t)[yu1(t) + yu2(t)]

+ JS1S1(t)[θ(t− l)yu1(t− l)− y12(t)θ(t)]

+ JS1S2(t)[yu2(t− 1)θ(t− l) + y12(t− l)θ(t− l)− q(t)]

+σ212θ(t)2JS1θθ(t) +

σ2x2x(t)2JS1xx(t)dt

+ σ1θ(t)JS1θdz1 + σxx(t)JS1xdzx.

(3.15)

Applying Ito’s differential generator, 1/dtEtd(•), 9 on both sides

of (3.15):

1

dtEd[JS1(t)] = JtS1 − JSuS1(t)θ(t)[yu1(t) + yu2(t)]

+ JS1S1(t)[θ(t− l)yu1(t− l)− y12(t)θ(t)]

+ JS1S2(t)[yu2(t− l)θ(t− l) + y12(t− l)θ(t− l)− q(t)]

+σ212θ(t)2JS1θθ(t) +

σ2x2x(t)2JS1xx(t).

(3.16)

Substituting (3.16) into (3.14) leads to,

1

dtEtd[JS1(t)] = −φ1(t), (3.17)

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

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25

5 Appendix A

5.1 A1

Suppose (2.7) and (2.8) hold as equalities but (2.9) holds as strict

inequality. Subtracting (2.8) from (2.9) yields

e−rt(c2 − c1) + λu(t)− λ1(t) > 0. (5.1)

Lag (2.7) by l periods as,

λ1(t) = λu(t− l) + e−r(t−l)c1. (5.2)

Substituting it into (5.1) and recognizing λu(t) = λu(t− l) yields

c2 > (1 + erl)c1. (5.3)

So if c2 > (1 + erl)c1, yu1(t) > 0, y12(t) > 0, but yu2 = 0.

Thus only two-step transfers occur. However if (2.7)-(2.9) hold as

equalities, they reduce to c2 = (1 + erl)c1, However the existence of

adjustment cost implies c2 > 2c1 and thus (1 + erl)c1 > 2c1. This

cannot hold for any l ≥ 0. Thus it is impossible for both two-step

and one-step transfers to occur simultaneously.

On the other hand, suppose (2.7) and (2.8) hold as strict inequal-

ities but (2.9) hold as equality. Rewrite (2.7) and (2.8) as

λu(t− l) + e−r(t−l)c1 > λ1(t), (5.4)

λ1(t) > λ2(t+ l)− e−rtc1. (5.5)

They imply that

λu(t− l) + e−r(t−l)c1 > λ2(t+ l)− e−rtc1. (5.6)

Substituting (2.9) into the inequality above to eliminate λ2(t+ l)

yields the result as,

λu(t− l) + e−r(t−l)c1 − ertc2 − λu(t) > e−rtc1. (5.7)

26

Note that λu(t) = λu(t− l) and the equation above reduces to

c2 < (1 + erl)c1. (5.8)

Under this condition, yu1(t) = yu2(t) = 0, but yu2 > 0. Thus,

only one-step transfers occur.

5.2 A2

This section offers the proof by contradiction for S1(t) = S2(t) = 0

along the equilibrium path.

Suppose at some period t = t∗, S1(t∗) > 0 and S2(t

∗) > 0. Then

φ1(t∗) = φ2(t

∗) = 0. By (2.11) and (2.12), λ1(t∗) = λ2(t

∗) = 0.

Rewrite (2.7) as

λ1(t∗ + l) ≤ e−rt∗c1 + λu. (5.9)

Note that λu is constant over time because of (2.10). The left

hand side of (5.9) denotes the marginal benefits of moving extra one

unit of oil from stock Su to S1, while the right hand side captures

the full marginal cost of moving extra one unit of oil from Su to S1.

It consists of two components: e−rt∗c1 denotes the present value of

marginal cost of the transfer; while λu is opportunity cost of trans-

ferring extra one unit of oil from Su to S1.

Suppose λ1(t∗ + l) = e−rt

∗c1 + λu. Then in the following period

t > t∗, λ1(t+ l) > e−rtc1 + λu because λ1(t∗) = 0 and thus λ1(t

∗) =

λ1(t), and e−rt < e−rt∗. This implies that the marginal benefits of

moving extra one unit from Su to S1 is greater than the full marginal

cost of this transfer. Given this situation, the firm will liquidate all

the resource in Su and move them to S1 at time t [12]. This radical

action contradicts to the fact that Su(t) > 0 and thus λu(t) = 0.

On the other hand, suppose λ1(t∗+ l) < e−rt

∗c1+λu. If the value

of λ1(t∗ + l) − λu is sufficiently small such that in the next period

t > t∗, λ1(t + l) < e−rtc1 + λu still holds. However, if the value

of λ1(t∗ + l) − λu is not small such that in the some period t > t∗,

λ1(t + l) = e−rtc1 + λu since the right hand side of (5.9) is lowered

by t > t∗. This leads to the previous case again in which the firm

27

liquidates all the resource in Su.

In sum, it must be true S1(t) = 0 along the equilibrium path.

Similarly, S2(t) = 0 along the equilibrium path.

28

6 Appendix B

This section offers the proof of proposition 2.

Suppose (3.11) and (3.12) hold as equalities and (3.13) holds as

inequality. Subtracting (3.12) from (3.13) yields

e−rt(c2 − c1) + θ(t)Jsu(t)− θ(t)Js1(t) > 0. (6.1)

Lagging (3.11) by l periods and solving for Js1(t),

Js1(t) =e−r(t−l)c1θ(t− l)

+ Jsu(t− l). (6.2)

Substitute it into the (6.1) to eliminate Js1(t) and yields,

e−rt(c2−c1)+θ(t)[Jsu(t)−Jsu(t− l)]− θ(t)

θ(t− l)e−r(t−l)c1 > 0. (6.3)

Take expectation Et over (6.3) and yields

e−rt(c2 − c1)− e−r(t−l)c1Et[θ(t)

θ(t− l)] > 0, (6.4)

while Et[Jsu(t) − Jsu(t − l)] = 0 because integrating both sides of

(3.18) yields Et[Jsu(t)] = Et[Jsu(t− l)] = Et[Jsu(0)]. Moreover, θ(t)

can be solved for12,

θ(t) = e[−12σ21t+σ1z1(t)], (6.5)

So,

θ(t)

θ(t− l)= e−

12σ21 leσ1[z1(t)−z1(t−l)]. (6.6)

Since dz1 is a Wienner process, z1(t)−z1(t−l) obeys a distribution

as N ∼ (0, l). So σ1[z1(t) − z1(t − l)] obeys a distribution as N ∼(0, σ21l) and its log normal distribution eσ1[z1(t)−z1(t−l)] has a mean as

Et[eσ1[z1(t)−z1(t−l)]] = e

12σ21 l. Thus, by using (6.6),

Et[θ(t)

θ(t− l)] = e−

12σ21 l+

12σ21 l = 1. (6.7)

12see Øksendal [17], page 64-65

29

Substitute (6.7) into (6.4), multiplies through ert and yield,

c2 > (1 + erl)c1, (6.8)

when (6.8) holds, only two-step route occurs.

Similarly, if (3.11)-(3.13) hold as equalities, they reduce c2 =

(1 + erl)c1. However, it contradicts to c2 > 2c1. Therefore, two-step

and one-step route cannot occur simultaneously.

On the other hand, Suppose (3.11) and (3.12) hold as inequalities

and (3.13) holds as equality. Lagging l periods of (3.11) as,

Js1(t) < Jsu(t− l) +e−r(t−l)c1θ(t− l)

, (6.9)

Rewrite (3.12) as,

Js2(t+ l)− e−rtc1θ(t)

< Js1(t). (6.10)

Therefore,

Js2(t+ l)− e−rtc1θ(t)

< Jsu(t− l) +e−r(t−l)c1θ(t− l)

. (6.11)

Multiplying θ(t) on both sides yields,

θ(t)Js2(t+ l)− e−rtc1 < θ(t)Jsu(t− l) + e−r(t−l)c1θ(t)

θ(t− l). (6.12)

Substituting (3.13) into (6.12) to eliminate θ(t)Js2(t+l) and then

applying Et on both sides of (6.12) yields,

c2 < (1 + erl)c1. (6.13)

So under this condition only one-step route occurs.

30

7 Appendix C

Applying Ito’s lemma on (3.5) yields,

dp = xf ′(q)dq + f(q)dx+1

2xf ′′(q)(dq)2 + f ′(q)dqdx. (7.1)

Note that the optimal production q = q∗(Su, S1, S2, x, θ) along

the equilibrium path. Now expanding dq using Ito’s lemma:

dq =−qsu(t)θ(t)[yu1(t) + yu2(t)]

+ qs1(t)[θ(t− l)yu1(t− l)− θ(t)y12(t)]

+ qs2(t)[θ(t− l)yu2(t− l) + θ(t− l)y12(t− l)− q(t)]

+σ212θ2(t)qθθ +

σ2x2x2(t)qxxdt

+ σxx(t)qx(t)dzx + σ1θ(t)qθ(t)dz1.

(7.2)

Thus

(dq)2 = [σ2xx2(t)q2x(t) + σ21θ

2(t)q2θ(t)]dt, (7.3)

and,

dqdx = [σ2xx2(t)qx(t)]dt. (7.4)

Substituting (7.3) and (7.4) into (7.1) and then applying Ito’s

differential operator on (7.1) yields,

1

dtEt(dp) =xf ′(q)

1

dtEt(dq) +

1

2xf ′′(q)[σ2xx

2(t)q2x(t) + σ21θ2(t)q2θ(t)]

+ f ′(q)[σ2xx2(t)qx(t)].

(7.5)

Substitute (3.28) and (3.29) into (7.5) respectively and then solve

for 1dtEtd[q] as

1

dtEt[dq(t)] =

erl(σ21 − r)c1[ 1θ(t−l) + erl

θ(t−2l) ] + r(p(t)− c0)xf ′(q)

−12xf

′′(q)[σ2xx2(t)q2x(t) + σ21θ

2(t)q2θ(t)] + f ′(q)[σ2xx2(t)qx(t)]

xf ′(q)(7.6)

31

1

dtEt[dq(t)] =

erl(σ21 − r) c2θ(t−l) + r(p(t)− c0)xf ′(q)

−12xf

′′(q)[σ2xx2(t)q2x(t) + σ21θ

2(t)q2θ(t)] + f ′(q)[σ2xx2(t)qx(t)]

xf ′(q)(7.7)

In a case of the linear demand curve f(q) = p − q(t), (7.6) and

(7.7) reduce to,

1

dtEt[dq(t)] =

r(p(t)− c0) + σ2xx2(t)qx(t)

−x

+erl(σ21 − r)c1[ 1

θ(t−l) + erl

θ(t−2l) ]

−x

(7.8)

1

dtEt[dq(t)] =

r(p(t)− c0) + σ2xx2(t)qx(t)

−x

+erl(σ21 − r) c2

θ(t−l)

−x

(7.9)

In next step, we solve the equilibrium level of expected production

E[q(t)]. Substitute p(t) = x(t)[p− q(t)], 13x(t) = e−12σ2xt+σxzx(t) and

θ(t) = e−12σ21t+σ1z1(t) into (7.8) and (7.9) and then take expectation

over both sides and yields,

1

dtEt[dq(t)] =− rp− σ2xqx + rEtq(t) + rc0e

σ2xt

− e(r−σ21)l(σ21 − r)c1(1 + e(r−σ

21)l)e(σ

2x+σ

21)t,

(7.10)

and,

1

dtEt[dq(t)] =− rp− σ2xqx + rEtq(t) + rc0e

σ2xt

− e(r−σ21)l(σ21 − r)c2e(σ

2x+σ

21)t.

(7.11)

Note that we are using the fact that Et[x(t)] = 1 and 14Et[e−σzx(t)] =

e12σ2t.

(7.10) and (7.11) are first-order linear differential equations and

13A general solution to x(t) can be found at Øksendal [17], page 65.14see Øksendal [17], page 65.

32

their are solutions15 can be solved for,

Et[q(t)] = K − ηeσ2xt − βe(σ2

x+σ1)t − Zert. (7.12)

where η, β, and Z are positive constants defined as K = p + σ2xqxr ,

η = rc0r−σ2

xand

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

. 2c1 < c2 < (1 + erl)c1

(7.13)

15For a first-order differential equation y + αy = g(t), its general solutionis y = e−αt

∫eαsg(s)ds + ce−αt, where c is a constant pinned down by initial

conditions. See Boyce and Diprima [1], page 33.

33