Monopoly Extraction of a Nonrenewable Resource Facing
Capacity Constrained Renewable Competition
Min Wang∗
Peking University
Jinhua Zhao†
Michigan State University
March 23, 2013
Abstract
This paper studies monopoly extraction of a nonrenewable resource with the presence of
a competitively supplied renewable substitute. We show that if the renewable substitute has
a capacity constraint, the monopolist would keep the price flat for a while to stave off the
renewable substitute when the latter becomes competitive. The monopolist would then let the
resource price jump up so that the renewable substitute produces at full capacity. The price
and quantity paths thus are discontinuous.
∗
National School of Development, Peking University, Beijing, China. Email: [email protected].
Department of Economics, Department of Ag., Food and Res. Economics, and Environmental Science and Policy
Program, Michigan State University, East Lansing, MI 48824, USA. E-mail: [email protected].
†
1
1
Introduction
When studying the interactions between renewable and nonrenewable resources (e.g., renewable
energies and fossil fuels), resource economists commonly treat the renewable resource as a pure
backstop without capacity constraints: once the resource price reaches its marginal cost, the backstop floods the market and drives out the nonrenewable resource (e.g., Dasgupta and Stiglitz (1981),
Dasgupta et al. (1982, 1983) and Chakravorty et al. (2006, 2008)).1 However, most renewable
resources are not pure backstops since they have capacity constraints so that by themselves they
cannot supply the entire market. For example, the production capacity of biofuels is limited by
land availability and increasing demand for food and feed. The prime wind sites and rivers that
are available for generating wind power and hydro power are limited by natural landscapes and
endowments. In all cases, the marginal production costs are expected to increase sharply after
certain thresholds, e.g., if these energies replace fossil fuels as dominant energy sources.
In this paper, we study a monopolist owner of a nonrenewable resource facing competition from
a renewable resource with a capacity constraint. We find that given the capacity constraints, the
monopolist has incentive to stave off the renewable resource even when the latter becomes competitive: when resource price rises to the marginal cost of the renewable resource, the monopolist
“floods” the market, so that the market share of the renewable is still zero.2 This “stave-off” period can last a long time. Further, when the renewable resource does enter the market, the resource
price jumps up. That is, when the monopolist finds it not desirable to continue staving off the
renewable resource, it will allow the price to jump up so that the renewable substitute produces at
full capacity, and act as a monopolist off the residual demand.
Our study is the first one examining the price and extraction paths of a nonrenewable resource
when the production of the renewable substitute is capacity constrained. To our knowledge, it is
also one of the few studies in the literature showing that the price path can be discontinuous in a
deterministic setting.3 The rest of the paper is organized as follows. Section 2 presents the model
1
Amigues at.al (1998) and Holland (2003) are the few studies that highlight the important roles of capacity
constraints. They show that capacity constraints could change the order of extraction of heterogeneous resources, for
the scarcity rent generated by the capacity constraints changes the cost order of different resources.
2
This result is similar to that in Hoel (1978, 1983) which study the case where the monopolist faces competition
from a renewable substitute that is a pure backstop (without capacity constraints). The author shows that when
the resource price rises to equal the marginal cost of the backstop, the monopolist would flood the market until its
nonrenewable resource stock is exhausted. The resource price is still continuous over time.
3
Groot et al. (1992) show that resource price can jump when a cartel and a competitive fringe, both owners of
2
and Section 3 discusses the linearity assumptions of the model. We conclude in Section 4.
2
The Model
Consider two substitutable resources: a nonrenewable resource (indexed by  = 1) with a starting
reserve of 0  0 and constant unit extraction cost of 1 , and a renewable resource (indexed by
 = 2) with a constant marginal production cost of 2 . We assume that 0  1  2 . Let  ()
be the remaining reserve of the nonrenewable resource in period . We assume a stable resource
demand function  =  (1 + 2 ) with 0 (·) ≤ 0, where  ≥ 0 is the output of resource ,  = 1 2.
Further, we assume the renewable resource has a production capacity constraint of ̄2 and that
by itself, the renewable resource cannot supply the whole market at the moment that it becomes
competitive: 2  −1 (2 ). Finally we assume the sector of the renewable resource is competitive
(i.e., when  = 2 ), but the nonrenewable resource is owned by a monopolist.
First we derive the renewable resource supply function. Since the production of the renewable
resource is perfectly competitive with a constant marginal cost, its supply follows
⎧
⎪
⎪
if  ()  2
⎪ = 0,
⎨
2 ()
∈ [0  2 ] ,
if  () = 2 
⎪
⎪
⎪
⎩ = ,
if  ()  2
2
(1)
Let  be the depletion time of the nonrenewable resource. The monopolist takes the supply
function (1) as given and maximizes its own discounted profit:
Z 
− [ (1 () + 2 (())) 1 () − 1 1 ()] 
max
1 ()2 ()
0
s.t. ̇() = −1 ()  1 () ≥ 0 (0) = 0  ( ) ≥ 0 and (1)
where  is the market interest rate. We assume that the revenue function  (1 + 2 ) 1 is concave in
1 . Let  be the present shadow value of the nonrenewable resource stock. Then the Hamiltonian
at time  can be written as
 =  (1 () + 2 (())) 1 () − 1 1 () −  1 () 
(2)
nonrenewable resources, play an open-loop von Stackleberg game. However, these open-loop strategies involving price
jumps are not dynamically consistent. (Groot et al. (2003) identifies which of the strategies in Groot et al. (1992)
are dynamically consistent.) In constrast, the equilibrium in our model is subgame perfect, which, of course, is also
dynamically consistent.
3
The optimality condition on 1 () involves comparing the marginal revenue  (1 ()) with 1 + ,
which, following Holland (2003), is called the “augmented marginal cost” and is denoted as  ().
In calculating  (1 ()), the monopolist takes into consideration the reaction function (1), and
since 2 () is discontinuous in price  (),  (1 ()) is also discontinuous. We will discuss the
formula of  (1 ()) later, and for now, it suffices to note that the optimal 1 () is determined by
a Kuhn-Tucker condition comparing  (1 ()) with  ().
Since   0, the transversality condition means that ( ) = 0, i.e., the nonrenewable resource
will be exhausted. The free choice of exhaustion time  implies  = 0. Notice that at the
exhaustion time  , the renewable supplies the entire market, so that  ( ) =  (2 ). Hence  = 0
is equivalent to
 ( 2 ) = 1 + 
(3)
∞
The market equilibrium is the sequence of {1 ()  2 ()   ()}∞
=0 such that (a) given {()}=0 ,
{1 ()  2 ()}∞
=0 satisfy (1) and the Kuhn-Tucker condition on 1 (); (b) the resource market is
clear at every moment:  () =  (1 () + 2 ()); and (c) the transversality conditions ( ) = 0
and  = 0 are satisfied.
We now proceed to characterize the marginal revenue function  (1 ). Figure 1 illustrates the
residual demand for the monopolist, obtained by subtracting supply function (1) of the renewable
from the demand curve, and Figure 2 shows the corresponding marginal revenue function. When
  2 , the renewable is not competitive and (1) implies 2 = 0. In this case, the monopolist’s
marginal revenue function is
 1 (1 ) ≡  (1 )|2 = 0 (1 )1 + (1 )
(4)
When   2 , the renewable supplies at its full capacity and (1) implies that 2 = ̄2 . Then the
marginal revenue is
 3 (1 ) ≡  (1 )|2 = 0 (1 + ̄2 )1 + (1 + ̄2 )
(5)
When  = 2 , 2 ∈ [0 ̄2 ] and there is a range of 1 within which the monopolist can increase its
output without driving down the market price since the renewable supply will decrease accordingly.
This is represented by the flat segment of the residual demand curve in Figure 1, with the range
of 1 being [−1 (2 ) − ̄2  −1 (2 )] since the total market demand is −1 (2 ) and 2 ∈ [0 ̄2 ]. Since
1 = 0 in this interval,
 2 (1 ) ≡  (1 )|=2 = 2 
4
(6)
and the marginal revenue jumps at the two end points of the range, i.e., at 1 = −1 (2 ) and at
1 = −1 (2 )− ̄2 (because 1  0 when 1 is outside the interval). We denote the two marginal
revenue levels at the jump points as  1 and  2 in Figure 2.4
p
MR
hq2 
hq2 
c2
c2
MR2
MR1
AMCt 
0
h1c2   q2 h1c2 
q1
0
1
h1c2   q2 h c2 
q1
Figure 1: Residual demand for the monopolist Figure 2: Marginal revenue for the monopolist
Based on Figure 2, we study the price and extraction paths of the resources by comparing
 (1 ) given in (4) - (6) with  (), which is increasing in time. In Figure 2,   () is a
horizontal line and continuously moves up as time goes by. Let 1 be the time when  (1 ) =
 1 , which is also the first moment when () reaches 2 . For  ∈ [0 1 ], resource price ()
continuously increases but is less than the cost of the renewable so that the monopolist supplies
the whole market. Output 1 () is determined by  1 (1 ()) =  (), i.e.,
0 (1 ()) 1 () +  (1 ()) = 1 +  
(7)
Let 2 be the first moment when  () reaches  2 :  (2 ) =  2 . For  ∈
[1  2 ],   () ∈ [ 1   2 ], and the marginal revenue, which equals 2 , is higher than
  (). The monopolist produces at the corner: it floods the market and thus staves off the
renewable substitute while maintaining the market price of () = 2 :5
1 () = −1 (2 ) 
(8)
As long as  (1 ())   (), the monopolist has incentive to flood the market. When will
the monopolist stop staving off the backstop and let the price increase? As   () increases
further so that   () ∈ [ 2  2 ],  () also intersects the downward slopping portion of





Specifically 1 =0 −1 (2 ) −1 (2 ) + 2 , and 2 =0 −1 (2 ) −1 (2 ) −  2 + 2 
5
It produces at the corner because if it produces more than the corner solution, the price will be lower than 2
4
and the marginal revenue drops below the augmented marginal cost.
5
the marginal revenue curve in Figure 2. There exist two locally optimal solutions of 1 (): the
corner solution as in (8) and an interior solution determined by setting  3 (1 ()) =  ().
Let 2 be the time when  () rises to 2 :   (2 ) = 2 . If the monopolist floods the market
until 2 , its profit (after deducting the Hotelling rent) decreases to zero at 2 , while if it chooses
the interior solution, it has a positive profit since the price strictly exceeds the augmented marginal
cost (although its output is lower). Therefore the monopolist must stop staving off the backstop
at some time before 2 . Let 2 be that time and we determine 2 as follows (the proof is given in
the Appendix).
Proposition 1 As the resource price rises to the marginal cost of the renewable resource, the monopolist will stave the renewable off until 2 , when the economic value (represented by the Hamiltonian) under the interior solution of the monopolist’s resource supply is equal to that under the
corner solution of the monopolist’s resource supply, i.e.
£
¤ £
¤¡
¢
1 (2 )  (1 (2 ) +  2 ) − 1 − 2 = −1 (2 ) − 2 2 − 1 − 2
(9)
where 1 (2 ) is determined by  2 (1 (2 )) =  (2 ).
H t 
Interior solution for q1 t 
Corner solution for q1 t 
0
T1 TMR2
T2
Tc2
T
t
Figure 3: The path of Hamiltonian
Figure 3 illustrates the intuition of Proposition 1. During [2  2 ], the Hamiltonian  ()
has two paths corresponding to the two possible solutions of 1 (). Proposition 1 states that the
monopolist should choose the one that generates a larger economic value, i.e. a higher  (). As
the monopolist switches from the corner solution to the interior solution at 2 , resource price jumps
up from 2 to (1 (2 ) + ̄2 ), as shown in Figure 4. Note that the optimal () is still continuous
in time.
6
Stage three [2   ] starts when the renewable resource begins to supply the market at full
capacity and the optimal supply rule of the nonrenewable resource is again determined by an
interior solution equating  3 (1 ()) with  ():
0 (1 () +  2 ) 1 () +  (1 () + 2 ) = 1 +  
(10)
As in the first stage, resource price keeps rising and the supply of nonrenewable resource continuously decreases. At the end of this stage  , resource price reaches  (2 ) and the supply of
nonrenewable resource falls to zero, depleting the reserve.
During the final stage of [ ∞], resource price () keeps constant at  ( 2 ), 1 () = 0 and the
renewable resource supplies the whole market at its full capacity of production.
The equilibrium path is completely characterized by { 1  2   1 (2 )}, and given 0 ,  2 and
 ,  = {1 2}, the solutions of { 1  2   1 (2 )} are determined by (3), (9) and
Z 
Z 1
¤
£
1 ()  + −1 (2 ) −  2 (2 − 1 ) +
1 ()  = 0
2
0
¢
¡
0 −1 (2 ) −1 (2 ) + 2 = 1 + 1
0 (1 (2 ) +  2 ) 1 (2 ) +  (1 (2 ) + 2 ) = 1 + 2 
(11)
(12)
(13)
where (11) is the resource exhaustion condition for the nonrenewable resource, 1 () for  ∈ [0 1 ]
in (11) is derived from (7), and 1 () for  ∈ [2   ] is given in (10). Equations (12) and (13)
determine 1 and 2 , the switch times between interior and corner solutions.
Figure 4 and 5 respectively illustrate the price path and extraction path of the nonrenewable
resource, and we summarize the main results in the following:
Summary 1 Under the monopoly extraction of a nonrenewable resource, if the renewable substitute is capacity constrained, the resource price will first continuously increase until the renewable
substitute becomes competitive. Then the monopolist will stave off the renewable substitute for a
certain period of time while keeping the resource price flat at the renewable’s marginal cost. Eventually, the monopolist will discretely reduce its own production, let the renewable substitute produce
at full capacity and let the resource price jump above the cost of the renewable substitute. Thereafter
the resource price continuously increases until the monopolist’s stock is exhausted.
7
p t 
q1 t 
hq2 
h 1 c2 
c2
0
T1
T2
T
t
0
T1
T2
T
t
Figure 4: Price path of nonrenewable resourceFigure 5: Extraction path of nonrenewable resource
3
More General Cost Functions
Price jumps when the monopolist switches from a corner solution to an interior solution. Corner
solutions are usually driven by linearities in optimization problems, and we do assume linear production costs for both the nonrenewable and renewable resources. In this section, we show that the
price discontinuity in our model is not sensitive to the linearity assumptions.
We make two simplifying assumptions about the extraction cost of the nonrenewable resource:
the marginal extraction cost is constant and is independent of the stock level. If we instead assume
˜ () =
convex costs, e.g., 1 (1 ) with 01  0 and 001 ≥ 0, the augmented marginal cost becomes 
01 (1 ()) +  , which is an upward slopping curve in Figure 2 that moves up as time goes by. It is
then evident that there is still a time interval during which the marginal revenue of 1 () strictly
˜ (), resulting in a corner solution.
exceeds 
The easiest way to see what happens if the extraction cost is stock dependent is to follow
Hartwick (1978): we divide the region [0 0 ] into multiple segments, i.e., [  −1 ],  = 1     ,
with  = 0 and −1   , and assume constant marginal cost 1 on segment [  −1 ]. By
allowing 1 to increase in , this setup approximates the stock dependence of extraction costs, and
the approximation becomes more accurate as the number of segments  rises. Then the augmented
¯ () = 1 +   for  such that () ∈ [  −1 ], where  is the
marginal cost becomes 
(present value) Hotelling rent of stocks in segment [  −1 ], with 1  2  · · ·   . In Figure
¯ () would still be a horizontal line, and the only difference now is that as the stock moves
2, 
¯ () increases goes down as  goes down. Note that
to the next “segment,” the rate at which 
¯ () is still continuous in time (even though  values can jump as the stock moves to a new

8
¯ () and the monopolist
segment),6 and thus there is still a time period during which 2  
chooses a corner solution.
We next relax the assumption of constant marginal cost for the renewable resource while maintaining the assumption of a production capacity. Suppose the production cost is increasing and
convex. In particular, let 2 (2 ) be the marginal cost function, with 2 (0)  1 and 02 (2 )  0 for
all 2 ∈ [0  2 ]. In addition, we assume 2 ( 2 )   (2 ) so that when it becomes competitive, the
renewable resource by itself cannot satisfy the entire market demand. Then the supply function of
the renewable resource becomes
⎧
⎪
⎪
0,
⎪
⎨
2 () =
−1
2 ( ()) 
⎪
⎪
⎪
⎩  ,
2
if  ()  2 (0)
(14)
if  () ∈ [2 (0)  2 (2 )]
if  ()  2 (2 )
Corresponding to Figures 1 and 2, Figures 6 and 7 illustrate the residual demand and the marginal
revenue curves of the monopolist. The flat segment of the residual demand in Figure 1 is replaced
by a downward slopping segment, but the change in slope of the residual demand function implies
that the marginal revenue still jumps when the renewable becomes competitive and when it supplies
at full capacity. (The derivation of the marginal revenue function is given in the Appendix.)
p
MR
hq2 
hq2 
c2 q2 
MR4
c2 0
MR3
MR2
MR1
0
h1c2 q2   q2 h1c2 0
q1
0
Figure 6: Residual demand for the monopolist
6
AMCt 
h1c2 q2   q2 h1c2 0
q1
Figure 7: Marginal revenue for the monopolist
The details are available upon request from the authors, and can also be found in Hartwick (1978).
9
pt 
hq2 
c2 0
0
T1
Tm T2
T
t
Figure 8: Price path of the nonrenewable resource
As before, the monopolist’s decision is determined by comparing   () and the marginal
revenue in Figure 7, and the resulting price path is illustrated in Figure 8. 10 corresponds to 1
in Figure 4, and is the time when () rises to 2 (0) (or when  () rises to  10 in Figure 7).
After the renewable becomes competitive, the monopolist staves off the renewable for the period
0 ], where  0 is the time when  () rises to  0 in Figure 7. The monopolist chooses a
[10  

2
corner solution during this time period since the marginal revenue strictly exceeds the augmented
0 ,  () crosses with the marginal revenue curve and the monopolist
marginal cost. After 
chooses an interior solution. That is, it gradually lets the renewable into the market. As  ()
further rises to lie in [ 30   40 ] in Figure 7, the monopolist has two interior solutions, and by
the same logic as before, it will switch from one to the other when  () becomes sufficiently
close to  40 , resulting in a price jump. Time 20 corresponds to 2 in Figure 4, and is the time
when the monopolist switches solutions. Therefore, the discontinuity of the price path still holds
when the renewable’s cost function is convex.
4
Conclusion
In this paper we study the price and extraction paths of a monopolist owned nonrenewable resource
when the competitively supplied renewable substitute has capacity constraints. We show that
monopoly and capacity constraint together lead to a discontinuous price path. In particular, when
resource price rises to the marginal cost of capacity constrained renewable substitute, the monopolist
would keep the price flat for a while to stave off the renewable resource, and then would let the
price jump up and thus the renewable substitute produces at full capacity. These results are robust
to the cost structures of the renewable and nonrenewable resources.
10
Although our model is highly stylized, these findings do correlate well with some recent phenomena of the energy market. First, many renewable resources are “almost” competitive for long
periods of time, despite major technological advances that significantly reduced their costs. Second,
the entry of renewable resources into the energy market is often associated with upward jumps in
energy prices. For example, the large scale entry of corn based ethanol in U.S. occurred when oil
price jumped up significantly.
Appendix
Proof of Proposition 1. We can rewrite the monopolist’s optimization problem as
Z 
Z 1
Z 2
Z 
max
−  ()  =
− 1 ()  +
− 2 ()  +
− 3 () 
0
1
0
2
¡
£
¢
¤¡
¢
where 1 () = 1 () (1 ()) − 1 −  , 2 () = −1 (2 ) − 2 2 − 1 −   3 () =
£
¤
1 ()  (1 () +  2 ) − 1 −   and 1 ()  2 ()   1 and  are at their optimal values. The
Proposition follows directly by the first order condition on 2 .
Technical details of Figure 7. Using (14), we
⎧
⎪
0 (1 ) 1 +  (1 )
⎪
⎪
⎨
0 (1 +−1
2 ( (1 )))1
 =
+  (1 )
0
1− (1 +−1
( (1 )))02 ( (1 ))
⎪
2
⎪
⎪
⎩
0 (1 +  2 ) 1 +  (1 +  2 )
can derive the monopolist’s marginal revenue as
for 1 ≥ −1 (2 (0))
£
¤
for 1 ∈ −1 (2 ( 2 )) −  2  −1 (2 (0))
(15)
for 1 ≤ −1 (2 (2 )) − 2
¡
¢
where  =  (1 ) is the residual demand function derived from  =  1 + −1
2 () when the
renewable’s supply is 2 = −1
2 () (i.e., when () ∈ [2 (0) 2 (̄)]). Substituting in the critical
values of 1 , we obtain the four critical marginal revenues in Figure 7 as
¢
¡
 10 = 0 −1 (2 (0)) −1 (2 (0)) + 2 (0)
¢
¡
0 −1 (2 (0)) −1 (2 (0))
0
+ 2 (0)
 2 =
1 − 0 (−1 (2 (0))) 02 (0)
¢
£
¡
¤
 30 = 0 −1 (2 ( 2 )) −1 (2 ( 2 )) −  2 + 2 (2 )
¢£
¡
¤
0 −1 (2 ( 2 )) −1 (2 ( 2 )) −  2
0
+ 2 ( 2 )
 4 =
1 − 0 (−1 (2 (2 ))) 02 ( 2 )
References
[1] J. Aabakken, Power technologies energy data book, fourth edition. Technical report NREL/TP620-39728, National Renewable Energy Laboratory August 2006.
11
[2] J.P. Amigues, P. Favard, G. Gaudet, and M. Moreaux, On the optimal order of natural resource
use when the capacity of the inexhaustible substitute is limited. Journal of Economic Theory
80 (1998) 153-170.
[3] U. Chakravorty, B. Magne, and M. Moreaux, A Hotelling model with a ceiling on the stock of
pollution. Journal of Economic Dynamics & Control 30 (2006) 2875-2904.
[4] U. Chakravorty, M. Moreaux, and M. Tidball, Ordering the extraction of polluting nonrenewable resources. American Economic Review 98 (2008) 1128-1144.
[5] P. Dasgupta, R.J. Gilbert, and J.E. Stiglitz, Invention and innovation under alternative market
structures - the case of natural resources. Review of Economic Studies 49 (1982) 567-582.
[6] P. Dasgupta, R. Gilbert, and J. Stiglitz, Strategic considerations in invention and innovation
- the case of natural resources. Econometrica 51 (1983) 1439-1448.
[7] P. Dasgupta, and J. Stiglitz, Resource exhaustion under technological uncertainty. Econometrica 49 (1981) 85-104.
[8] F. Groot, C. Withagen and A. De Zeeuw, Note on the open-Loop von Stackelberg Equilibrium
in the Cartel versus fringe model, Economic Journal 102 (1992) 1478-84.
[9] F. Groot, C. Withagen and A. De Zeeuw, Strong time-consistency in the cartel-versus-fringe
model, Journal of Economic Dynamics and Control, 28 (2003) 287-306.
[10] J. M. Hartwick, Exploitation of Many Deposits of an Exhaustible Resource, Econometrica, 46
(1978) 201-217.
[11] M. Hoel, Resource extraction, substitute production, and monopoly. Journal of Economic
Theory 19 (1978) 28-37.
[12] M. Hoel, Monopoly resource extractions under the presence of predetermined substitute production. Journal of Economic Theory 30 (1983) 201-212.
[13] S. Holland, Extraction capacity and the optimal order of extraction. Journal of Environmental
Economics and Management 45 (2003) 569-588.
[14] S. Salant, Exhaustible resources and industrial structure: a Nash-Cournot approach to the
world oil market, Journal of Political Economy 84 (1976) 1079-1093.
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