1
Updated 16.02.04
ECON4925 Resource economics, Spring 2004
Olav Bjerkholt:
Lecture notes on the Theory of Non-renewable Resources
1. Gray's problem: the discovery of resource rent
In his early contribution Gray (1914) made a point of considering a non-renewable
resource as an asset. The problem of extraction and utilization of the resource can
accordingly be regarded as a portfolio choice problem. A general result in this theory is
that optimal allocation of assets requires marginal returns to be equalized. As we shall
see, this is also the essence of the optimal path of a resource extracting firm.
Gray (1914) assumed the resource to be coal which could be sold on the market at a given,
constant price. It does not make much difference if the price is assumed to vary exogenously
over time, known to the resource owner. The resource owner thus takes the exogenously
given price, p, as given. The total stock of the resource is also known and given at the outset
as S . The rate of depletion is Rt and the cost of extracting the resource in this amount per
unit of time is given by b=b(Rt). We define the net marginal price as qt  p  b( Rt ) .
The problem for the resource owner is to choose an extraction path {Rt} for the resource
stock that maximizes total, discounted profits. The discount rate is r, which can be thought
of as the return on financial investments.
Formally, this can be stated as
T
(1.1)
Max  [ pRt  b( Rt )]e  rt dt
Rt ,T
0
given
T
(1.2)
 R dt  S
t
0
How much is it profitable to extract per unit of time (year)? Assume that the stock of
extracted resource is large relative to this amount. Try to reason intuitively about the
depletion profile. What is the role of the discount rate, what would be the solution if the
rate of discount was zero, rather than a positive discount rate?
Part of the problem is also to determine the length of the depletion period. In
mathematical terms, (1.1) conditioned by (1.2) is an optimal control problem that can be
solved by well-known principles, see Hammond et al. (2003), Sydsæter, Seierstad and
Strøm (2002), Sydsæter, Strøm and Berck (2000). We will first try to give an intuitive
explanation for conditions that must be fulfilled along the optimal extraction path.
2
When to extract a natural resource
We start with a very simple situation. We assume constant unit costs, i.e. b  b  Rt and that
the resource owner has just one unit of the resource. When is it optimal to extract the
resource?
Clearly, if the price is given as p, then it must either hold that p  b or p  b . In the first
case the answer is to extract now, in the latter case never! Let us then consider a resource
price that increases with a constant rate  , i.e. pt  p0 e t . We must assume that  < r ,
otherwise we would never extract. It is straightforward to show that the resource should be
extracted at a time when the price path reaches the "trigger price" given by
(1.3)
p*  rb /(r   )
Why? We simply maximize ( p0e t  b )e rt and solve for the optimal t=t*, which is when
the net marginal price increases at a rate equal to the return on financial investments, r .
(1.3) is equivalent to  p* /( p*  b )  r . As p   p this again states that the marginal net
return on holding back on extraction should equal the interest rate.
Until the "trigger price", p* is reached, the return on postponing extraction exceeds the
interest rate. After t* the return on keeping the resource unexploited is less than the interest
rate.
Depletion profile with increasing marginal costs
Let us then consider the problem as stated in Gray (1914). The problem is that of an
owner of a stock of coal that can be extracted with decreasing returns to scale and the
usual convexity properties of the cost function, b > 0 and b > 0 and facing a given
constant price p. How to extract the resource? Would it be optimal to extract such that
b  p , as for ordinary goods, or, perhaps at the level which minimizes the average cost (i.e.
b( Rt )  b( Rt ) / Rt ), such that the resource can be extracted at the lowest possible cost?
Neither is true.
One way to look at it is that the owner of the coal mine at any point in time is faced with
a portfolio choice that involves two asset options:
1)
He can choose to extract one more unit of coal. If so, he will be able to earn a
return of r by depositing the earnings from the sale of coal in the bank.
2)
Alternatively, he can leave the coal in the ground. The resource rent or the
marginal profit of extracting one more unit of coal at time t, i.e. qt  p  b( Rt ) .
The return on leaving the resource in the ground is then q/q . By postponing
extraction, the return on the unit left in the ground is q/q .
How can the resource rent vary when the price is constant? By adjusting extraction.
Along the optimal path, extraction is adjusted so that the returns on the different assets
are equalized at each point in time, i.e.
(1.4)
qt / qt  r
3
(1.4) expresses a general result from the theory of optimal portfolio choice that total
wealth should be allocated on different assets so that (marginal) returns are equalized. If
(1.4) is not fulfilled, the resource owner can increase his wealth by reallocating between
different assets. In the present case, one of the "assets" is the value of keeping the
resource in the ground, and the choice problem simply consists of deciding on the timing
and speed of resource extraction.
Then, what is the depletion pattern? For the resource rent, i.e. the difference between the
given price and the marginal cost, to increase over time the extraction rate must be
reduced over time. If we knew the initial extraction rate, R0, we could work out the entire
extraction path down the marginal cost curve. What would the end point look like? A
conclusion close at hand is to guess that the last unit of the resource must be extracted at
the lowest possible cost, i.e. at the rate that minimizes the average cost curve.
In other words we balance off the need to get the resource out quickly, forced upon us by
the discounting and at the same time extract at the lowest possible cost.
The optimal control solution
The problem stated in Gray (1914) can be restated as an optimal depletion programme for
a profit-maximizing natural resource firm (mine), facing a given price (or, possibly, a
price path) for the extracted resource and having marginal extraction costs varying with
the amount of depletion with the usual U-shaped marginal cost and average cost curves.
Using optimal control theory we could proceed as follows:
S 0  S is the amount of resource in the ground at time t=0. The amount depleted (per unit
of time) at time t is Rt while the remaining amount of resource at time t is St. The cost of
extraction is given by the function b(Rt) with b’>0 and b’’>0. The price of the extracted
resource, p, is constant over time. The rate of discount is r. The problem is to find the
depletion profile {Rt*} and the time T* maximizing the present value of total profit.
T
(1.5)
Max  [ pRt  b( Rt )]e  rt dt
Rt ,T
0
St   Rt , S0  S , ST  0, Rt  0
The state variable in this problem is the amount of remaining resource St, while the
control variable is the rate of depletion Rt. The Hamiltonian of this problem is the
integrand, plus the product of the adjoined price (shadow price), t, of the state variable
and its rate of change.
(1.6)
H (t , St , Rt , t )  [ pRt  b( Rt )]e  rt  t Rt
In this formulation the shadow price is the present value shadow price. Assume that Rt*
and T* solves the problem. Then it follows from the maximum principle that Rt*
maximizes the Hamiltonian for each t, which implies that when continuity,
differentiability and concavity of the Hamiltonian hold, we have
(1.7)
H
 [ p  b( Rt )]e  rt  t  0
Rt
4
Furthermore, the rate of change of the adjoined price is given by
t  
(1.8)
H
0
S
Alternatively, the Hamiltonian can be formulated in current value terms. The current
value shadow price t can be solved in a similar way from the Hamiltonian in current
values given by
(1.9)
H C (t , St , Rt , t )  p  Rt  b( Rt )  t  Rt
The first order conditions now becomes
(1.10)
H C
Rt
 p  b( Rt )  t  0
and
(1.11)
t  r t  
H C
0
S
Corresponding to the condition ST*  0, we must have T*  0 ( =0 when ST* > 0 ). [If
the end point condition had been ST given , there would have been no constraint on T* ,
while if ST is free, we must have T* = 0 .]
Corresponding to the free end point, we have the transversality condition
(1.12)
H (T *, ST * , RT * , T * )  0
Or, in current value terms,
(1.13)
H C (T * , ST * , RT * , T * )  0
By combining the first order condition at the end point with the transversality condition,
we easily find, that at the end point we must have
(1.14)
b( RT * )  b( RT * ) / RT *
Depletion with stock dependent costs
For many exhaustible natural resources a realistic description of the extraction
technology, implies increasing extraction costs as the stock is depleted. One reason may
be that the resource deposit consists of a series of layers of different quality, and
accordingly it is more costly to extract from the deeper layers. For crude oil and natural
gas another reason is that the pressure in the reservoir decreases as the accumulated
extraction increases. In the literature, a common representation of such phenomenon is to
specify a more general cost function
(1.15)
b  b( Rt , St ), bR  0, bS  0
The optimization problem facing the producer is the same as before, i.e. maximizing
discounted profits, which in this case is expressed as
5
T
(1.16)
Max  [ pRt  b( Rt , St )]e  rt dt
Rt ,T
0
again constrained by (1.2). By solving we find that the resource rent must obey, cf. (1.4),
(1.17)
qt p  bR
b

r S
qt p  bR
p  bR
As bS  0 , we get qt / qt  r . Thus, when extraction costs vary negatively with remaining
reserves, optimal extraction implies that the resource rent should increase at a rate less than
the rate of interest.
This result is rather intuitive: By extracting a marginal unit of the resource, the net marginal
value of remaining reserves is reduced by bS . Or, holding back a marginal unit at time t,
yields a return given by the last term in (1.17), in addition to the increase in the net price of
the resource. This implies that the increase in the resource rent required to equalize the
returns on different assets is less than the interest rate.
The end point condition (1.14) is not affected by introducing a more general cost
function; the production intensity at the terminal point is at the intersection between
marginal and average costs. However, with the cost function (1.14), as the resource is
depleted, the marginal cost schedule shifts upwards. As a consequence, it could now well
be the case that the deposit becomes unprofitable at the margin before the stock is
exhausted. This gives rise to a meaningful definition of economically exhaustible
resources, as distinct from the amount of resources that are physically exhaustible.