ECON4925 – 2010
LECTURE 3 (OB), SEPT. 7, 2010
INTRODUCTION TO EXHAUSTIBLE RESOURCES (AND TO OPTIMAL
CONTROL)
THE OPTIMAL CONTROL PROBLEM AND ITS SOLUTION (CF. SYDSÆTER &
HAMMOND)
t1
max  f (t , x, u )dt ,
u
u U
t0
x&
x(t0 )  x0
t  g (t , x(t ), u (t )),
with one of three terminal conditions imposed:
(i) x(t1 )  x1
(ii) x(t1 )  x1
(iii) x(t1 ) free
x(t ) is called the state variable, while u U is the control variable.
Suppose ( x* (t ), u* (t )) is an optimal solution. Then there exists a continuous
function p(t) such that for all t in [t0, t1], the control function u * (t ) maximizes
the Hamiltonian, defined as
H (t , x(t ), u, p(t ))  f (t , x, u )  p(t )  g (t , x, u )
Thus
H (t , x* (t ), u, p(t ))  H (t , x* (t ), u* (t ), p(t )) for all u U
In practice we apply the first order condition
H / u  0 .
Furthermore, p (t ) fulfills p&(t )   H x (t , x* (t ), u* (t ), p(t )) .
For each terminal conditions there is a corresponding transversality condition:
(i’) p(t1) no condition
(ii’) p(t1 )  0 (with p(t1)=0 if x*(t1)>x1)
(iii’) p(t1)=0
If the problem is redefined with t1 free, then the conditions are satisfied on [t0,
t1*], and in addition
H (t1* , x* (t1* ), u * (t1* ), p(t1* ))  0
THE OPTIMAL CONTROL PROBLEM FOR A NONRENEWABLE RESOURCE
St = Stock of nonrenewable resource
Rt = Depletion of
nonrenewable resource
T
Objective
function
W  max  F ( St , Rt , t )e  rt dt
Rt
S&
t   Rt
System
S0  S
ST  S
Terminal
state
Terminal
point
0
T fixed
ST  S
T free
T fixed
Present-value
Hamiltonian
H  H (S , R, t ,  )  F (S , R, t )e rt   R
[Currentvalue
Hamiltonian]
[ H C  (S , R, t ,  )  F (S , R, t )   R]
Equations of
motions
Max H [ H C ]
Transversality
conditions
S&  R
H
& 
S
T free
H C
[ & r   
]
S
H
H C
 0 (=0 for Rt  0) [
 0 (  0 for Rt  0)]
R
R
ST  S
T  0 (with T  0 if ST  S )
No condition on T [T ]
[ T  0 (with T  0 if ST  S )]
H T  0 [H TC  0]
H T  0 [H TC  0]
Characteristic features of resource markets are related to the
exhaustibility
Firstly, when resource scarcity prevails, the market price of the resource will
typically exceed marginal extraction costs to reflect the opportunity cost of
using a resource unit today rather than conserving it for the future. This gives
rise to a resource rent in the extraction activity.
Secondly, due to property rights and limited access, oligopolistic behaviour
typically prevails in resource markets. This leads to a monopoly rent, i.e. an
additional mark-up. It is difficult to separate empirically between the various
kinds of “rents” in resource markets. We shall discuss imperfect competition in.
A third feature of natural resource markets is uncertainty. For instance,
individual agents have to make assessments of remaining reserves of the
resource. Demand shifts and intertemporal adjustments in such markets
complicate market equilibrium and induce uncertainty regarding the actual
behaviour and functioning of the market. We shall look at some aspects of
uncertainty.
We shall look at taxation of non-renewable resources from the viewpoint of
whether tax rules interfere with optimal depletion.
We shall also look at climate issues related to nonrenewable resources.
When to extract a natural resource
Let us start the simplest possible way. Let the resource owner have one unit of
the resource with the cost of extracting equal to b and his concern is to maximize
the present value of his asset. The ban gives the resource owner an interest rate r
and he thus takes r as his rate of discount. When is it optimal to extract the
resource unit?
Let us for the moment assume that the price is given and constant. What then?
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. Why?
It can be shown that the resource should be extracted when the price path
reaches the trigger price
p*  rb /(r   )
Why? Maximizing ( p0e t  b )e rt wrt. t and solving for the optimal t=t*, we find
that this is when the rate of increase of the net price (which we shall also
denote as the resource rent qt  pt  b ) is equal to the interest rate r , i.e.
d ( pt  b ) / dt
r
pt  b
Before the “trigger price”, p* is reached, the return on postponing extraction
exceeds the discount rate, after t* the return is less than the interest rate.
The Hotelling rule derived from profit maximization when unit costs
are constant
Let S0 be the total resource stock in the economy and Rt the total extraction at
time t. We assume constant unit costs equal to b . The market price is pt . The
resource market consists of many small agents each owning a small part of the
resource stock an taking the price as given.
The optimal extraction path for the society as a whole is found by maximizing
(with T free)
T
[ p R
t
 bRt ]e  rt dt , Rt  0
0
with respect to Rt, over a possibly infinite time horizon [0, T ] , constrained by the
condition
T
S0 given & St   Rt or
 R dt  S
t
0
0
The Hamiltonian is
H (t , St , Rt , t )  [ pt Rt  bRt ]e  rt  t Rt
The first order condition is
H / R  0  ( pt  b )e  rt  t  0
or, more completely,
[ pt  b ]e rt  t (=t for Rt  0)
The shadow price of the resource stock turns out to be constant.
&
t  H / S  0 , i.e. t  0  
For positive extraction, i.e.
pt  b  qt   e rt
This is the Hotelling rule in the simplest form: along the optimal path the net
price, i.e. the resource rent, should increase at the rate of discount.
The interpretation of the Hotelling condition as a market equilibrium condition.
All resource owners are indifferent wrt. time of depletion. The price profile is
given but its absolute level must still be determined, say by determining p0 .
The length of the horizon is still undetermined.
The Demand Side
We make three alternative stylized assumptions about the demand side.
1) Let assume that the demand for the resource is given by a (linear) demand
function, say
pt  d ( Rt )     Rt
Demand falls to zero when price at T reaches pT  p max   , which we shall call
the choke price. The transversality condition yields [ p0  b ]erT  [ p max  b ] . Solving
the demand function wrt. Rt and using the expression
pt  b  ( p0  b )e rt  b  ( p max  b )e r (t T )
the resource constraint gives us one equation in one unknown, namely T.
2) We assume that at price equal to c a perfect substitute is available. We call
this a backstop assumption.
3) We assume that the demand function has constant elasticity, i.e. Rt  pt
Shift analysis in the simple Hotelling model
We have found characteristic properties of the Hotelling price path and how
knowledge about the demand side helps us to determine the exact location of
the price path.
Let us consider the impact on the price path of changes in
 The discount rate
 The unit cost
 The amount of resource
 Demand shifts
Depletion with stock dependent costs
For many exhaustible natural resources a realistic description implies increasing
extraction costs as the stock is depleted. The resource deposit may consist of a
layers of different quality, and more costly to extract from deeper layers. A cost
function representation of such cases might be
b  b( Rt , St ), bR  0, bS  0
Maximizing discounted profits is in this case expressed as
T
Max  [ pRt  b( Rt , St )]e  rt dt
Rt
0
By solving we find that the resource rent must obey

q&t p& b&
b
R

r S
qt p  bR
p  bR
As bS  0 , we get q&t / 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 above.
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.
Constant, but different extraction costs
Above we assumed price taking firms with homogenous and identical resource
stocks and cost structure. In practice, extraction costs may vary. With
heterogeneous producers and stocks, an interesting question relates to the
succession of depletion of the various resource deposits.
Assume two resource owners, with unit costs b1 and b2 , respectively. From the
Hotelling Rule derivation it should be clear that market equilibrium must fulfill
the following relations:
pt  bi  e rt i (i  1, 2)
where λi is the shadow price related to the resource stock of producer i. As
before, the strict inequality prevails for Ri  0 .
With constant unit costs the equilibrium requires sequential extraction of each
resource stock. Assume e.g. that we have simultaneously R1 , R2  0 , implying
b1  e rt 1  b2  e rt 2 
b1  b2  (2  1 )e rt
What can we deduce from this?
At first, in [0, T1 ] , the market is supplied by the cheapest resource (say producer
1). Producer group 2 with the higher costs reaps a return on leaving the resource
stock in the ground that, since we have
p&t
p&t

r
pt  b2 pt  b1
The price path is continuous, but with a "kink" in T1, when producer 2 "takes
over" the market.