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Hotelling’s Rule
In what follows I will use the term “price” to denote unit profit. That is, the nominal
money price minus the average cost of production.
We begin with competition.
Suppose that a firm owns a small part, a, of the total amount of an exhaustible
resource. This small competitive firm can sell some or all at the current price, Po, in
a competitive market. Now consider a future time, say t. The price at that time will
be P(t) and we can assume it is known with certainty. Now competitors have an
arbitrage situation. Suppose that the future price is know with certainty to be P(t) >
Poert. It follows that anyone could borrow Po at interest rate r and buy a unit of the
resource on the open competitive market now. This person would then hold the
resource until future time t and then sell it for P(t) at time t. He would then return
the borrowed money Po and the interest (ert-1)Po and would make a riskless profit of
(P(t) - Poert). Since anyone can do this and there are no transactions costs in our
simple example, the price P(t) must equal Poert. If P(t) < Poert, a riskless profit can
again be made by anyone owning the resource now. The resource owner would
merely sell the resource now and get Po and then lend the Po in credit markets at
interest rate r. By time t, the value lent would have risen to Poert. The resource
could then be bought back at P(t) and the speculator would have a profit of (Poert.P(t)).
Therefore arbitrage forces P(t) = Poert.
This is the proof of the P(t) = Poert competitive pricing relation in Hotelling’s paper.
This pricing relation does not hold for a monopoly since the monopolist must consider
that the amount he sells may influence the price of the product. A competitive firm
does not have this problem since each firm is small compared to the total industry.
Note that Hotelling does not tell us (at this time) how that Po is determined.
For the firm, the total amount of the resource that can be extracted is equal to a. This
means that we have the constraint
T
T
T
0
0
0
a   qdt   q( P(t )) dt   q( Po e rt )dt
where T is when we have finished extracting the resource. Now, Hotelling rightly
notes that at time T, the quantity extracted is zero – there is no more that can be
extract at time T. If there was any resource left at time T, it could then be sold for a
profit and therefore T would not be the culminating time of extraction.
All this means that q(PoerT) = 0, which is an important boundary condition
Hotelling generalizes the demand function q(P(t)) to allow the demand to change over
time. He writes it as q(P(t),t), and therefore the terminal condition becomes
q(P(T),T) = 0
This equation above holds for any demand function q and for any path of extraction.
Hotelling takes the example of
q(P(t),t) = 5 – P(t) = 5 - Poert
which implies at time T that
Po = 5e-rT
Plugging the demand function into the integral constraint above gives
T
 {5  P e
rt
o
0
}dt  5T 
Po
(1  e rT )  a
r
which simplifies to
5T  a 
Po
5e  rT
5 5e  rT
(1  e rT )  5T  a 
(1  e rT )  5T  a  
0
r
r
r
r
This last equation can be written as
a
e  rT  1  r (  T )
5
Hotelling then draws a graph to show how that T can be calculate from this equation.
(see next page)
Hotelling’s analysis thus far showed that competitive pricing would lead to
P ' (t )
r
P
where this differential equation has a terminal condition given implicitly by
q( P(T ), T )  0
This determines Po and the future path of P(t).
The pricing relation in the box above is called Hotelling’s Rule.
Hotelling next considers the socially optimum rate of extraction of the resource. To
do this, he makes liberal use of the concept of utility being the area under a demand
curve. Clearly, this is only an approximation.
Hotelling’s Rule -- the Optimal Extraction of a Nonrenewable
Resource
Hotelling next considers the problem of determining the socially optimal rate of
extraction.
He begins by noting that the area under a demand curve is closely related to the total
utility enjoyed by consumers. This is because in equilibrium
Marginal Utility of X = (Price of X)*(Marginal Utility of Income)
And, if the Marginal Utility of Income is relatively constant, then
N
TU   Pi X i
i 1
and therefore
q
TU = u (q)   P( s)ds
0
in differentiable form.
Consider the maximization of intertemporal utility subject to the exhaustible resource
constraint.
T
Max  u(q(t ))e
q (t )
 rt
T
subject to
 q(t )dt .
0
The Lagrangian can now be written as
dt
0
T
T
L   u (q(t )e dt   {a   q(t )dt
 rt
0
0
The first order Euler Equation is
u  rt
e   0
q
but
u
 p(q(t )) by the definition of utility u(q) above. Therefore, we can write
q
this as
p(q(t ))e  rt    0
and thus,
p(t )  e rt
or
p (t )  p o e rt
This shows that the socially optimum rate of extraction and use of an exhaustible
resource is the same as that found under competition. Hotelling cautions that this
does not mean that competition is clearly optimal, since we have made many
unrealistic assumptions in deriving the result. However, this is something which
makes competition look very favorable compared to other types of ownership and
usage.
Hotelling continues on to analyze monopoly. It is not difficult to understand his
analysis. The student should try to calculate the optimal price and quantity that
results from a monopoly owning a fixed resource.
Assume demand is given by P = 5 – q, as before.