Seminar Solutions
ECON4925 Resource Economics, Autumn 2011
Seminar 4: Hotelling 2
November 3, 2011
1
Exercise Part 1 – Iso-elastic demand
A non renewable resource of a known stock S0 can be extracted at zero cost. Demand
for the resource is of the form: D(pt ) = p−ε
t
ε > 0 At a known future date t = s a com-
petitively supplied perfect substitute becomes available. Its production causes constant
unit costs c.
1.1
Competitive market resource market
(1) Derive the time paths for resource extraction and price when the resource market is
perfectly competitive and:
(a) s = 0
(b) s → ∞.
I understand the case (a) as that the backstop is always there, so that the inverse
demand function has a kink at p = c, and I understand the case (b) as that the backstop
is never there, so that there is always some demand. In other words, even as p → ∞,
demand will approach, but never actually reach zero. It is useful to start deriving the
price path for this case.
Backstop is never there
The problem for a supplier of the non-renewable resource under competitive market
conditions is:
Z
T
pt Rt e−rt dt
mx
Rt ,T
0
s.t. − Rt = S˙t
S0
given
Rt ≥ 0
ST ≥ 0
T
free
1
∀t
Seminar Solutions
ECON4925 Resource Economics, Autumn 2011
Solve by maximum principle, using current-value Hamiltonian H = (pt − μt )Rt . Necessary
conditions for optimality include:
HR t = 0
pt = μt
⇔
μ˙t = rμt − HSt
μT ≥ 0
(= 0
⇔
if
(A)
μ˙t
μt
=r
(B)
ST > 0)
From (B) we know that μt = μ0 ert and hence pt = p0 ert So we know that the price is
rising at rate r. Moreover, we know that it will rise without bounds. What we don’t know
yet is the initial level from which it starts, p0 .
In order to pin down the initial price level, we can use the condition that the entire
resource will be used up. Technically, this follows from the (C). Intuitively, it cannot be
optimal to leave something of value in the ground. Since there are no extraction costs
and the price can rise without bound, every unit of the resource is valuable. So, we have:
S0 =
∞
Z
Rt dt
(1)
0
= [p0 ert ]−ϵ :
Now, use the market equilibrium condition: Rt = D(pt ) = p−ϵ
t
S0 =
∞
Z
[p0 ert ]−ϵ dt
(2)
0

S0 = 
S0 =
p−ϵ
0
−ϵr
p−ϵ
0
ϵr
∞
e
−ϵrt 
(3)
0
1
p0 = (ϵrS0 )− ϵ
⇔
(4)
Backstop from beginning
Let us now turn to the case when there is a backstop-technology from the beginning.
The first-order conditions will be the same. Only now, there is an upper bound to the
price that the producer can ask. Due to the presence of the backstop technology, the
resource owners cannot sell anything at a price above p = c. Again, (A) and (B) imply
that the price rises at rate r, and hence it will eventually hit the level c. What we have
to find, is the point in time when this happens, and the (new) initial level that the price
starts from. But first, we should argue that indeed the price reaches the level c at time
T when the last unit of the resource is extracted.
It cannot be optimal to exhaust the resource before the price has reached c (a higher
price path resulting in more profits is feasible). But it can also not be optimal to have
resource stock remaining after the price has reached c (these resource units can only be
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Seminar Solutions
ECON4925 Resource Economics, Autumn 2011
sold at the price c, but due to discounting, their present value is declining the later they
are sold). Hence, as before
S0 =
T
Z
Rt dt
(5)
0
Note however that now T is finite. To pin down T, we can use the fact that we know
pT = c. Once we know T, we know p0 since:
pt = p0 ert = pT e−r(T−t)
pT = p0 erT = c
⇔
⇔
p0 = ce−rT
So, again using the market equilibrium condition Rt = D(pt ) and inserting it in (5), we
get an equation with T as the only unknown variable:
S0 =
Z
T
”
ce−r(T−t)
—−ϵ
dt
0
So, T and consequently p0 are uniquely defined.1 What is left is to argue that the initial
price p0 will be lower than in the case with no backstop, as the same amount of resource
is supplied to the market, only during a shorter time, so at each point in time demand
has to be more, and hence price has to be lower.
1.2
Monopolist
Now turning to the monopolist, it will again be pedagogical to first consider the case
when the backstop is never there, and then to see how the existence of a backstop
changes things. The essence of monopolistic behavior is to try to extract rents by suppressing supply. However, in the non-renewable resource case, total supply is given by
RT
the resource constraint S0 ≥ 0 Rt dt, and the only thing that the monopolist can do is to
shift supply over time, from relatively price-elastic periods to relatively less price-elastic
periods.
1 In
fact, closed form solutions can be found: T =
1
ϵr
ln
3
€
ϵr
c−ϵ
S0 + 1
Š
Seminar Solutions
ECON4925 Resource Economics, Autumn 2011
backstop never there
Formally, the monopolists problem is:
T
Z
pt (Rt )Rt e−rt dt
mx
pt ,T
(6a)
0
s.t. − Rt = S˙t
S0
given
(6b)
(6c)
ST ≥ 0
(6d)
T
(6e)
free
Rt =
p−ε
t
(6f)
For a change, we will use the present value Hamiltonian H = pt (Rt )Rt e−rt − λt Rt . The
corresponding first order conditions include:
HRt = (pt + p0t Rt )e−rt − λt = 0
−HSt = 0 = λ̇
(A’)
(B’)
Again (and for the same reasons) it is true that all resources will be used and that extraction will go on forever. From inserting the demand function in (A’) we get:
(pt + p0t Rt )e−rt = λ
(pt + p0t Rt ) = (p0 + p00 R0 )ert
‚
Œ
1 − 1ε −1
1 − 1ε −1
− 1ε
− 1ε
Rt = R0 − R0
R0 ert
Rt − Rt
ε
ε
‚
Œ
‚
Œ
1
1
− 1ε
− 1ε
Rt
1−
= R0
1−
ert
ε
ε
− 1ε
Rt
−1
= R0 ε ert
pt = p0 ert
Looks familiar? The reason why the monopoly follows the standard Hotelling rule is that
it can do no better. Demand elasticity is constant – hence shifting is of no meaning and
the monopolist is left with no power to play around with.
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Seminar Solutions
ECON4925 Resource Economics, Autumn 2011
backstop there from the beginning
When the backstop is there from the beginning, the problem (6) remains the same, only
condition (6f) is changed to:

Rt =
p−ε
t
for
pt < c

(0, c−ε )
for
pt = c

0
for
pt > c
(6f’)
The necessary conditions from maximizing the present-value Hamiltonian are:
HRt = (pt + p0t Rt )e−rt − λt ≤ 0
(< 0
if
Rt = c−ε )
(A”)
−HSt = 0 = λ̇
λT ≥ 0
(= 0
H(T ∗ ) = pT ∗ RT ∗ e
(B”)
ST > 0)
if
−rT ∗
− λT ∗ RT ∗ = 0
(transvers.)
Due to the changed constraint (6f’), the first-order-condition (A”) now features an inequality sign. For an interior solution we have – as before – that the marginal revenue
should rise at the rate of interest. But due to the backstop, this is no longer feasible
when the price would have to be above c. We have two cases:
(1) The initial stock of the resource is so low that the monopolist would want to set an
initial price above c. As this is not feasible, the best that the monopolist can do is to sell
all resource at price c, until it is empty, so that Rt = c−ϵ .
(2) The initial stock is “large” so that there will be an initial phase where (A”) holds
with equality, followed by a phase where the price stays at c and the monopolists supplied his resource until it is empty. Denote the end of the first phase, THot , marking the
end of Hotelling’s days. Hence there are now three unknowns: p0 , T and THot .
Start by using (transvers.). We know that pT = c and hence RT = c−ε > 0. We get:
c = λerT
(7)
From (A”) we also know that at THot it is the case that:
(pTHot + p0T
Hot
RTHot )e−rTHot = λ
(8)
Inserting for the demand function and the limit price:
– ‚
c 1−
1
ε
Ϊ
e−rTHot = λ
5
(9)
Seminar Solutions
ECON4925 Resource Economics, Autumn 2011
Putting (7) into (9) lets us solve for the length of the second phase (T − THot ):
– ‚
c= c 1−
Ϊ
e−rTHot erT
ϵ
ϵ
erT−THot =
T − THot
1
ϵ−1
1
ϵ
= ln
r
ϵ−1
(10)
The total amount extracted during this second phase is:
Smt = (T − THot )c−ϵ
Deducting this amount from the initial stock we can find the length of the ‘Hotelling
phase’ in the ‘ususal way’ by:
S0 − Smt =
=
=
Z
THot
Rt dt
0
Z THot
Z
p−ϵ
dt
t
0
THot
€
ce−r(THot −t)
Š−ϵ
dt
(11)
0
Hence, we have two equations for the two unknowns T (equation 10) and THot (equation
11) and since p0 = ce−rTHot , the problem is solved. The initial price with a backstop will be
higher than when the monopolist is unconstrained: As “such a maximal price depresses
the monopolist’s future price, some of this price reduction may be compensated for by
raising the the price early in the period of extraction” (Hoel, 1978, p.35).
The intermediate case: the backstop appears
Sketch of solution: Demand from the monopolist’s viewpoint takes the form:

−ε
 pt
Rt =
p−ε
 t
0
t≤s
for
for
pt ≤ c
t>s
for
pt > c
t>s
Similarly to 2(a), that means there is an additional constraint Rt ≥ c−ε , but now only
for t > s. How could we solve this? In two steps. First we solve the problem from t = s
onwards. This is almost the same as in 2(a), except that we do not yet know how much of
the resource there is initially (Ss ). So we solve the problem conditional on some unknown
Ss and get a function V(Ss ) describing what any size of stock is worth to the monopolist
at t = s.
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Seminar Solutions
ECON4925 Resource Economics, Autumn 2011
V(Ss ) = mx
pt ,T
Z
T
pt (Rt )Rt e−rt dt
(15)
s
As argued above, it might either be that there is only a limit pricing phase (Ss ‘small’)
or that there is a Hotelling phase followed by a limit pricing phase. In the former case
we have that marginal revenue at t = s is equal to c, in the latter to p0 ers (1 − 1ε ).
Next then, we solve the fixed horizon problem until t = s taking V(Ss ) into account.
V(Ss ) is a so called scrap value.
Z
s
pt (Rt )Rt e−rt dt + V(Ss )e−rs
mx
pt
0
s.t. − Rt = S˙t
S0
given
Ss ≥ 0
Rt = p−ε
t
The present value Hamiltonian looks as usual. But we get an additional condition for
the scrap value, namely:
λs ≥ V 0 (S∗
)
s
(= V 0
if
Ss > 0)
(16)
This is quite naturally interpreted: on the LHS we have what an additional unit of
the resource would be worth if used at s(−), on the RHS what it is marginally worth if
transferred – and those should be equal if anything is transferred. Otherwise, it must be
better to use everything now.
We know from the FOCs that λs = λ = (pt + p0t Rt )e−rt . The envelope theorem tells us
that V 0 (S∗
) = μ, where μ is the shadow price on the resource for the second problem
s
(t > s) which again from the FOC we know to be μ ≥ (pt + p0t Rt )e−rt .
Informally, two things can happen. (I) It is optimal to leave so much resource that
when t = s(−), ps(−) < c and hence we have the same solution as in 2(a) – that the
constraint comes later does not matter, there is nothing to gain from shifting production
to earlier periods. The second case (II) is ps(−) > c. In that case, we must have ps(+) = c
Here we can get two sub-cases: (IIA) suppose ps (1 −
1
)
ε
> c. Then marginal revenue
right before we reach t = s is higher than right after – there is no point in extracting
anything after that point in time. Or we get (IIB) that c(1 − 1ε ) < ps (1 − 1ε ) < c. In that
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Seminar Solutions
ECON4925 Resource Economics, Autumn 2011
case marginal revenue right before t = s is lower than right after and hence extraction
continues and price “jumps” down to c from something c < ps < c(1 − 1ε )−1
2
Exercise Part 2 – Pollution
This problem is very vague and specifies no model. Hence, do it yourself! The easiest
would be to take the case of no extraction cost and a choke price b. The main point with
question (1) is to recognize that the producers will not take the externality into account
RT
and hence produce as if there was no pollution. Since P˙t = Rt and 0 Rt dt = S0 we will
have that PT = S0 . All of the resource will be extracted and end up as pollution. The
main point with (2) is to recognize that avoiding P to go beyond P̄ essentially means
prohibiting that the entire resource is extracted, but leaving S̄ = S0 − P̄ in the ground.
This can e.g. be achieved by levying a tax of at least the size of the present value of
the resource rent at the time when Pt has reached P̄. The effect on the price path is the
same as a reduction of the initial reserve size: The new price path still grows at the rate
of interest, but from a higher starting point, and extraction ends earlier.
Using the model with stock depending costs makes your life a lot harder. An good
discussion of this case can be found in Hoel (2012, section 3 and 4).
References
Hoel, M. (1978). Resource extraction, substitute production, and monopoly. Journal of
Economic Theory, 19(1):28 – 37.
Hoel, M. (2012). Carbon taxes and the green paradox. In Hahn, R. and Ulph, A., editors, Climate Change and Common Sense: Essays in Honour of Tom Schelling. Oxford
University Press.
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