Download Solutions Seminar1 Exercise 1 – Competitive market with

Solutions Seminar1
Exercise 1 – Competitive market with exponential demand
Consider a competitive resource market with a total stock S0 of a non-renewable resource. The stock
is distributed among resource owners who pursue profit maximization and have the same unit cost b.
The rate of discount is r. What characterizes the market price path? Let the demand function for the
resource be Rt  pt i.e. constant elasticity , >1. Explain how the depletion profile is determined.
Discuss the effect on the price and depletion profile of
a. an increase in the rate of discount;
b. an increase in the initial size of resource:
c. an increase in .
Reconsider the problem in exercise 2 when there is a backstop technology, that is perfect substitute
available at cost c = p*. Consider also the effect on the price and depletion profile of a decrease in c.
Assume in a further elaboration of this problem that the backstop technology does not become
available until t1. This is known by all market participants at t = 0.
Solution
Here we solve the standard simple Hotelling problem as in lecture 3 and find the shape of the price
curve as determined by the Hotelling Rule. With a demand function with a choke point we can easily
express pt as a function of the choke price and the unknown (free) horizon T. The resource constraint
equation will then determine T. But with a demand side without choke point such as the constant
elasticity demand function given here, we cannot use the same method to anchor the price. Instead
we express pt as a function of p0 and use the resource constraint to determine p0.
By reasoning directly (and counterfactually) we easily find that a) requires a lower initial price (higher
depletion) and a price curve cutting through the original one.
b) requires a lower initial price and the new price curve will accordingly be lower (and depletion
higher) for all t.
c) represents a negative shift in demand and works out to be similar to b).
If there is a backstop the final price will be the backstop cost at the free horizon. Then pt can be
expressed in terms of the backstop price and T just as for the case with a choke price. Generally the
existence of a backstop substitute will push the price path down (and depletion up).
But if the backstop arrives at a given, known date the problem becomes a little more complicated.
No resource producer can then produce after t1 at a higher price than p*. Generally if the amount of
resource is “small” the production will terminate at t1 with price above p*, while if the amount of
resource is “large”, the terminal price will be p* and we can just disregard that the backstop
substitute does not become available until t1.
Exercise 2 – Competitive market with linear demand; shifts in r and b
Suppose there is a known initial stock S0 of a non-renewable resource, which can be extracted at
constant unit costs b. The demand schedule for the depleted resource, R, is linear with negative
slope and invariant over time. The interest rate r is constant over time. Production is undertaken by
competitive firms with well-defined property rights to deposits. What characterizes the time path of
the price for the depleted resource? Explain the economics of your answer!
a) Explain how the initial price, p0, is determined from the parameters of the problem.
Discuss how p0 will change by hypothetical shifts in r and b, respectively.
b) Consider negative shifts in both r and b, such that p0 remains the same as before. How
will the price path, resulting from these shifts, differ from the original.
Solution
A confident students may go straight to the Hotelling Rule, but a more elaborate approach would be
to formulate the profit maximization problem as a Hotelling control theoretic problem a sin lecture 1,
trying to get everything right. The price path should be described or explained, which is most
conveniently done by using an explicit expression for the demand side and setting out the expression
for the price as a function of the choke price and the length of the depletion period. The key
characterization is of course that the net price increases by rate r. For the economics of the answer it
is to look at the depletion process from a capital asset equilibrium point of view, requiring that the
return is the same for resource-in-the-ground and financial assets.
To determine the initial price requires knowledge of the length of the depletion period and that can
only be determined by using the resourse constraint, as sketched in lecture 1.
A negative shift in r increases p0 because it means that the net price increases at a lower rate and an
unchanged p0 would imply that the total demand would be higher than the original case, i.e. a
contradiction. Thus the new p0 must be higher and the new price path cut through the original one
from above. A negative shift in b decreases p0 (but less than the shift in b), because unchanged p0
would imply a price path above the original one and thus lower totals demand. Hence, the new p0
must be lower and the new price path cut through the original one from below. (If p0 shifted as much
or more as b, the price paths would not cut and the resource constraint cannot hold in both cases.
It follows from the argument that a (small) negative change in r can be counteracted by a suitable
negative change in b such that p0 unchanged. In other words for a given small change r <0 there
must a corresponding b  0 such that p0 is unchanged.
We thus have two price paths starting at p0, namely the original path pˆ t ,
pˆ t  b  ( p0  b)e rt
and the new one,
pt  b  b  ( p0  b  b)e( r r )t
So far it should be easy for all who have got basic knowledge about the Hotelling theory. But now
what, will the new path differ from the original one and in what way?
It may seem tempting to guess that as the starting point is the same, the price paths will be
the same throughout. But that would to give up too early. Just by inspection of the two expressions
above one can easily conclude that they are not mathematical identical and thus represent different
paths.
In what way the two expressions above are different is easy to answer if approached in the right way,
but difficult if one gets entangled in the mathematics. As soon as one has recognized that the paths
must be different one can use the resource constraint argument to prove that there must be (at least)
another crossing point. The new path cannot be wholly above or below the original one.
As each expression is a constant plus an exponential factor, eventually, the expression with the
highest exponential factor will grow quickest. Hence, with a little mathematical intuition one could
argue that the new path must increase quicker than the original one to begin with, but eventually the
original path will cut through it from below and reach the choke point earlier than the new path.
Another way to argue intuitively is that the cost reduction which goes directly into a corresponding
increase in resource rent will by the Hotelling rule push up the price to begin with, even with a little
smaller discount rate. Hence, the new price path increase at a faster pace to begin with.
These argument are good, but if one is highly matematically inclined one might look for convincing
mathematical corroboration. To attack the problem mathematically, one runs the risk, as the founder
of our department put it many years ago, of just chasing one variable at the time to the left-hand
side of the equation. It is tempting to try to work out the trade-off between decrease in r and
decrease in b. The hard way to approach this problem is to start with the equations representing
p0  b  ( pT  b)e  (T t )
price in terms of the choke price
T

and the resource constraint S  Rt dt . Specifying the invariant linear demand function as
0
pt     Rt the resource constraint can be solved explicitly as
S
 b
 (rT  1  e rT )
r
Then the price path equation with  instead of pT can be solved wrt. T and used to write the
resource constraint as function of r, b og p0, men ikke av T. The equation thus arrived at is the right
one for finding r / b , but this is neither necessary nor sufficient to answer the question
convincingly. Remember you cannot differentiate the expression for the original price path directly
wrt. b or r without also differentiating to get the effect via T, and the expression then become very
untractable.)
In fact, the answer by means of mathematics can be found in a deceptively simple way. Let
us calculate growth along the new price path, relative to the original one
p&t ( p0  b  b)(r  r )e( r r )t
b
r

 (1 
)(1  )  ert
rt
&
( p0  b)re
p0  b
r
pˆ t
For given b and r the first two factors in this expression are just numbers. The third factor
declines over time from being 1 for t  0 . As we have tried to look into r / b , we don’t know the
magnitude of the product of the first two factors. We see that one is bigger than one and one is
smaller than one. Let us assume for a moment that the product is equal to one.
In that case the whole expression states that the growth of the price along the new path is always
less than along the original path. In other words the new price path is in its entirety below the
original one. This cannot be consistent with the resource constraint! If the product of the first two
factors is less than one, the same situation arises. Hence, by invincible logic, the product of the first
two factors must be bigger than one! (This can be interpreted as for the changes in b and r to give
unchanged p0, the change in b must in this sense be bigger than the change in r.)
It then follows that the new path increases more to begin with. The exponential factor will however
counter this difference and reverse it such that the price path sooner or later crosses. If they didn’t
the resource constraint would not hold.
Simple, isn’t it!
Exercise 3 - Gray’s problem
Consider a problem discussed by L.C. Gray in 1914 (QJ E 28, 466-489):
The resource considered is coal. A resource owner has a known total stock of coal given at the outset
as S . The resource owner takes the price, p , as given and constant. 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 ) . The cost
function has the usual properties, such as a U-shaped average cost function. The rate of interest is r.
Solve the problem of the determining the resource owner’s optimal extraction program. Try to give an
intuitive answer before embarking on a formal solution. Give economic interpretation of your solution.
Reconsider Gray’s problem for the case when the extraction cost instead of b(Rt) is b(Rt,St)=c(St)Rt
with c’(St)<0, thus the marginal extraction cost is a function of the remaining stock, but independent
of the rate of depletion. In other words, the extraction unit cost is “constant, but increasing”.
Solution
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. Gray’s problem can thus be stated as
T
(0.1)
given
Max  [ pRt  b( Rt )]e  rt dt
Rt ,T
0
T
 R dt  S
(0.2)
t
with Rt  0
0
How can we solve this problem? How can we reason intuitively to find out how much is it profitable
to extract per unit of time (year)? At what cost level should we expect the optimal depletion to be
when the stock of extracted resource is very large, and when it is very small? Can we 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 as stated in (0.1) is also to determine the length of the depletion period. In
mathematical terms, (0.1) with condition (0.2) is an optimal control problem that can be solved by the
textbook theorem. Let us first try to give an intuitive explanation for conditions that must be fulfilled
along the optimal extraction path.
The problem, as stated in Gray (1914), 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? Another idea would perhaps be to deplete 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?
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 (we define the net marginal price as qt  p  b( Rt ) ):
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 .
But how can the resource rent vary when the price is constant? The answer is: 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.
(0.3)
q&/ q  r
(0.3) 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 (0.3) 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 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 problem of Gray (1914) can, as mentioned, be stated 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 is constant over time,
pt  p . The rate of discount is r. The problem is to find the depletion profile {Rt*} .
T
(0.4)
Max  [ pRt  b( Rt )]e  rt dt
Rt
0
S&
t   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
shadow price, t, of the state variable and its rate of change.
(0.5)
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*} 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
(0.6)
H
 [ p  b( Rt )]e  rt  t  0
Rt
Furthermore, the rate of change of the shadow price is given by
(0.7)
&
t  
H
0
S
From this follows that t is constant, t   .
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
(0.8)
HC (t , St , Rt , t )  p  Rt  b( Rt )  t  Rt
The first order conditions now becomes
(0.9)
H C
 p  b( Rt )  t  0
Rt
and
& r  
(0.10)
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
H (T *, ST * , RT * , T * )  0
(0.11)
Or, in current value terms,
HC (T * , ST * , RT * , T * )  0
(0.12)
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
(0.13) b( RT * )  b( RT * ) / RT *
For the additional question we note that there is now no reason to delay production, no cost can be
saved by doing that, and there is no price increase to wait for. It is only a questionof how much to
produce. The answer would be to produce nothing if
c( S )  p
produce everything now of
c(0)  p
and in the remaining case, one can easily find that the optimal amount to leave in the ground is given
by
c( S # )  p
Exercise 4 - Monopoly
A non-renewable resource is available at the initial level S0 (at time zero) and extraction costs are
zero. The demand for the resource is D(pt) where pt denotes the resource price at time t. The
demand function is given as
D( p)  p 
with  > 0.
At a future date t1 a perfect substitute becomes available at a constant unit cost c. The substitute is
supplied competitively. Both c and t1 are known at t=0.
1. Assume   1 . What is the price and extraction path when the resource is owned by a
monopoly (assuming there is no limit on the production of the substitute) when (a) t1=0, (b)
t1   , (c) t1 is positive and finite.
Assume that   1 , t1=0, and that the substitute can only be supplied up to a limit L  c . Try to
derive the monopolist’s optimal extraction and price path.
Solution
re 1 (a)
t1=0 means that we have the standard monopoly model with backstop, i.e. the two-phased solution.
1
From maximizing the Hamiltonian in current values we get the first-order condition pt (1  )  t

and also that t  0 e .
rt
The transversality condition is H c  0 , i.e. pT RT  T RT . As we know that RT must be non-zero (in
fact RT  p  ) this implies that pT  T . Phase 1 lasts until the price starting at p0 hits the
1
backstop ceiling c. Let us call the length of phase 1 for T1. The value of T1 is c(1  ) . The length of

1
the second limit-pricing phase, denoted T2, is the time it takes for  to grow from c(1  ) to c, i.e.

T2 is determined by
1
1
1
c(1  )e rT2  c , or T2  ln(1  ) . We may note in passing that the length of the second phase is

r

under the given assumptions independent of the initial amount of the resource. We can easily work
out how much stock is depleted in the second phase, namely T2  c  . If we deduct this amount from
the initial stock S0 we get the amount to be depleted in the first phase, S1. We can then finally
determine T1 by the resource constraint
T1
S1   (ce r (T1 t ) ) dt
0
p0 follows easily when T1 is known. The depletion profile can then be expressed in exact terms from
the demand function.
re 1 (b)
t1   means that we have the standard monopoly problem of the simplest kind. Price increase at
rate r, the depletion decreases at rate  r . The initial price is determined from the resource
constraint integral, which even can be solved analytically.
re 1 (c)
0  t1   . This intermediate case may seem a little tricky, but can be easily sorted out. Generally,
for problems with “large” resource stock the solution 1 (c) applies, we can just ignore that the
backstop is not available from the beginning. (It may not be obvious at the outset, but nevertheless
true, that it is not more profitable to sell part of the stock at high prices before the backstop arrives.)
For “small” stocks the problem is that of maximizing profit at a fixed horizon equal to t1. The tricky
middle case here is when the stock is such that the terminal price (of course bigger than c) for the
fixed horizon problem is low enough for the marginal revenue to be below c. In that case the optimal
solution is a price path hitting the vertical barrier at a price higher than c followed by a limit pricing
period such that the  reaches c at the terminal point.
re 2
Here we have inelastic at least below c. Hence it can never be optimal to produce at a price lower
than c. Hence, we can reasonably assume an initial period of limit pricing (at least unless the price is
very small).
Thus also here we have two phases. What then about phase 2?
When the price is above the competitive backstop producers will always be able to unload their total
production L. The residual demand directed towards the monopoly and is thus Rt  pt  L . We

1
notice that this residual demand has a chokepoint, namely for p*  L  . The elasticity of this
demand wrt. price (as a positive number) is
ˆ 
We find that ˆ  1 for Rt 

1

1 
pt
R L
  t

pt  L
Rt
L , i.e. for a total demand of
L
. Hence for lower demand, i.e.
1 
1  1
) (if this is higher than c) and up to the choke point the residual
1 
for prices from pt  L  (
demand is elastic and the monopoly will sell along this price path. The total stock that can be sold
along this price path can be deducted from the total stock to determine how much of the stock that
has to be depleted in the phase 1 and hence the length of this phase.