2
(d) Explain the economic logic underlying the Hotelling Rule?
(e) Assume a production cost of $c per unit of the resource extracted, as in (c). To find the values of the
four variables identified in (a), we need four equations. These equations are (i) the two demand functions,
(ii) the Hotelling rule, and (iii) an adding up condition that the total extracted in the two periods just
equals the total stock. Write down those four equations.
(f) Assume that c = $3. Solve for the equilibrium values of P
1
, P
2
, Q
1
and Q
2
.
(g) In this solution, does price rise at the rate of the interest rate? If not, why not?
(h) In this solution, does the quantity extracted decline over time?
(i) What is the discounted present value of the profit over the two periods?
4. [Lecture 3-31]
(a) If an industry owning a non-renewable resource acts monopolistically, how does the time pattern of
resource price, p
t
, differ from that when the industry acts as a price-taking, competitive industry? How
does the time pattern of extraction, y
t
, differ? What about the date when all of the resource has been
extracted – does that differ? How about the total discounted present value of profit to the industry over
the life of the resource – does that differ?
(b) A non-renewable resource firms owns two deposits of copper, where the potential amount of copper in
the deposits are the same but one is located at a shallower depth and has a lower cost of extraction. How
would you exploit those two resources? Would you extract some copper from each deposit every year, or
would you do something different? Describe the extraction strategy you would employ, and the
underlying economic rationale.
(c) In the context of a non-renewable energy resource, what is a “backstop” technology? How does the
existence of a backstop technology for generating energy affect the price today of coal?
[Lecture 4-2, slides33-38]
(d) Has the world price of oil risen, from 1950 to now, approximately at the interest rate, as a simple
version of the Hotelling rule would suggest?
(e) If not, what is an economic explanation of the price of oil over the last 15 years?
[Lecture 4-7 & 12]
5. There is a fishery for sardines in Monterey Bay. The growth equation for change in the stock of
biomass over time, S
t+1
– S
t
,, is given by the equation
1
( )
0.2
1
500
t
t
t
t
t
S
S
S
G S
S
Let H
t
denote the harvest in year t.
(a) When S
t
= 250, what is the annual growth in the stock?
When S
t
= 300, what is the annual growth?
When S
t
= 350, what is the annual growth?
When S
t
= 400, what is the annual growth?
When S
t
= 450, what is the annual growth?
When S
t
= 500, what is the annual growth?
(b) What is the carrying capacity of the resource?