The Hotelling Model

Question 1

The hotelling model (harold hotelling)

Answer 1

Depletion of a fixed stock over time (producers want to choose the depletion path which will maximise returns)
present value of returns from production in each time period will equalised (profits cannot be increased further) - long term prices should therefore increase per year
only produce a limited supply of their product if it will yield more than bonds or interest bearing instruments
initially costs are constant (AC= MC)
P0 - C0 = P1 - C1 / (1 + r)^1 (max returns)
to maintain equalities net prices will rise at the rate of discount
prices rise at the discount rate
(P - C) increases until the stock is exhausted
if costs are not changed, prices will continue to rise - demand falls as price goes up
deposits are exhausted when Q demanded = 0
Pt - Ct = (Pi - Ci) (1 + r)^T shows present value
V0 = Ri / (1 + r)^i
Pi = P0 (1 + g)^i
V0 = R0 in equilibrium
r > g oil earns less than the market rate of return investor will sell more oil (increases g over time)
if g < r will sell less (to reduce g over time)
R = resource rent or royalty - rent grows at the same rate as the discount rate

Question 2

Alternative model

Answer 2

competitive source of renewable resources (backstop) technology a close subsitute e.g. solar panels
rent in the initial period is the price of the backstop technology minus production cost discounted)

Question 3

Effects of a decline in backstop price

Answer 3

when backstop starts competing with the price of oil
shortens the planning horizon - accelerates extraction and lowers spot prices (some of the oil resource remains unexploited)
life of oil deposits is reduced
demand goes up now exhuasted earlier
FOLLOWS ORIGINAL PRICE PATH (equalises returns)

Question 4

Frequent discoveries/increase in resource stock

Answer 4

P = price path when no discoveries
one discovery does not affect total
extend the time to exhaustion or the time to the displacement of oil by the backstop technology
decrease price now and extend period by which price = that of backstop technology
price drops
steeper as rent increases in the initial period

Question 5

Path of resource price over time

Answer 5

discoveries come unexpectedly

huge discoveries price may not rise at all

Question 6

Fall in extraction costs

Answer 6

when price falls consumption rises (less left over so will reach depletion earlier) e.g. in the north sea
discount is unchanged

Question 7

Changes in demand

Answer 7

An increase in demand so new path moves to the right

price path is now above original (no change in reserves) higher production levels - exhaust resource earlier

Question 8

Monopoly

Answer 8

hotelling model is an example of a competitive market
depletion will be at a socially optimal rate
monopoly price is higher so price path is less steep extends the life of reserve (producer maxes returns)

Question 9

Increased discount rate

Answer 9

changes whole path
initially p is reduced as a consequence, this entails a higher growth rate (steeper)
more rapid exhaustion
lower price so more demand then adopt a higher price to discourage demand
Owners seek to secure benefits earlier as future income is less valuable
this result is uncertain - higher r discourages consumption (deter new fields)
existence of external effects such as pollution imply that social rates of depletion are lower than private