Natural Resource Theory
Copyright, 1998 by Peter Berck
Introduction
Natural Resource Theory is the economic
theory of exhaustible and renewable
resources. These resources last for more than
one period of time and so function as a type
of capital. They are also used for food, fiber
and energy and so function as ordinary goods
Exhaustible Resource
• Oil, Coal, etc.
• Old growth trees
• These provide value by being used up.
Hotelling’s Model: 3 Equations
• 1. Capital Market Equilibrium
• 2. Feasibility
• 3. Flow Market Equilibria.
The Capital Market
• Assets Earn a return through
• Dividends
• Dividends paid out from earnings of firms.
• Car factory adds value to steel, labor etc.
• Capital Gains
• Stock can also have capital gain: its price goes up.
• All Exhaustible Nat Resource returns must come from
price change
Price Goes Up a Rate of Interest
• Hotelling’s Rule
• Price of resource is P(t)
• price at time 1 is P(1). (e.g. $700/th bd ft for
redwood.)
Put P(1) $ in bank
Buy one unit of
Resource
Period 1
have $P(1)
have $P(1) worth
of resource
Period 2
have $(1+r) P(1)
have $P(2) worth of
resource
so…
• When is it a good idea to by the resource?
• when p(2) >= (1+r) p(1)
• If >, then everyone would want the resource and
nobody would want anything else in their
portfolio.
• If <, then nobody would want the resource
• So p(2) = (1+r) p(1)
• And in general P(t+1) = P(t) (1+r).
Real Old Growth Redwood Price
450
400
350
$/MBF
300
250
200
150
100
50
0
53
55
57
59
61
63
65
67 69
year
71
73
75
77
79
81
83
Use no more than there is
• Second, the sum of the stumpage cut, q(t),
over time equals the original stock of
stumpage,
X = Q0 + Q1 + . . . + QT.
Demand and Supply
• D(p, h) is the demand when price is p and
some shift variable (housing starts if this is
oldgrowth stumpage) is h.
• (3) Q(t) = D(p(t) , h).
•
Solving the Model
•
•
•
•
p = p0(1+r)t, where p0 is initial price
Q(t) = D(p(t) , h).
SO
Q(t) = D(p0(1+r)t, h).
X
 D( p (1  r ) , h)
t
t o ,T
0
Left Over
• Why is it equality?
• Why not >
• Why not <
• Not so far fetched. Suppose global warming bites
and we give up coal mining. What will coal then
be worth?
X
 D( p (1  r ) , h)
t
t o ,T
0
The solution
•So all we need to do is to find p(0)
•and possibly T if its not infinity
•Then we will know p and Q for every time.
X

t o ,T
D( p0 (1  r )t , h)
Backstop
• Linear demand curve (and any other one that
hits the axis) has a choke price.
• Choke price is the price that chokes off
demand for the resource.
• At choke price some other technology is used
to meet demand (e.g. coal instead of oil.)
• P0 (1+r)T= Choke
• If you know p0, and choke, you know T,
exhaustion time.
Hotelling in 4-Quadrants
•Note: choke price, T.
demand
p
P0(1+r)t
p0
q
450 line
t
q
Hotelling in 4-Quadrants
demand
p
P0 ert
p0
q
450 line
t
q
Equilibrium: Green is X(0)
•Area under curve is sum of all Q’s.
demand
p
P0 ert
p0
q
450 line
t
q
A choice of Po below
the equilibrium value
leads to more q in each
period, which is more
than X(0) byP0the
ert red.
Too Low a P0
demand
p
p0
q
450 line
p0
t
q
Increase in r
Red Bounded area must
p
equal green area
Initial price lower
T sooner
Two Price-time paths
p0
must
q cross
450 line
P0 ert
p0
t
q
Taking of the Redwood Park
• In 1968 and again in 1978 the US took a total
of 3.1 billion bd ft of standing timber from
private companies to form the Redwood
National Park
• The amount by which the price of Redwood
went up as a result of the take is called
enhancement
Enhancement: Lowering X(0)
p
Red price path is
result of red X(0)
Arrow shows size
of enhancement
p0
P0 ert
p0
q
450 line
t
q
Recall
X t  
 D( p (1  r ) , h)
j
j t ,T
0
•Lets call the solution to this P(x(t)), the price
•When stock is x at time t.
•P(x(0) )= p0
•P(x(t)) = p0 (1+r)t
Folded Diagram Model
• p(x(t)) = p0(1+r)t
•price as function of stock is same as price as function of time
•price in year t + 1 is just p(x(t) – q(t)) which is also p0(1+r)(t+1)
n
•p(x(t) –  q(t ) ) = p0(1+r)(t+n)
t
•price after n years of cutting equals the price at
• time t ,(p0(1+r)t) times the interest factor nfor n years (1+r) n.
• Choose n so that the Park taking equals  q(t )
t
Enhancement: Years Method
p
X(0) is again red area.
Arrow shows number ofp
0
years need to wait to
find equivalent
P0 ert
p0
q
450 line
t
q
Value of Enhancement
• The 1978 Park taking was 1.4 billion board
feet, which is the equivalent of 2.26 years of
cutting.
• price 1978, was $311 per MBF.
• real interest rate—7 percent
• 2.26 years at 7 percent real per year or
17 percent of price
Conclusion
• Gov’t paid $689 million for second take
• enhancement was $583 million
• Therefore the US paid nearly twice for the
park
Continuous time example
• Interest is exp(-rt) rather than (1+r)-t
• Integral replaces summation
• Demand function has no choke price (makes it
easier)
Example: Q = p-

x  0    Q(p 0 e rt )dt.
0

x0 p e
-
0
- rt
dt.
0
- rt
- e
x0  p 0
 r

0
p -0

r
Example Concluded:
Reduced Form
- rt
e
x0  p
 r
-
0


0
 1 

p 0  
  r x0  
p
-
0
r
1


ln(  )  ln( r )  ln( x(0) 

ln(p 0 )   1