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 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) 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
Use no more than there is • Second, the sum of the stumpage cut, q(t), over time equals the original stock of stumpage, X = Q 0 + Q 1 +. . . + Q T.
Demand 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 = p 0(1+r)t, where p 0 is initial price Q(t) = D(p(t) , h). SO Q(t) = D(p 0(1+r)t, h).
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?
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.
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. ) • P 0 (1+r)T= Choke • If you know p 0, and choke, you know T, exhaustion time.
Hotelling in 4 -Quadrants • Note: choke price, T. demand p P 0(1+r)t p 0 q 450 line t q
Hotelling in 4 -Quadrants demand p P 0 ert p 0 q 450 line t q
Equilibrium: Green is X(0) • Area under curve is sum of all Q’s. demand p P 0 ert p 0 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) by 0 the red. P ert Too Low a P 0 demand p p 0 q 450 line p 0 t q
Increase in r Red Bounded area must p equal green area Initial price lower T sooner Two Price-time paths p 0 must cross q 450 line P 0 ert p 0 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 p 0 P 0 ert p 0 q 450 line t q
Recall • Lets call the solution to this P(x(t)), the price • When stock is x at time t. • P(x(0) )= p 0 • P(x(t)) = p 0 (1+r)t
Folded Diagram Model • p(x(t)) = p 0(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 p 0(1+r)(t+1) • p(x(t) – ) = p 0(1+r)(t+n) • price after n years of cutting equals the price at • time t , (p 0(1+r)t) times the interest factor for n years (1+r) n. • Choose n so that the Park taking equals
Enhancement: Years Method p X(0) is again red area. Arrow shows number of p 0 years need to wait to find equivalent P 0 ert p 0 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-
Example Concluded: Reduced Form
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