# Natural Resource Theory Copyright, 1998 by Peter Berck.

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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 bankBuy one unit of Resource Period 1have \$P(1)have \$P(1) worth of resource Period 2have \$(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 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 = 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.) P0 (1+r) T = Choke If you know p0, and choke, you know T, exhaustion time.

Hotelling in 4-Quadrants t p q q demand p0p0 P 0 (1+r) t 45 0 line Note: choke price, T.

Hotelling in 4-Quadrants t p q q demand p0p0 P 0 e rt 45 0 line

Equilibrium: Green is X(0) t p q q demand p0p0 P 0 e rt 45 0 line Area under curve is sum of all Q’s.

Too Low a P 0 t p q q demand p0p0 P 0 e rt 45 0 line p0p0 A choice of P o below the equilibrium value leads to more q in each period, which is more than X(0) by the red.

Increase in r t p q q p0p0 P 0 e rt 45 0 line p0p0 Red Bounded area must equal green area Initial price lower T sooner Two Price-time paths must cross

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) t p q q Red price path is result of red X(0) Arrow shows size of enhancement p0p0 P 0 e rt 45 0 line p0p0

Recall 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)) = p 0 (1+r) t price as function of stock is same as price as function of time 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 t p q q X(0) is again red area. Arrow shows number of years need to wait to find equivalent p0p0 P 0 e rt 45 0 line p0p0

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