R=P-MC PV [R] = R 0 + R 1 /(1+r) + R 2 /(1+r) 2 +… Optimal extraction quantity R 0 = R 1 /(1+r) = R 2 /(1+r) 2 =… Hotellings Rule - net price rises over time with the rate of interest.
The Two-Period Model Revisited Dynamic efficiency is the primary criterion when allocating resources over time. Recall the two-period model from Chapter 5. This model can be generalized to longer time periods. An n-period model presented uses the same numerical example from Chapter 5, but extends the time horizon and increases the recoverable supply from 20 to 40.
Assumptions Fixed supply of certain depletable resource Consider two time periods only Total supply is 20 units Demand (marginal WTP) is constant:
A higher discount rate will favor the present. The amount allocated to the second period falls as the discount rate rises.
Marginal user cost rises over time at the rate of discount causing efficient prices to rise over time and thus reflecting scarcity.
With constant marginal extraction cost, total marginal cost (or the sum of marginal extraction costs and marginal user cost) will rise over time. The graph shows total marginal cost and marginal extraction cost. The vertical distance between the two equals the marginal user cost. The horizontal axis measure time. Rising marginal user cost reflects increasing scarcity and the intertemporal opportunity cost of current consumption on future consumption.
As costs rise, quantity extracted falls over time. Quantity extracted falls to zero at the point where total marginal cost reaches the maximum willingness to pay (or choke price) for the resource such that demand and supply simultaneously equal zero.
Choke price – the minimum price of a good or service that would result in a zero quantity demanded Price path – the price of a resource over time Extraction path – the extraction rate of a resource over time
Technology Decreases marginal cost of extraction Higher quality resources will be extracted first. => subeconomic resources may become economic when the price rises or technology improves
http://en.wikipedia.org/wiki/The_Limits_to _Growth http://en.wikipedia.org/wiki/The_Limits_to _Growth Written in 1972, predicting over use of resources http://en.wikipedia.org/wiki/The_Populatio n_Bomb http://en.wikipedia.org/wiki/The_Populatio n_Bomb Written in 1968, predicting a population crash due to resource scarcity The wager : http://en.wikipedia.org/wiki/Simon%E2%80%93Eh rlich_wager
Technological progress would also reduce the cost of extraction. Lowering the future marginal cost of extraction would move the transition time further into the future. Total marginal cost could actually fall with large advances in technology.
An efficient allocation thus implies a smooth transition to exhaustion and/or to a renewable substitute. The transition point to the renewable substitute is called the switch point. At the switch point the total marginal cost of the depletable resource equals the marginal cost of the substitute.
The transition can for two depletables with different marginal costs will also be a smooth one. The rate of increase of total marginal cost slows down after the time of transition because the marginal user cost represents a smaller portion of total marginal cost for the second, higher cost resource.
For this case, the marginal user cost declines over time and reaches zero at the transition point. The resource reserve is not exhausted. The marginal cost of exploration can be expected to rise over time as well. Successful exploration would cause a smaller and slower decline in consumption while dampening the rise in total marginal cost.