Resources for the Future
Krautkraemer and Toman
technological and geological information. Returns across periods are linked in this model by the simple fact that any energy resource produced in some period is by definition not available for production in future periods (energy resources obviously cannot be recycled). Only a finite amount of energy resource is available in this framework, and all of it will eventually be extracted. But the finite availability of fossil fuels is not the only key feature of energy supply, even in the simplest framework.
In this framework, the present value of net returns is maximized when the “equi-marginal principle” is satisfied. This principle requires that the marginal net return from the extraction and sale of the energy resource be the same in every time period with positive extraction. This, in turn, requires the current value of marginal net profit to be increasing at the rate of discount applied to future returns.
T h i s p r i n c i p l e c a n b e s t a t e d i n s i m p l e m a t h e m a t i c a l t e r m s . L e t P t d e n o t e t h e p r i c e o extracted energy resource at time t, qt the quantity extracted at time t, C(qt ) the cost of f e x t r a c t i n g P t , a n d δ t h e d i s c o u n t r a t e . T h e n , u n d e r t h e e q u i - m a r g i n a l p r i n c i p l e , i t m u s t b e t h a t :
= ′ − q C P ) ( 0 0
′ − q C P ) ( 1 1 (1+ δ )1
′ − q C P ) ( 2 2 (1+ δ )2
′ − t t q C P ) ( (1+ δ )t
=L= λ ,
where λ denotes the common present value of marginal net profit. This number can be interpreted as the present value shadow price or in situ value of the resource stock, since it reflects the incremental increase in present value that would be enjoyed if a resource owner could experience an augmentation of resource stock. It also is referred to in the literature as the user cost” or “scarcity rent” associated with depletion of the resource stock.
At any point in time, then:
λ δ t t t q C P ) 1 ( ) ( + + ′ =
In any successive time periods,
Pt−1 −C′(q (1 + δ) t−1 t−1
P t − ′ C ( q t )
(1 + δ)t
P t − ′ C ( q t ) = ( 1 + δ ) [ P t − 1 − ′ C ( q t − 1 ) ] .