For this discussion, assume that there is only one class of a mineral
reserve, that extraction costs are constant, and that the unit value of the
reserve rises at the social rate of discount. Variables are:
R_t_ = total quantity of reserves of the mineral commodity at year end
H_t_ = unit value of the reserves (say, petroleum reserves), which equals
Hotelling rent under the above assumptions
A_t_ = quantity of new reserves discovered during the year
q_t_ = quantity of extraction or production during the year
V_t_ = total value of the reserves at year end
In a given year, petroleum firms might discover new reserves totaling A_t_.
Then the additions are given by:
|
additions_t_ = H_t_ A_t |
(3.1) |
During that year, petroleum production, and therefore depletion
of existing reserves, is measured by q_t_. Depletion is, under the special
assumptions listed above, quantity times the value of reserves:
|
depletions_t_ = H_t_ q_t_ |
(3.2) |
The total value of reserves at year end is:
|
value of reserves = V_t_ = H_t_ R_t_ |
(3.3) |
The change in the value from the end of year t-1 to the end
of year t is given by:
|
change in value of reserves V_t_ - V_t-1_ = H_t_ R_t_ - H_t-1_ R_t-1_ |
(3.4) |
Revaluations are the change in the value corrected for the value of additions and depletions:
|
revaluation = H_t_ R_t_ - H_t-1_ R_t-1_ - H_t_ A_t_ + H_t_ q_t_
|
(3.5) |