Non-renewable Resource Extraction and the Right to Sell You are the lucky owner

Question

Non-renewable Resource Extraction and the Right to Sell You are the lucky owner of an oil deposit. However, somewhat sadly you only live for two periods. This means that you have only two periods during which you can extract your oil, periods 0 and 1 Due to customary practice, you cannot resell your deposit to anyone else a) Let So be the total initial quantity of oil barrels in the deposit Each barrel fetches a fixed per-period price po and pi on the market and the total cost of extraction per period is c(Rt), where Rt de- notes the quantity of barrels extracted at period t, with c'(Rt)>0, c" = 1/(1 + r), write down the present-value profit maximizing problem and derive the corre- (R) > 0. Using a discount factor of sponding optimal arbitrage condition between Ro and R b) Calculate Ro and R, assuming So-500, po-Pi-50, r 10%. and c(Rt)R2/20. Calculate the marginal rent at each period. At what rate is it changing? c) Do the same exercise as the previous assuming now So Interpret your result. d) Suppose now that after you sadly leave us at the end of the pe riod 1, life on Earth fortunately goes on, say for one more period to simplify. The government decides that due to intergenerational equity considerations, you are forced to leave 200 barrels of oil in the ground for the generation living during period 2, with P2 50 Assuming So 1200 still, how would that affect your extraction rates Ro and R1? e) The government considers amending customary law in order to allow you to resell your deposit. We would like to investigate whether the possibility of selling the oil deposit can affect your extraction choices R0 and R1, if at all. Analyze this question assuming So-1200 and 250

Explanation / Answer

Static (single-interval) model
suppose you possess a mine containing 20 million Kg of turbidium that's sold in a aggressive market for a steady fee
P = MR = $20 per Kg.
The marginal price of extracting the resource from your mine is
MC = $8 + $zero.30Q the place Q is the extraction range (thousands of Kg).
Your marginal useful resource employ is
MRR = MR - MC = $12 - $0.30Q.
The complete resource hire is the discipline of the trapezoid under the MRR time table, identical to the discipline between the cost and MC schedules.
For those who could extract and promote all 20 million Kg of your turbidium stock today, what would your marginal resource rent be?
What would your total useful resource employ be? (natural the lengths of the 2 parallel facets and multiply by Q, the space between them.)

Two-interval model
k, think you want to maximise your complete rents by using promoting off your 20 million Kgs over a two-12 months time interval. Graphically, you would plot MRR0 left-to-proper and MRR1 right-to-left with the horizontal axis restrained to 20 items complete.
You could in general intuit that maximizing whole useful resource rents over two time periods (TRR0 + TRR1) manner equating the marginal rents across the two time durations. Without discounting, you might split the resource equally between the 2 time durations the place MRR0 = MRR1. However managing this useful resource like another monetary asset, you need next year's discounted marginal appoint to equal this year's marginal hire. Graphically, this discounting shifts the MRR1 time table downward, which moves the intersection of MRR0 and MRR1 rightward, implying an most useful Q0 > Q1.
Utilising reduction expense r = 0.05, clear up for the extraction quantities in the two time periods, Q0 and Q1, that satisfy each the most efficient allocation situation

MRR(Q0) = MRR(Q1)/1.05
and the stock constraint
Q0 + Q1 = 20.

(trace: by using substituting 20 - Q0 for Q1 within the first equation, you get 12 - 0.30Q0 = (12 - zero.30[20-Q0])/1.05 which which you can solve algebraically for Q0.)
Calculate the values of MRR0 and MRR1.
Calculate the total resource appoint in every time period--the subject underneath the MRR0 schedule between zero and Q0, and the discipline underneath the MRR1 agenda between Q0 and 20. Examine the sum of these total useful resource rents versus the only-interval whole appoint you calculated in part 1.
The foremost extraction schedule requires that the present values of MRR must be the same in each time period. This means that marginal resource rents will have to upward thrust at the cost of discount over time. This "step rule" between time intervals was once first articulated by using Harold Hotelling (JPE, 1931), and is referred to as "Hotelling's Rule." It defines the rent-maximizing strategy for any two adjoining time durations:
MRRt = MRRt+1/(1+r)
and by using extension, across all time durations except depletion time T:
MRR0 = MRR1/(1+r) = MRR2/(1+r)2 = ... = MRRT/(1+r)T.

When your MRR is rising on the cost of reduction by way of time, you're indifferent between promoting an additional unit of the resource in one time interval versus some other.
The two-interval drawback illustrates Hotelling's Rule, but it surely's unlikely that you would arbitrarily come to a decision to fritter away your inventory in just two durations. Extra realistically, your goal can be to maximise the reward price of your complete resource employ movement via choosing both the number of years within the most effective depletion agenda as well because the optimal quantities to extract in every year. Problems like these mostly require some lovely fancy math, but that you would be able to practice the superior step rule bought within the two-period case to clear up this situation quite straightforwardly. The trick is to solve it backwards!
Multi-period mannequin
feel your preliminary inventory of the useful resource is X0 = 300 items (millions of Kg), and the marginal useful resource appoint operate is the identical as earlier than: MRRt = $12 - $0.30Qt.
Hotelling's Rule says the top-rated MRR trajectory should follow the trail
MRR0 = MRR1/(1+r) = MRR2/(1+r)2 = ... = MRRT/(1+r)T

however you do not know the commencing worth for MRR0. You do know what the ending price for MRRT shall be, however. At depletion time T, XT = 0 and QT = zero, so MRRT = 12 - zero.30(0) = $12.
So from this ultimate time interval you can use Hotelling's Rule to clear up for MRR within the subsequent-to-final time period:
MRRT-1 = MRRT/(1+r). And from the inverse of the MRR system, Qt = [12-MRRt]/zero.30, that you can resolve for the value of QT-1 that yields MRRT-1.
This step rule applies to every backward step in time from t to t-1: MRRt-1 = MRRt/(1+r). And that you can solve for Qt-1 from MRRt-1.
As you clear up the most desirable trajectories of MRR and Q backward in time, that you can keep a walking complete of the extraction quantities. This often is the stock X integral to provide these quantities from time t unless depletion:
Xt = Qt + Qt+1 + ... + QT-1 + QT

and then that you could respect which era interval is "in these days." The time period when the gathering stock worth QT + QT-1 + QT-2 + QT-3 + ... Excellent approximates your present inventory level is the gift.
This may occasionally make a lot more sense whilst you work by way of a numerical instance in an Excel spreadsheet.
At the top of a blank worksheet, enter the column headings 12 months, MRR, Q, inventory and PVTRR within the first row of columns A by way of E.  
At the high of the MRR column, enter the quantity 12 in mobilephone B2, that allows you to represent the final MRR as the last bit of stock is depleted. (As Q zero, MRR = 12 - zero.30Q 12.)
in the mobilephone under (B3), enter the method to reduction the above telephone by using one year at a discount expense of 5 percentage, e.G.: =B2/(1.05). Copy this components down the column so that you obtain MRR values for about 25 years.
Within the Q column, calculate the implied price Qt for every MRRt: Qt = (12 - MRRt)/zero.30.
Within the stock column, calculate the cumulative sum of the values of from Qt as much as QT. That is the inventory Xt that would be required to provide the entire annual extraction quantities Qt + Qt+1 + ... + QT except depletion.
At this factor the highest three rows of your spreadsheet should seem like this:
Now compare the stock column. In what row does the cumulative stock worth fit your present inventory of 300 models most closely?
Let this spreadsheet row corresponds to "at present;" enter "zero" in that mobilephone and depend upward in the TIME column to the depletion yr. How many years away is the depletion time T?
What's the most reliable wide variety Q0 to extract in these days? What is MRR0?
In the PVTRR column calculate the reward value of the total resource rent acquired in every year. As partly 1(b) above, each and every year's total hire can also be calculated because the discipline of a trapezoid: (12 + MRR)/2 Q.
The reward price of year t's complete resource hire could be [(12 + MRRt)/2 Qt]/(1+r)t.
Sum the PVTRR column to acquire the complete rent worth of the mine. What is it (in $thousands)?
Create XY scatter-plots (now not Line plots!) of Q, MRR and inventory versus yr, formatted as traces.
These XY-plots must exhibit the trajectories from present time 0 to depletion time T, left to correct.
(MRR should be rising while Q and X decline.)
results of an unanticipated discovery
feel an unanticipated discovery increases trendy inventory to X0 = 380 units. What are the most fulfilling values of Q0 and MRR0 now?
(trace: with a better initial stock level, "today" is quite simply a few rows extra down on your common spreadsheet.) Insert a 2nd TIME column counting from zero upward from this row.
Create XY scatterplots to examine the most suitable time trajectories of Q, MRR and stock for beginning values X0 = 300 vs. X0 = 380. (Excel trick: Plot MRR versus the common TIME column for X0 = 300. Then plot MRR versus TIME when X0 = 380. Choose and duplicate (ctrl-C) such a plots, and paste (ctrl-V) itinto the opposite. The comparison of the MRR trajectories underneath the two inventory sizes will have to appear like this:

make certain all of your graphs are clearly labeled. Turn to your written solutions to components 1 and 2, and your spreadsheets and XY-plots for ingredients three and 4.

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