Optimal depletion of a
finite resource by a
competitive market
RESOURCE RENTS
David Ricardo (1772-1823)
The “Curse of Natural Resources”
A rich resource endowment often yields more misery than growth.
Technical explanations:
• internal market distortions
• low labor quality
• growth-inhibiting customs and institutions
• failure to exploit true comparative advantages (e.g., cheap labor)
etc.
Much of this is simply "blaming the victim."
This "natural resource curse" hypothesis is counter-intuitive for most
economists.
Conventional international trade theory does really explain why so many
resource-rich countries remain poor.
While development economists can identify market failures that contribute to
economic stagnation in many such countries, the fundamental causes are
mostly government and political failures.
A strong comparative advantage in resource extraction provides immediate
revenues from exports, but does not necessarily promote capital
accumulation or economic diversification.
Since the government is funded by resource royalties, the citizens aren't
taxed and don't demand government accountability.
Income inequality between the resource extraction sector and other
economic sectors increases.
As ethnic or political factions challenge the government for control of the
resource, the government turns authoritarian and uses the army to maintain
domestic control.
Export royalties are diverted from social programs to purchases of military
hardware.
Human rights suffer, and political opposition is suppressed.
If there are elections, they are typically rigged.
The government falls prey to corruption and covert influence by foreign
interests and multinational corporations.
Why Nations Fail (2013) by Daron Amoceglu and James Robinson:
Case studies of nations with "extractive" economies that fail to develop
compared to nations with "inclusive" economies and better development.
The requisites for long-term economic growth include
• secure property rights
• inclusive, accountable and reasonably transparent political institutions
• a rule of law that prevents monopolies, coercion of labor, etc.
The over-specialized mono-export economy is destabilized by resource price
fluctuations.
If the government borrows heavily against its future resource revenues, a decline
in the resource price may cause its credit to collapse.
As the exchange value of its currency becomes tied to the resource price, high
prices for its resource exports drive up the value of its currency and make its
other exports uncompetitive.
Basic Ricardian theory:
Landowners pay farm workers a subsistence wage,
keep the production surplus from their land as “Rent”
The low-cost resource has the slowest
increase in marginal rent—extract it first.
How a competitive resource extraction firm depletes a finite
reserve over time — numerical example:
Suppose you have 20 units of some resource to extract and sell at market price P = $20.
The marginal cost of extracting quantity in any time period Q is MC = 5 + 0.2*Q
The marginal rent from quantity Q is M.RENT = $20 – 5 – 0.2*Q = 15 – 0.2*Q
Q0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
PRICE
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
MC0
$10.00
$10.20
$10.40
$10.60
$10.80
$11.00
$11.20
$11.40
$11.60
$11.80
$12.00
$12.20
$12.40
$12.60
$12.80
$13.00
$13.20
$13.40
$13.60
$13.80
$14.00
M.RENT0
$15.00
$14.80
$14.60
$14.40
$14.20
$14.00
$13.80
$13.60
$13.40
$13.20
$13.00
$12.80
$12.60
$12.40
$12.20
$12.00
$11.80
$11.60
$11.40
$11.20
$11.00
If you extracted and sold 12 units, for example…
How would you apportion 20 units of the resource over two years (t=0 and t=1) to
maximize the present value of the total rent where r = 3%?
Q0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
PRICE
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
MC0
$10.00
$10.20
$10.40
$10.60
$10.80
$11.00
$11.20
$11.40
$11.60
$11.80
$12.00
$12.20
$12.40
$12.60
$12.80
$13.00
$13.20
$13.40
$13.60
$13.80
$14.00
M.RENT0
$15.00
$14.80
$14.60
$14.40
$14.20
$14.00
$13.80
$13.60
$13.40
$13.20
$13.00
$12.80
$12.60
$12.40
$12.20
$12.00
$11.80
$11.60
$11.40
$11.20
$11.00
T.RENT0
$0.00
$14.90
$29.60
$44.10
$58.40
$72.50
$86.40
$100.10
$113.60
$126.90
$140.00
$152.90
$165.60
$178.10
$190.40
$202.50
$214.40
$226.10
$237.60
$248.90
$260.00
Q1=20-Q0
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
r=
MC1
$14.00
$13.80
$13.60
$13.40
$13.20
$13.00
$12.80
$12.60
$12.40
$12.20
$12.00
$11.80
$11.60
$11.40
$11.20
$11.00
$10.80
$10.60
$10.40
$10.20
$10.00
0.03
M.RENT1/(1+r)
$10.68
$10.87
$11.07
$11.26
$11.46
$11.65
$11.84
$12.04
$12.23
$12.43
$12.62
$12.82
$13.01
$13.20
$13.40
$13.59
$13.79
$13.98
$14.17
$14.37
$14.56
T.RENT1/(1+r) SUMTOTRENTS
$252.43
$252.43
$241.65
$256.55
$230.68
$260.28
$219.51
$263.61
$208.16
$266.56
$196.60
$269.10
$184.85
$271.25
$172.91
$273.01
$160.78
$274.38
$148.45
$275.35
$135.92
$275.92
$123.20
$276.10
$110.29
$275.89
$97.18
$275.28
$83.88
$274.28
$70.39
$272.89
$56.70
$271.10
$42.82
$268.92
$28.74
$266.34
$14.47
$263.37
$0.00
$260.00
PV of total rent is maximized where the discounted marginal rents
are equal:
A higher discount rate (r = 6%) penalizes future rents more and accelerates depletion:
Q0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
PRICE
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
MC0
$10.00
$10.20
$10.40
$10.60
$10.80
$11.00
$11.20
$11.40
$11.60
$11.80
$12.00
$12.20
$12.40
$12.60
$12.80
$13.00
$13.20
$13.40
$13.60
$13.80
$14.00
M.RENT0
$15.00
$14.80
$14.60
$14.40
$14.20
$14.00
$13.80
$13.60
$13.40
$13.20
$13.00
$12.80
$12.60
$12.40
$12.20
$12.00
$11.80
$11.60
$11.40
$11.20
$11.00
T.RENT0
$0.00
$14.90
$29.60
$44.10
$58.40
$72.50
$86.40
$100.10
$113.60
$126.90
$140.00
$152.90
$165.60
$178.10
$190.40
$202.50
$214.40
$226.10
$237.60
$248.90
$260.00
Q1=20-Q0
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
r=
MC1
$14.00
$13.80
$13.60
$13.40
$13.20
$13.00
$12.80
$12.60
$12.40
$12.20
$12.00
$11.80
$11.60
$11.40
$11.20
$11.00
$10.80
$10.60
$10.40
$10.20
$10.00
0.06
M.RENT1/(1+r)
$10.38
$10.57
$10.75
$10.94
$11.13
$11.32
$11.51
$11.70
$11.89
$12.08
$12.26
$12.45
$12.64
$12.83
$13.02
$13.21
$13.40
$13.58
$13.77
$13.96
$14.15
T.RENT1/(1+r) SUMTOTRENTS
$245.28
$245.28
$234.81
$249.71
$224.15
$253.75
$213.30
$257.40
$202.26
$260.66
$191.04
$263.54
$179.62
$266.02
$168.02
$268.12
$156.23
$269.83
$144.25
$271.15
$132.08
$272.08
$119.72
$272.62
$107.17
$272.77
$94.43
$272.53
$81.51
$271.91
$68.40
$270.90
$55.09
$269.49
$41.60
$267.70
$27.92
$265.52
$14.06
$262.96
$0.00
$260.00
…and here’s the solution for r = 10%:
Q0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
PRICE
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
MC0
$10.00
$10.20
$10.40
$10.60
$10.80
$11.00
$11.20
$11.40
$11.60
$11.80
$12.00
$12.20
$12.40
$12.60
$12.80
$13.00
$13.20
$13.40
$13.60
$13.80
$14.00
M.RENT0
$15.00
$14.80
$14.60
$14.40
$14.20
$14.00
$13.80
$13.60
$13.40
$13.20
$13.00
$12.80
$12.60
$12.40
$12.20
$12.00
$11.80
$11.60
$11.40
$11.20
$11.00
T.RENT0
$0.00
$14.90
$29.60
$44.10
$58.40
$72.50
$86.40
$100.10
$113.60
$126.90
$140.00
$152.90
$165.60
$178.10
$190.40
$202.50
$214.40
$226.10
$237.60
$248.90
$260.00
Q1=20-Q0
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
r=
MC1
$14.00
$13.80
$13.60
$13.40
$13.20
$13.00
$12.80
$12.60
$12.40
$12.20
$12.00
$11.80
$11.60
$11.40
$11.20
$11.00
$10.80
$10.60
$10.40
$10.20
$10.00
0.10
M.RENT1/(1+r)
$10.00
$10.18
$10.36
$10.55
$10.73
$10.91
$11.09
$11.27
$11.45
$11.64
$11.82
$12.00
$12.18
$12.36
$12.55
$12.73
$12.91
$13.09
$13.27
$13.45
$13.64
T.RENT1/(1+r) SUMTOTRENTS
$236.36
$236.36
$226.27
$241.17
$216.00
$245.60
$205.55
$249.65
$194.91
$253.31
$184.09
$256.59
$173.09
$259.49
$161.91
$262.01
$150.55
$264.15
$139.00
$265.90
$127.27
$267.27
$115.36
$268.26
$103.27
$268.87
$91.00
$269.10
$78.55
$268.95
$65.91
$268.41
$53.09
$267.49
$40.09
$266.19
$26.91
$264.51
$13.55
$262.45
$0.00
$260.00
As these three examples show, higher rates of discount shift extraction from the
future to the present:
Maximizing the present value of the total rents means
equating the present value of the marginal rents:
M.RENT0 = M.RENT1/(1+r) = M.RENT2/(1+r)2 = …
Profit-maximizing resource extraction firms will manage their
reserves so that their marginal resource rents rise at the rate
of discount over time.
How a competitive resource market depletes a finite reserve
over an indeterminate time horizon — numerical example:
Suppose total reserves are 1000 units, market demand is P = 100 – 0.2Q
and marginal extraction cost is MC = $25
so Marginal Rent = P – MC = 75 – 0.2Q.
Use a discount rate r = 6%.
(less is sold in each year, price rises)
marginal resource rents rise at the rate of discount over time
(annual extractions decline and price rises)
Time trajectories:
How I solved this problem…backwards:
(method for HW#4)
We know what the depletion point looks like: since Marginal Rent R = 75 – 0.2Q,
when the resource runs out, Q = 0 and the final R = $75.
I discounted $75 backward in time at discount rate r = 0.06 to get the
Marginal Rent trajectory:
$75  $75/1.06  $75/1.062  …
= $75  $70.75  $66.75  …
Inverting the Marginal Rent function yields the corresponding Quantities
Q = 5*(75-R):
0  21.2  41.3  …
…and accumulating the quantities yields the necessary reserves to supply those
quantities:
0  21.2  (21.2+41.3)  (21.2+41.3+…)
= 0  21.2  62.5  …
“Today” (year 0) is when the accumulated reserve matches 1,000 (approximately):
r=
Year
10
9
8
7
6
5
4
3
2
1
0
0.06
Marginal
Rent Quantity Reserves
$75.00
0.0
0.0
$70.75
21.2
21.2
$66.75
41.3
62.5
$62.97
60.1
122.6
$59.41
78.0
200.6
$56.04
94.8
295.4
$52.87
110.6
406.0
$49.88
125.6
531.6
$47.06
139.7
671.3
$44.39
153.0
824.4
$41.88
165.6
990.0
Price
$100.00
$95.75
$91.75
$87.97
$84.41
$81.04
$77.87
$74.88
$72.06
$69.39
$66.88
MC
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
Incorporating risk into the model:
Now suppose this resource is located in Berzerkistan, where the
Prime Minister is talking about nationalizing it and taking over the
American companies that have been extracting it.
Suppose there’s a 14% annual risk of nationalization.
Including this risk, the discount rate would now be 6% + 14% = 20%.
How does this risk affect the optimal depletion schedule?
Optimal depletion schedule under expropriation risk:
P(t) = 100 – 0.2Q(t); MC = $25; Initial Stock = 1,000; r = 0.20
r=
Year
6
5
4
3
2
1
0
0.2
Marginal
Rent Quantity Reserves
$75.00
0.0
0.0
$62.50
62.5
62.5
$52.08
114.6
177.1
$43.40
158.0
335.1
$36.17
194.2
529.2
$30.14
224.3
753.5
$25.12
249.4
1002.9
Price
$100.00
$87.50
$77.08
$68.40
$61.17
$55.14
$50.12
MC
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
$25.00
The risk of expropriation makes US firms accelerate their extraction
schedules… which angers the Berzerki government and makes
expropriation more likely… which makes firms accelerate extraction
even more…a prisoners’ dilemma!
Those US firms would really like to get rid of that troublesome Prime
Minister! Maybe call up a friendly Congressman…?
Here are six of many true stories showing how this works:
Iran:
1951: Mohammed Mossadeq elected prime minister; Iranian
Parliament nationalizes Britain's Anglo-Iranian Oil Company.
1953: Britain's MI6 and the CIA orchestrate the overthrow of
Mossadeq and install Reza Pahlavi as Shah, backed by his
secret police (SAVAK).
1979: Shah flies to NY for cancer treatment; Iranian popular
revolution coopted by Islamic hard-liners who make the
Ayatollah Khomeini Supreme Leader of an Islamic state;
Iranians seize the US embassy and hold 54 Americans
hostage for 15 months, damaging Jimmy Carter’s political
credibility. Carter defeated by Reagan in 1980. Reagan
tarnished by Iran-Contra scandal; 11 White House officials
convicted, but pardoned.
Guatemala:
The United Fruit Company (aka Chiquita) dominated Guatemala's
economy for decades, expropriating small landholdings and
consolidating them into enormous plantations.
1952: Jacobo Arbenz Guzman, the democratically-elected
president, institutes a land reform program to redistribute land to
small farmers. Compensation to large landowners would be the
assessments on which they had been taxed; Chiquita had grossly
understated the values of its holdings to minimize its tax payments,
and refused to accept those valuations as compensation.
1954: Arbenz overthrown in a CIA-orchestrated military coup,
Carlos Castillo installed as dictator, cancels the land reform,
restores the secret police, and established the "National Committee
of Defense Against Communism," a death squad to squelch unions
and political opponents.
After Castillo's assassination in 1957, a succession of military
dictators drove Guatemala into a 36-year civil war (1960-1996) in
which about 200,000 civilians were killed and another 50,000
people "disappeared."
Congo:
Brutal exploitation Belgium's Leopold II in the late 19th century.
Independence from Belgium in 1960: Patrice Lumumba elected
Prime Minister.
With US and Belgian support, the southern province of Katanga,
(where the country's mineral resources were located) rebelled.
Lumumba deposed by Col. Joseph Mobutu, arrested, taken to
Katanga province where he was executed.
Mobutu ruled “Zaire” until his death in 1997, embezzled about $5
billion of oil and mineral royalties. Mobutu backed the Hutu
perpetrators of the 1984 Tutsi genocide in neighboring Rwanda.
Tutsis in Zaire joined with other rebel groups to capture Kinshasa in
1997. Mobutu fled and died in exile soon after.
The country was renamed the Democratic Republic of the Congo
(DRC). Endemic civil war continues over its oil and mineral
resources.
Despite its resource wealth, DRC ranks close to last in the world in
per-capita income.
Chile:
For several decades, any foreign leader who challenged US
interests and talked about land reforms or capital
redistributions might be labeled a Communist.
But in 1970 Salvador Allende was the first self-avowed
Marxist to be elected president of a western-hemisphere
country.
1973: Allende attempts to nationalize Chile's copper mines
(the largest owned by Anaconda and Kennecott);
Is overthrown by CIA-backed military coup, assassinated or
committed suicide in the presidential palace.
Gen. Augusto Pinochet ruled Chile from 1974-1990,
suppresses political opposition with disappearances,
internment and torture. He was later prosecuted for human
rights abuses and died in 2006.
Indonesia:
After Japanese occupation in World War II, the nationalist
leader Sukarno prevented the Dutch from re-establishing
colonial control. But Sukarno’s government ran up massive
foreign debt, inflation rose to 600% per year, and the
economy foundered.
1965: the army crushed a Communist coup attempt with CIA
support, army chief Gen. Suharto launched a purge of
Communists, killing about 1.5 million people. Sukarno was
placed under house arrest and died 3 years later.
Suharto’s “New Order” government enjoyed strong US
support for his anti-communism. He invaded East Timor in
1975 (100,000 killed). Indonesia’s economy foundered
during the 1990’s, his government was seen as increasingly
corrupt, and he left office in 1998.
Panama:
The US supported Panama's secession from Colombia in 1903
in exchange for control of the Canal zone, and completed
construction of the Canal in 1914.
Panama was a constitutional democracy until 1968, when a
military coup ousted President Arnulfo Arias.
Gen. Omar Torrijos instituted land reforms and public works
programs, and got the US to cede control of the Canal to
Panama. He was killed in a plane crash in 1981 (the CIA
appears to have been involved).
Gen. Manuel Noriega, a graduate of the US's "School of the
Americas" counter-insurgency training program in the Canal
Zone and a long-time cooperator with the CIA, ruled Panama
from 1983 until 1989, when the US invaded, captured him
and convicted him of drug trafficking in US District Court in
Miami.