An introduction to



A depletable resource is not naturally replenished or is replenished at such a low rate that it can be exhausted.  The depletion rate is affected by demand, durability and

reusability.  Example: Oil, Natural Gas and Coal. 

A recyclable resource has some mass that can be recovered after use.  Copper and Aluminum are an examples of depletable but

recyclable resource. 

A renewable resource is one that is naturally replenished.  Example: Water, Fish, Forests and Solar Energy.



Mineral: inorganic solid substances found in or on the ground.  Fuel (Crude Oil, Natural Gas)  Non Fuel (Metals (e.g. iron), Industrial Minerals (e.g. cement) ▪ Metals ▪ Ores like Iron, Nickel, Bauxite (which is the main resource for Aluminum), and other precious metals.

▪ Industrial minerals ▪ Cement, Minerals used in Fertilizers, Abrasives used in, say, industrial polishing and even some Gem Stones.



A mineral is a non-renewable resource  The change in stock (ΔS) is zero: ΔS=0 ▪ Given a fixed quantity of non-renewable resource, how fast should it be extracted and used? ▪ How much should be spent on finding new stocks? ▪ Can we find a (sustainable) way to recycle these stocks?



The management problem for depletable resources is how to allocate dwindling stocks among generations while transitioning to a renewable alternative (e.g. Energy)



The management problem for renewable resources is in maintaining an efficient and sustainable flow (e.g. Fisheries)



A resource taxonomy is a classification system used to distinguish various categories of resource availability.  Current reserves are resources that can be

extracted profitably at current prices.  Potential reserves resources potentially available. They depend on people’s willingness to pay and technology.  Resource endowment represents the natural occurrence of resources in the earth.

• Efficient Inter-temporal Allocation of Depletable

Natural Resources – a 2 periods model. • A Rule for Non-renewable Resource Extraction

• With MC=0. • With non-zero (and increasing) MC.



The Two-Period Model  Scarcity and trade-off between today and tomorrow.  Instead of Static Efficiency, which has been already

reviewed in details, the Dynamic Efficiency is the primary criterion when allocating resources over time.  Thus, the need to use discounting, as appropriate. ▪ Thought: If I have a three cookies for the next two days, how many should I eat each day?

Assume that MWTP curves and MC curves are constant across today and the future (next year); which implies  no change in preferences during the two periods  no change in technology during the two periods  (Socially) optimal extraction is 200 units in each period.  Suppose only 300 units is available though at current technology.  How do we determine optimal extraction today and optimal extraction in the future? 

$

MWTP0

$

MWTP1

MC0

MC1 p1

p0

100

200

Quantity, current period

100

200

Quantity, next period

Let MWTP curves be P = 410 – Q and MC curves be MC=110 + ½Q for both periods. Draw the graphs and show that the (socially) optimal extraction each period is 200 units. Suppose only 300 units are available. Fix the discount rate at, say, 10% How do we determine optimal extraction today and optimal extraction in the future?

    

$

MWTP0

$

MWTP1

MC0

MC1 p1

p0

100

200

Quantity, current period

100

200

Quantity, next period

 

NB0=TB0-TC0 and NB1=TB1-TC1 Discount Factor: ß=1/1+r where r is the discount rate (e.g. r=10%, ß=90%)



Present Value of Net Benefits is defined as: PVNB=NB0 + ß (NB1)=NB0 + (1/1+r)*(NB1)  If r=0 and there is no discounting (i.e. tomorrow is

as important as today), then PVNB=NB0+NB1  As r→∞ tomorrow becomes less important such that at its limit PVNB=NB0





At r=10%, PVNB=NB0+(0.9)NB1, we can maximize the PVNB with the choice variable being the extracted quantities. MAX [NB0+(0.9)NB1] subject to Q0+Q1=300  In case you recall the Lagrangian method from

your previous calculus courses in college, this would be a very good example for you to practice. 

The result of the maximization: Changes in NB today should be equal to the discounted changes in NB tomorrow.

Assuming only two periods: Present Value of Net Benefit= (net benefits, year 0) + 1/(1+r)*(net benefits, year 1) 

The Trade Off: Increase in extraction today implies less available for extraction tomorrow. 



Present value of net benefits maximized when Change in net benefits in year 0 IS EQUALT TO

   

1/(1+r)*(change in net benefits in year 1) Why?!? Solve for the Maximization Problem in the previous slide. Can be expressed as MWTP0-MC0= (1/(1+r))*(MWTP1-MC1) Given that market clearing price is MWTP we get P0-MC0= (1/(1+r))*(P1-MC1) Note that (1/(1+r))*(P1-MC1) is the user cost of today extraction.



Let’s put all the pieces together:  The consumption decision will be made based on: 



P0-MC0= (1/(1+r))*(P1-MC1) The Demand and MC functions are the following: P0 = 410 – Q0 and MC0=110 + ½Q0 P1 = 410 – Q1 and MC1=110 + ½Q1 Can we solve for the quantities when r=10%?!? What seems to be the problem? One equation, two unknowns: Q and Q ! Our consumption is constrained with Q0+Q1=300 Solve for Q0 and Q1 0

 

1



Inter-temporal efficiency requires  Quantity today is greater than quantity tomorrow

To double-check this, solve for Q0 and Q1 for the set-up above  Price today is lower than price tomorrow Sounds like law of demand to me; based on the difference in quantities above 

Why is quantity higher today?  Discount rate. If rate was zero extraction would be equal today

and tomorrow 

Resource rent: difference between price and MC at extraction level: Rent1 = (1+r)*(Rent0)  rent rises at the rate of discount



Motivating question: Suppose you own an oil field, and the cost of extracting oil from the field is zero.  How should you extract oil from the field over time?  What does it depend on?



Let’s bring back the inter-temporal decision making: Suppose we have a firm trying to decide how much oil to extract this year and how much to extract next year.  And assume: ▪ Profit-maximizing firm. ▪ Competitive industry. ▪ Costless extraction. ▪ Perfect information on future prices.



Consider the situation where the producer is indifferent between selling the last unit of oil this year or next year.  The present value of a barrel of oil would have to be

the same in both periods. i.e. MR0=(1/1+r)MR1 => p0=p1 / (1+r) => (1+r)p0=p1 => (p1– p0)/p0=r : Hotelling’s Rule



Hotelling’s rule: (p1 – p0) / p0 = r  The price of oil increases at a rate equal to the discount rate.  Oil prices will rise through time.  In the context of our simple model, what is the intuition behind

this result:

▪ Suppose p0 >p1 / (1+r) => p0(1+r)>p1 ; extract all the oil in the current year ▪ Suppose p0 p0(1+r)
But what about the case where the costs of extraction are increasing as the resource stock declines?



Example: Suppose the price of oil next year (year 1) is $110 per barrel. According to Hotelling’s Rule, what must be the price today (year 0) if the discount rate is 10%?

If we introduce costs, marginal profit (Mπ): Mπ = p – MC where MC is the marginal cost of extraction.  The term marginal profit is also called economic rent (as defined on a previous slide too).  A modified Hotelling’s rule follows: (Mπ1 – Mπ0) / Mπ0 = r 

 Marginal profit (marginal net benefit to the firm) increases over

time at a rate equal to the discount rate.  This is the efficient (optimal) equilibrium extraction.  An important implication: The value of the mineral asset to appreciate at the same rate as other forms of capital.



Example: Suppose there are 100 tons of Milwaukonite, a newly discovered non-renewable mineral! Assume the following:  Two-period model.  Discount rate = 10%.

 The marginal cost of producing Milwaukonite is $10.  Inverse demand curve for Milwaukonite: p = 80 – q, where p is

price and q is quantity.  Let’s suggest that 50 tons should be extracted in the first period and 50 tons produced in the second period.

80

Demand

80

Demand

Supply $

Supply $

30

30

10

10 25

50

25

Tons / Year

Tons / Year

Period 0 1

Period 12

50



Is this equal division of Milwaukonite a market equilibrium? Does it maximize social net benefits?  Initially, net revenue for all producers is $20 per unit in both

periods ▪ p in both periods is $30 ▪ NPV of $20 in period 1 is 20/1.1 = 18.18 < 20 ; market incentive to extract more in period 0 – WHY IS THIS CONCEPT IMPORTANT? ▪ NPV of the mineral = ½ x (50 x 50)+(20 x 50) + (½ x (50 x 50)+(20 x 50) )/1.1 = $4295.45

 Suppose one ton of Milwaukonite is shifted from extraction in

period 1 to period 0: Is this socially better? Will the market be in equilibrium NOW? ▪ p0 falls to $29 and p1 increases to $31. ▪ Net revenue per unit is $19 in period 0, 21 in period 1 ▪ Total welfare increases: ½ x (51 x 51)+(19 x 51) + (½ x (49 x 49)+(21 x 49) )/1.1 = $4296.32 ▪ But market will judge this as going too far: NPV of $21 => 21/1.1=19.09 > 19; so incentive for firms to reduce extraction in period 0.



What is the equilibrium allocation?  (Mπ1– Mπ0) / Mπ0 = r (Hotelling’s Rule)  p –mc = 80 – q -10  Hotelling’s rule: p0 – mc0 = (p1 –mc1 )/1.1 ▪ 70-q0 = (70-q1)/1.1 (Hotelling’s Rule) ▪ q1 = 100-q0 (Resource Constraint) ▪ 70-q0 = (70 –100+q0)/1.1 ▪ q0 = 50.95  p0=19.05; p1=20.95; NPV of p1 = 20.95/1.1 = 19.05



By the Welfare Theorem, this allocation is efficient (maximizes the present value of social net benefits)



Graphically with two periods, the socially efficient allocation of extraction is the following: $ 70

$ Marginal Net Benefit in 0

Quantity in 0

PV of Marginal Net Benefit in 1

q1= 50.95; q2= 49.05 49

Actually, doing the math gives q1=50.95

70/1.1 = 63.6

Quantity in 1



Two observations of the model:  First, notice that the efficient solution does not require maximizing

current social net benefits. Why? ▪ Because there is an opportunity cost of consuming resources today, namely the value of consuming those resources in the future. ▪ This is called the marginal user cost.  Second, notice that efficient allocation requires consuming more

in the first period than in the second. ▪ This reflects the opportunity cost of holding Milwaukonite in the ground. ▪ Opportunity cost of foregone investment.



What happens to the two-period model with higher discount rates?  Suppose the discount rate is 20% rather than 10%.  Would there be incentive to produce more than 50.95 tons in period 0?

▪ p –mc = 80 – q -10 ▪ Hotelling’s rule: p – mc = (p-mc)/1.2

▪ ▪ ▪ ▪ 

70-q0 = (70-q1)/1.2 q1 = 100-q0 70-q0 = (70 –100+q0)/1.2 ; 2.2q0/1.2 =70+30/1.2 q0 = 51.82

Higher discount rates imply faster extraction.



What happens to the two-period model with higher discount rates? Graphically: 70

Marginal Net Benefit in 0

PV of Marginal Net Benefit in 1

70/1.1 = 63.6 70/1.2 = 58.3

$

$

Quantity in 0

q0= 51.82; q1= 48.18

Quantity in 1

$

$ Quantity PV of MNB Period 1

MNB Period 0

B

E D

F

A H Q0

Questions…

Q*

G Q1

Resource price over a 50-year horizon with costless extraction, p0=$10, and r=0.05: 140 120 100 80 60 40 20

Year

48

44

40

36

32

28

24

20

16

12

8

4

0

0

Price





Emprical test of Hotelling’s Rule (Krautkramer 1998):  Tracked prices for 9 non-renewable resources from

1967 to 1994.  There was no upward trend in prices. Why? …To be examined next time

7-32

7-33

The member countries of the international cartel called the Organization of Petroleum Exporting Countries (OPEC) collude in order to gain monopoly power.  Members:Algeria, Indonesia, Iran, Iraq, Kuwait, Libya, Nigeria, Qatar, Saudi Arabia, United Arab Emirates, and Venezuela  What is collusion? 

 Withholding oil from the market  If it works, it keeps prices higher than they would be in a free market.

What does this mean for efficiency?

7-34

Compatibility of Member Interest  Individual cartel members have incentives to cheat on production agreements.  If everyone else keeps oil off the market, then the individual who chooses to increase production gains a LOT.  Enforcing the collusive agreement is essential for the success of the cartel. 7-35

Price inelasticity of demand measures the sensitivity of demand to price changes.  If demand is not sensitive to price changes, then there are big potential gains from cartelization.  In the long run, price elasticity of demand depends in part on the availability of substitutes. Thus in the long run, price elasticity of demand is usually larger.  Substitutes for oil are expensive and transition times are long. Solar

energy sets a long-run upper limit on the ability of OPEC to raise prices.

7-36

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