1. Chapter 4 Dynamic Efficiency: Oil and Other Finite Energy Resources: Part A
Peter M. Schwarz
Professor of Economics and
Associate, Energy Production and Infrastructure Center (EPIC),
UNC Charlotte

2. Outline Introduction Dynamic Efficiency Competition Monopoly
Other Factors
Sustainability
Risk

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Introduction (1)
Will we run out of oil?
Hubbert and other physical scientists say yes.
Economists say no.
Is our energy use sustainable?
Ecologists and other physical scientists say no.
Economists say yes.
Both questions require a dynamic model
Consider current vs. future use.
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Introduction (2)
Start with simplest setting
Add complexity as needed
As always, balance the ease of simplicity with greater predictive accuracy from a more complex model
Marginal cost
Zero
Positive
Increasing
Backstop technologies
Alternative technology available above some price
Competition
Monopoly
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Dynamic Efficiency (1)
Compare value today to value in future.
Future dollars must be discounted
Forgone opportunity to invest today.
Present Value
𝑃𝑉= 𝐹𝑉 1+𝑟 𝑡
where r is (real, inflation-adjusted) discount rate,
t is the number of years in the future when the earnings will be received.
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6. Dynamic Efficiency (2): Example
Suppose we could redesign car engines to emit less carbon at a cost of $10 million today to prevent $50 million in damage 50 years from now. Alternatively, we could invest the money and get an average return of 4% per year over the 50 year period. Is the redesign of car engines dynamically efficient?
The present value of $50,000,000 to be received in 50 years, discounted at 4% is:
PV = $50,000,000/(1.04)50 = $7,035,630.67, which is less than $10 million.
So at 4%, or any higher interest rate, the answer is No.
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7. Dynamic Efficiency (3): When to Use
Renewable vs. non-renewable resources
Renewables
Depletable
wood
Non-depletable
Solar, wind
Non-renewables
Oil
Natural Gas
Coal
Uranium
The more years of reserves remaining,
the less difference it makes to use a
dynamic model.
Old-growth forests may take many years
to replace, while oil reserves are
increasing rapidly.
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Competition (1): MC = 0
Simplified Model: Assume producer must sell oil today or one year from now. Competitive market.
Maximize profit = maximize total revenue
TRt = Pt Qt
TR = TR0 + TR1 = P0 Q0 + P1 Q1
PV (TR)= P0 Q0 + (P1 Q1)/ (1 + r)
Equimarginal Principle
MR0 = PV (MR1) = (MR1)/(1+r)
Under competition, P = MR
P0 = PV (P1) = (P1)/(1+r)
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9. Competition (2): Example
Demand is P = 80 – 8Q
Total remaining supply of oil is 10 barrels
Interest rate is 10%.
How much oil should the producer sell today, how much one year from now?
We will present both an algebraic and a graphical solution.
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10. Competition (2): Example (cont.)
If static problem, P = MC.
P = 0, since MC = 0. Q = 10.
CS = 0.5* ($80)*10 = $400.
PS = 0.
SW = Total Surplus = CS + PS = $400.
But that solution leaves no oil for the future. Suppose we produce 5 barrels in each period.
P = $40.
CS0 = 0.5* ($80-$40)*5 = $100
PS0 = $40*5 = $200
SW0 = $300b
SW1 = $300/(1.1) = $272.73b
SWT = $572.73b
Does
this solution
maximize SW?
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11. Competition (4): Example (cont.) Graphical Solution
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12. Competition (5): Example (cont.)
To maximize SW (Equimarginal Rule):
P0 = P1 / (1+r)
80 – 8Q0 = (80 – 8Q1)/1.1
and Q0 + Q1 = 10
Substituting Q1 = 10 - Q0 and solving for Q0,
Q0 = 5.238, Q1 = 4.762
P0 = 80 – 8Q0 = $38.10, P1 = 80 – 8(4.762) = $41.90
SW0 = CS0 + PS0 = $ $ = $ billion
PV(SW1) = ($ $199.53)/1.1 ) = $290.24/1.1 = $263.86b
Total SW = $573.17b > $572.73b when output was allocated equally in each period.
Recall P0 = 80 – 8Q0
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Competition (4): MC = 20
Equimarginal Principle:
P0 – MC0 = (P1 – MC1) / (1 + r)
MNB = P - MC
MNB0 = MNB1 / (1 + r)
80 – 8Q0 – 20 = (80 – 8Q1 – 20) / (1.1)
Solving:
Q0 = 5.12, Q1 = 4.88, P0 = $39.04, P1 = $40.96
PV(SW) = TNB0 + PV(TNB1) = CS0 + PS0 + PV(CS1 + PS1)
TNB0 = 0.5( )(5.12) + ( )5.12 = $202.34b
TNB1 = 0.5( )(4.88) + ( )4.88 = $197.54/1.1 = $179.58b
TNB = $381.92b
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14. Competition (5): Example (cont.)
To obtain graphical solution: Replace P by Marginal Net Benefit (MNB)
MNB = P – MC
Efficient point is where MNB0 = PV (MNB1)
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15. Competition (6): Increasing MC
MC = 2Q
80 – 8Q0 – 2Q0 = (80 – 8Q1 – 2Q1) / (1 + r)
Solution:
Q0 = 5.14, Q1 = 4.86, P0 = 38.88, P1 = 41.12
MNB0 = – 2*(5.14) = 28.74
MNB1 = – 2*(4.86) = 31.40
PV(MNB1) = 31.40/1.1 = 28.55
MNB0 = PV(MNB1) = (Differences are due to rounding)
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16. Competition (7): Hotelling’s Rule
Hotelling’s Rule: The dynamically efficient allocation occurs when the PV of MNB for the last unit consumed is equal across time periods.
P0 – MC = PV (P1 – MC)
Equivalently:
(P0 – MC) (1+r) = (P1 – MC).
Marginal User Cost (MUC) = PV (P-MC) is opportunity cost of selling today, sacrificing what you could earn in the future.
Value of oil increases at the rate of interest, equal to what you would earn on alternative investment.
Unlike static model, P > MC, because of MUC.
MNB0 = – 2*(5.14) = 28.74
MNB1 = – 2*(4.86) = 31.40
28.74 = 31.40/1.1? Yes
(except for rounding errors).
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