Part I Hotellings Rule of Nonrenewable Resources

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Part IHotellings Rule of Nonrenewable Resources

• Content
• A simple partial analysis
• The monopoly resource supplier
• Extraction costs / different stocks

Gunter Stephan - Intertemporal Allocation of
Natural Resources 2008
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Different Resource Stocks

• Observation
• Nonrenewable such oil are extracted from
  different deposits
• Sustainable development requires to substitute
• Questions
• Do rules exist according to which different
  stocks will be exhausted?
• What does substitute for nonrenewable mean?

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Different Resource Stocks

• Assumption Different deposits (stocks) are
  characterized by different costs of extraction
• Simplification Stock A CA(xt) axt Stock
  B CB(xt) bxt
• Hotelling-rule ptA (1r)tqA a ptB
  (1r)tqB b

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Different Resource Stocks

• Questions
• Which stock will be exhausted first, which one
  second?
• How royalties of different stock are related to
  each other?
• What is a back-stop?
• Why dont we observe overshooting of prices?

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Different Resource Stocks
Stock A CA ax Stock BCB bx where a lt b
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Different Resource Stocks

• Thought experiment (b gt a)

stock Bstock A
time
t
t1
t2
(1r)-tpt - a (1r)-(t1)pt1
b (1r)-tpt - b (1r)-(t1)pt1
a ? (1r)b - a b a
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Different Resource Stocks

• Result
• (1r)b - a b a rb - a,gt 0, if r gt
  0
• If the interest rate r is positive, then the
  stock with lowest marginal costs of extraction
  will be exhausted first

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Different Resource Stocks
Price
Stock A
Stock B
Time
switch
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Different Resource Stocks

• Since
• ptA (1r)tqA a, ptB (1r)tqB b
• some date S exists such that
• (1r)SqA a (1r)SqB b(1r)tqA a lt
  (1r)tqB b if tltS(1r)tqA a gt (1r)tqB b
  if tgtS
• hence
• (1r)SqA -qB b a gt 0

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Different Resource Stocks

• Opening costs
• Suppose, opening a new stock causes costs d,
• This could be investment into a pipeline
• Hotelling rule without opening costs
• Hotelling rule with opening costs

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Different Resource Stocks

• This implies
• Hence, SWITCH gt switch
• since rents from exhausting the first deposit
  must cover opening costs of the second
• Therefore, resources from the first stock are
  extracted for a longer period, which implies a
  higher royalty

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Different Resource Stocks
Price
Stock A
Stock B
Time
switch
SWITCH
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Different Resource Stocks

• Interpretation and results
• If the interest rate is positive, stocks with the
  lowest extraction costs will be exhausted first
• The higher the marginal costs of extraction, the
  lower is the royalty
• There is no overshooting of prices, if there are
  no opening costs and date of switching stocks is
  correctly anticipated

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Different Resource Stocks

• Backstop Resource
• Concept was introduced by Nordhaus (1973)
• Perfect substitute,which has the properties
• Available at unlimited quantities
• Marginal costs are constant
• Supplied under complete competition

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Different Resource Stocks
Price
Marginal cost of backstop
nonrenewable
Marginal cost of nonrenewable
Time
switch
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Different Resource Stocks
Deflated prices of aluminum and quadratic trends
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Different Resource Stocks

• Explanation for the observed outcome
• Technological change reduces extraction costs
• Assumption
• technological change reduces marginal costs of
  extraction by a fraction ? lt 1 from period to
  period

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Different Resource Stocks
Price
Hotellings rule implies hence
Time