Price paths for optimal extraction over time

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Price paths for optimal extraction over time
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Dynamic efficiency rule

• Maximizing the present value of net benefits,
  PVNB, over time requires extracting quantities
  in each period, qt such that the present value of
  marginal net benefits is equal in all time
  periods
• Marginal benefits marginal WTP market price
  in a competitive market

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Equalization of PV(MNB) rule

• Marginal benefits marginal WTP market price,
  P in a competitive market
• Marginal net benefitst Pt MCt for any time
  period, t
• PV(MNB0) P0 MC0
• PV(MNB1) (P1 MC1)/(1r)1
• PV(MNBt) (Pt MCt)/(1r)t

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• Set the two equal as necessary for dynamic
  efficiency
• P0 MC0 (P1 MC1)/(1r)1
• Recall that Pt MCt Rentt
• Rent0 (Rent1)/(1r)1
• Rearrange
• Rent1 (1r)1 Rent0 , or in general,
• Rentt (1r)t Rent0 known as Hotelling rule

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Hotelling rule for resource rent

• Or in general
• Rentt (1r)t Rent0
• known as Hotelling rule

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Price path with Hotelling rule

• Replace Rent1 with P1 MC1
• Move MC to other side of the equal sign
• P1 (1r)Rent0 MC1
• Or in general,
• Pt (1r) Rent0 MCt
• The price in each period will increase as the
  value of rent increases

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Example

• If initial Rent0 1, MC 5 and r .05

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Price and rent over 25 years