ERE5: Efficient and optimal use of environmental resources

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ERE5 Efficient and optimal use of environmental
resources

• A simple optimal depletion model
• Resource substitutability
• Static and dynamic efficiency
• Hotellings rule
• Optimality
• An example
• Extraction costs
• Renewable resources
• Complications

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Last week

• Efficiency and optimality
• Static efficiency
• Optimality
• Dynamic efficiency and optimality
• Market efficiency
• Market failure public policy
• Externalities
• Public policy

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Exhaustion, Production Welfare

• One can construct production functions with
  various degrees of essentialness, but the
  question is of course empirical
• The question whether it matters that a resource
  gets exhausted depends on its substitutability
  or, if you cant do without, dont lose it
• So far, we have not run out of anything
  essential, but that does not mean we wont
• Human ingenuity is the ultimate resource, but
  also works to create problems

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Substitution possibilities and the shapes of
production function isoquants
K
? 0
0 lt ? lt ?
? ?
0
R
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Substitutability and Scarcity

• Feasibility of sustainable development depends on
  
• Substitutability
• Technical progress
• Backstop technology
• The magnitude of substitution possibilities
• Economists relatively high
• Natural scientists and ecologists limited
• However, it matters what services we look at

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Optimal Resource Extraction - Discrete Time
Social welfare function
Extraction
Investment
Production function
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Optimal Resource Extraction Discrete Time (2)
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Optimal Resource Extraction Continuous Time
Social welfare function
Extraction
Investment
Production function
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The Maximum Principle
Objective function
Subject to

• J depends on control variables (u), state
  variables (x), and time
• State variables describe the economy at any time
  the equation of motion governs its evolution over
  time
• Control variables are time-dependent policy
  instruments
• To obtain the solution we construct a current
  value Hamiltonian

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The Hamiltonian

• The Hamiltonian only contains the current state
  and controls current optimality is a necessary
  condition for intertemporal optimality
• The co-state variables (l) secure intertemporal
  optimality they are like Lagrange multipliers,
  indeed measure the shadow price

FOC
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Optimal Resource Extraction - Continuous Time (2)
Social welfare function
Equations of motion
Hamiltonian
Necessary conditions
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Static Efficiency

• Marginal utility of consumption equals the shadow
  price of capital
• Marginal product of the natural resource equals
  the shadow price of the resource stock

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Dynamic Efficiency

• The growth rate of the shadow price of the
  resource equals the discount rate
• The return to capital equals the discount rate

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Hotellings Rule

• Dynamic efficiency required
• The growth rate of the shadow price equals the
  discount rate
• An alternative interpretation
• The discounted price is constant along an
  efficient resource extraction path
• Thus, environmental resources are like other
  assets

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Growth Rate of Consumption

• The growth rate of consumption along the optimal
  time path
• Since ?gt0, consumption grows if the marginal
  product of capital exceeds the discount rate
• The intuition

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Hotellings rule and Optimality
Pt
PtB P0Be?t
PtA P0Ae?t
P0B
P0A
t
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Extraction Costs
Social welfare function
Constraints
Production function
Extraction costs
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Extraction Costs and Resource Stock
Gt (for given value
of Rt )
(iii)
(ii)
(i)
0
S0
Remaining resource stock, St
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Extraction Costs 2

• Hamiltonian
• Necessary conditions

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Resource Price

• Net price Gross price marginal extraction
  cost
• Gross price Marginal contribution to output,
  income (measured in utils)
• Net price Marginal value of the resource in
  situ
• Net price Rent Royalty

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Hotellings Rule -2

• The growth rate of the shadow price of the
  resource is lower if extraction costs rise with
  falling resource stocks
• The discount rate equals the rate of return of
  holding the resource, which equals its price
  appreciation plus the foregone increase in
  extraction costs

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Graphicalsolution
Net price Pt
PT K
Demand
P0
Pt
T
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R0
R
Time t
Rt
Area total resource stock
T
Time t
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Renewable Resources
Social welfare function
Constraints
Production function
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Renewable Resources -2

• Hamiltonian
• Necessary conditions

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Hotellings Rule -3

• The growth rate of the shadow price of the
  resource is lower for renewable resources
• The discount rate equals the rate of return of
  holding the resource, which equals its price
  appreciation plus the increase in the resource
  growth

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Complications

• The total stock is not known with certainty
• New discoveries increase the known stock
• There is a distinction between physical quantity
  and economically viable stock size
• Technical progress and RD
• Heterogeneous quality
• Extraction costs differ
• Availability of backstop-technology