Overview of Optimization and Resource Economics

1
Overview of Optimization and Resource Economics

• General discussion of literature and optimal
  control
• Application to simple resource problem
• Discussion of what these problems can provide in
  terms of analytical results
• Complications and Future Directions
• Solving Empirical optimization models

2
Role of Optimization in Natural Resource Economics

• Normative How should resources be managed to
  maximize utility?
• Most attention since Hotelling (1931) and Scott
  (1955) has focused here (see Deacon et al.,
  1998).
• Predictive How do changes in policy affect
  economic outcomes?
• Less attention focused here, even though most
  natural resources are not managed optimally
  (Gordon, 1954).
• Capital Theoretic Recognizing that resource
  stocks are similar to other capital assets
  (Clark, 1976).

3
Where is the literature today?

• Considerable effort has focused on building
  techniques to solve optimization problems.
• These use fairly simple characterizations of
  growth processes
• Today
• Integrating more complex ecological processes
• Recognizing that economics and ecology are
  endogenous.
• Gaining more insight into how real-world firms
  interact and respond to real-world conditions,
  including changes in policy.

4
What is Optimal Control?

• A set of methods to analyze natural resource
  allocation problems.
• Allows researchers to solve and analyze
  relatively simple problems analytically.
• For more complicated problems, provides
  researchers with information helpful for solving
  numerically.
• Note The economic literature trusts some
  complications, but not TOO much.

5
Consider the fishery problem
Density Dependent Growth Stock
X(t) Growth F(X(t)) gX(t)1-X(t)/K g
intrinsic growth rate
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Optimally controlling the fishery
Let h(t) be the annual harvest How much annual
Harvest would maximize the utility of this
fishery?
Subject to . dX/dt X F(X)
h(t) Initial Conditions
7
Setting up and solving optimal control problems

• Set up the Hamiltonian.
• Follows from Maximum Principle from the calculus
  of variations and from Pontryagin et al., (1961).
• Maximum Principle shows the conditions that need
  to be met to satisfy that you have a maximum.
• Clark (1976) Conrad and Clark (1987) Kamien and
  Schwartz Beavis and Dobbs.
• Solve for FOCs.
• Show the stability of the system.
• Phase diagram for simpler problems
• Show approach dynamics for steady state.

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Hamiltonian
Present Value
 
Current Value (multiply by ert)
Where
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FOCs
   
10
Rearranging Terms
Solve This for steady state
11
Phase Diagrams
. h 0
I
II
h
FX r
h
. X 0
III
IV
X
X
12
Phase Diagrams
. h 0
I
II
h
FX r
h
. X 0
III
IV
X
X
13
Making the FOCs Interesting
Add structure to the utility function
   
U(h)
U(h) A Bh Ch2 Note Uh B 2Ch P
h
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Combine equations
.
.
.
(1) In steady state, prices are constant P 0
? FX r (2) For Non-renewable resources
F(X) 0 FX 0 (Hotelling/Ramsey)
.
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Complications/Other

• Non-renewables
• Monopoly Hotelling (1931)
• Exploration Pindyck (1978)
• Open Access
• Gordon (1954) Scott (1955)
• Externalities
• Hartmann (1976) Berck (1981)
• Spatial
• Linking dynamics across cells
• Conrad (1985) Sanchirico and Wilen (1999 2001).

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Using Dynamic Models for Empirical Analysis

• Welfare impacts are more complicated than a set
  of static outcomes.
• Often interested in understanding the
  consequences of an environmental impact or policy
  on welfare outcomes.
• Stock externalites (climate change)
  Environmental impact changes and grows over time.
• Sustainability Issues.
• Bequests and terminal conditions.
• Dynamic models can be used to assess how humans
  adapt to impacts and to measure the consequent
  welfare impacts.

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Welfare Effects
18
Forestry Problem

• Biological Species that adds volume each year
• Different species grow at different rates
• Want to determine the optimal age to harvest the
  trees
• More Complex Problems consider optimal management
  over time.

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TerminologyAll Expressed as Volumes

• Yield The size of trees after a given time
  period of growth
• V(a) where V(a) Volume, and a age.
• Annual Growth The increase in volume of trees
  from one year to the next
• Average Annual Growth The average annual growth
  of trees up to a given time period
• AAG V(a)/a Volume at age a divided by a

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Southern Pine ExampleYield
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Southern Pine Growth
Annual Growth
MSY
Average Annual Growth
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Forestry Management
What is the optimal time to harvest and
regenerate forests? - Prices, tree
growth, interest rates, costs, and other
factors matter!
Maximize W PV(a)(1r)-a C 1
(1r)-a.
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First Order Conditions
Assuming Prices Constant
Allowing Prices to Adjust
Alternatively
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Southern Pine ExampleOne versus Infinite
Rotations
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Practically...
Tree Growth Rate
r
Age
10
0.49
0.04
20
0.12
0.04
30
0.05
0.04
40
0.03
0.04
50
0.02
0.04
60
0.01
0.04
70
0.01
0.04
80
0.01
0.04
90
0.01
0.04
100
0.00
0.04
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What if You get Non Market Values as a Forest
Ages? (Hartman, 1979)
NPV PV(a)e-ra ?PC?V(n)e-rndn C
(1-e-ra)
FOC
27
NPV Carbon Example
28
Forestry MarketsSeveral Stocks, Prices Endogenous
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Forestry Problem
D() Demand function Q(t) Quantity harvested
at t ?Hi(t)Vi(a(t),mi(t0)) (m3) Z(t)
Quantity of other goods consumed CH()
Harvesting cost function ( per m3) CG(?Gi(t),
?Ni(t), m(t)) Cost of regenerating ?Gi(t)
Ni(t) hectares with intensity m(t). Ri(Xi(t))
Rental cost of holding Xi hectares in i for 1
period.
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Maximization Problem
s.t.
Initial and terminal conditions
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Harvesting Accessible Forests
Harvesting Inaccessible Forests (Ramsey Rule)
Regenerating Timberland
Management Intensity
32
Forestry MarketsSeveral Stocks, Prices Endogenous
33
Do We Regulate the Forest

• Mitra and Wan (1985, 1986), Heaps (1984)
• Linear Utility Always optimal to harvest at
  Faustmann rotation, adjusted for price changes,
  therefore do not regulate.
• Concave Utility No regulation, steady state?
• Tahvonen and Salo (2002)
• Steady state occurs, but forest is not regulated.
  

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Example Climate Change ProblemGlobal Carbon
Cycle
IPCC, 2001, WG I
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The Economics of Climate Change
Business as Usual No Control
(1) Whats the damage Function look like? (2)
What level to Stabilize? (3) How to do it
most cheaply?
Conc. Of CO2
Control Emissions/ Stabilize Future Concentration
Baseline Human CO2 emissions remain at 1800
levels
1800
2100
Damage(t) F(Temperature Change(t)) G(CO2
Concentration(t))
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Historical DataCO2 and Temperature
Average Temperature
Global CO2 Emissions
37
Estimating Damages
Baseline Welfare
Climate Change causes a series of exogenous
impacts on forests (dieback, changes in growth,
etc.)
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Annual Welfare EffectsClimate Change Impact
Results for US Forests
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Climate Change as an Optimal Control Problem
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What Can be Done About it?Reduce Sources and
Increase Sinks

• Emission Sources
• Transportation 2.2
• Energy 2.1
• Industry 2.1
• Tropical Def. 1.6
• Total 7.9

• Decay and Sinks
• Temperate Ref 0.7
• Oceans 2.3
• Atmosphere 3.3
• Residual 1.6
• Total 7.9

Global, Billion Tons (Late 1990s)
US 25
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Optimal Control Model of Energy Abatement and
Forest SequestrationMinimize NPV of costs of
abatement and damages
Subject to
Change in atmospheric stock Baseline
Emissions - energy abatement decay of
carbon change in forest stock
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First Order Conditions
µ(T) CEA (marginal cost of abatement
shadow value of carbon) (MC of energy
abatement MC of storing a new ton in forests)
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Valuing Carbon in Forests

• ?(t) is the present value of future benefits of
  abating 1 ton of carbon emissions today
• ?(t) shadow value of a ton of carbon
• Each 1 ton deviation from the baseline stock of
  forest carbon in time t is valued at ?(t).
• New tons of sequestered carbon are valued
• ?(t) RC(t) PC(t)(r-n(t))

44
Integrating Forestry and Energy

• DICE/RICE Models
• (Nordhaus and Boyer, 2000)
• Dynamic growth theory model
• with a climate externality
• Predicts the marginal cost of
• optimal energy abatement.
• Incorporate carbon supply
• functions
• S(t) 0.042PC(t)0.870t0.706

• Dynamic Global Timber
• Market Model
• (Sohngen et al., 1999)
• Single world demand function
• with species quality adjustments.
• Forest yield and production
• functions for 50 forest types.
• 3.8 billion hectares.
• Rent carbon in forests.

• Policy Analysis
• Optimal Forest Policy
• Alternative Policies Benefits and Costs

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DICE Structure (Nordhaus and Boyer, 2000)With
Carbon Sequestration
Income Constraint
     
Production Function
Capital Stock
Emissions Process
Sequestration Costs
Carbon Accumulation Process (not Shown)
46
DICE/RICE Baseline
47
DICE Some Results
48
Integrated Energy-Forestry Results
49
Regional Carbon Storage
50
Change In Land Area in Forests
51
Change in Rotation AgeNA EU Comparison - 2100
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Benefit Cost Analysis
53
Alternative PoliciesReduction in Carbon Storage
Relative to Optimal Case
No Payments for Changes in Management
No Payments for Management/Rotations (Fixed
carbon budgets)
Fixed Annual Payments