Transcript Slide 1

Hotelling example (with small change)
1
Note on “shadow prices” or “dual variables” (π)
These are extremely important in economic modeling (and more generally in
economics).
Basic idea is that these represent opportunity costs. I will use the example of
cost minimization. In our Hotelling problem, we are minimizing costs (C)
subject to the various constraints. Let’s take the resource constraints (sum
production < resources = R1 ). If we do this as a Lagrangean, we get the
following interesting result:
∂C/∂R1 = π1 = shadow price on reserve grade 1 = $71.75.
This says that if we increase the quantity of reserves by 1 unit, this lowers
discounted cost by $71.75. (I changed the sign to positive.)
An important theorem from econ is that this equals the first-period royalty in
competitive markets (!). This grows with the interest rate until exhausted.
I calculated this numerically using a linear programming algorithm in the
GAMS program and got the πi shown in the table. We can actually do this
for 50,000 variables and constraints using a LP algorithm. This is done, for
example, in oil refineries to optimize yield from crude oil.
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Integrated Assessment Models
of Economics of Climate Change
Economics 331b
Spring 2010
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Integrated Assessment (IA) Models of Climate
Change
• What are IA model?
– These are models that include the full range of cause and
effect in climate change (“end to end” modeling).
– They are necessarily interdisciplinary and involve natural
and social sciences
• Major goals:
– Project the impact of current trends and of policies on
important variables
– Assess the costs and benefits of alternative policies
– Assess uncertainties and priorities for scientific and
project/engineering research
Major Components of Models
Behavioral and
Scientific
Equations
Identities
Value Judgments
(markets, policies,
ethics, etc.)
Person or
nation 1
Pareto Improvement from
Climate Policy
Bargaining
region (Pareto
improving)
Inefficient
initial (nopolicy)
position
Person or nation 26
Elements of IA Models.
To be complete, the model needs to incorporate the
following elements:
- human activities generating emissions
- carbon cycle
- climate system
- biological and physical impacts
- socioeconomic impacts
- policy levers to affect emissions or other parts of cycle.
Representative Scenarios for Models
“Baseline” or uncontrolled path:
- Set emissions at zero control or zero “tax” level.
- Business as usual
Alternative strategies:
- “Optimal” where maximize objective function
- Stabilize emissions, concentrations, or climate
- Kyoto Protocol/ Copenhagen Accord limits
There are many kinds of IA models, useful for different
purposes
Policy evaluation models
- Models that emphasize projecting the impacts of different
assumptions and policies on the major systems;
- often extend to non-economic variables
Policy optimization models
- Models that emphasize optimizing a few key control
variables (such as taxes or control rates) with an eye to
balancing costs and benefits or maximizing efficiency;
- often limited to monetized variables
Two Examples
1.Reprise on Hotelling example for pset 1
2. The Samuelson-Negishi equivalence for modeling
Maximize (producer + consumer surplus)
=
Maximize Discounted [U(c) – Cost(c)]
=
Competitive equilibrium
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1. Single market
q
q
0
0
U (q )   MU (q )dq ; C (q )   MC (q )dq ; W (q )  U( q )  C( q )
max W (q )  max CS+PS  competitive equilibrium
2. Single market over time: same with time-dated prices
(market discount rates)
3. Multiple market : simply have q is a vector of
goods, q  ( q 1 ,..., q n )
4. Multiple agents (Negishi): Need appropriate weights.
More complicated algorithm
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Economic Theory Behind Modeling
1. Basic theorem of “markets as maximization” (Samuelson, Negishi)
Outcome of efficient
competitive market
(however complex
but finite time)
Maximization of weighted
utility function:
=
n
W   i [U i (c ki ,s ,t )]
i 1
for utility functions U; individuals i=1,...,n;
locations k, uncertain states of world s,
i
time periods t; welfare weights  .
2. This allows us (in principle) to calculate the outcome of a market
system by a constrained non-linear maximization.
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How do we solve IA models?
The structure of the models is the following:
max W 
{  ( t )}
T max
 U[c(t ),L(t )]R(t )
t 1
subject to
c(t )  H[ (t ),s(t ); initial conditions, parameters]
(The H[...] functions are production functions, climate model,
carbon cycle, abatement costs, damages, and so forth.)
We solve using various mathematical optimization techniques.
1. GAMS solver (proprietary). This takes the problem and solves it
using linear programming (LP) through successive steps. It is
extremely reliable.
2. Use EXCEL solver. This is available with standard EXCEL and
uses various numerical techniques. It is not 100% reliable for
difficult or complex problems.
3. MATHLAB. Useful if you know it.
4. Genetic algorithms. Some like these.
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Can also calculate the “shadow prices,”
here the efficient carbon taxes
600
Marginal cost of Emissions Reductions ($)
Remember that in a constrained
optimization (Lagrangean), the
multipliers have the
interpretation of
d[Objective Function]/dX.
So, in this problem, interpretation
is MC of emissions reduction.
Optimization programs
(particularly LP) will generate
the shadow prices of carbon
emissions in the optimal path.
For example, in the problem we
just did, we have the following
shadow prices:
500
400
300
200
100
0
0
10
20
30
Period
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Basic economic strategy
1. Begin with a Solow-style economic growth model
2. Add the geophysical equations: note these impose an
externality
3. Then add an objective function to be optimized subject
to constraints:
-
1 + 3 = optimal growth model [Friday]
1 + 2 + 3 = integrated assessment model
4. Then estimate or calibrate the various components.
5. Then do various simulations and policy runs.
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Modeling Strategies (I): Emissions
Emissions trajectories:
Start with data base of 70 major countries representing 97 %
of output and emissions 1960-2004.
Major issue of whether to use PPP or MER (next slide)
Estimate productivity growth
Estimate CO2 emissions-output ratios
Project these by decade for next two centuries
Then aggregate up by twelve major regions (US, EU, …)
Constrain by global fossil fuel resources
This is probably the largest uncertainty over the long run:
σ(Q) ≈ .01 T, or + factor 2.5 in 100 yrs, +7 in 200 yrs
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CO2-GDP: Three countries (PPP v. MER)
Sigma: MER (DC)
Sigma: PPP (DC)
US
3.5
3.5
Russia
US
3
Russia
China
3
China
CO2/GDP
2.5
2
1.5
2
1.5
1
1
0.5
0.5
20
02
19
96
19
90
19
84
19
78
19
72
19
66
2000
1995
1990
1985
1980
1975
1970
1965
19
60
0
0
1960
CO2/GDP
2.5
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Modeling Strategies (II): Climate Models
Climate model
Idea here to use “reduced form” or simplified models.
For example, large models have very fine resolution and
require supercomputers for solution.*
We take two-layers (atmosphere, deep oceans) and decadal
time steps.
Calibrated to ensemble of models in IPCC TAR and FAR
science reports.
*http://www.aip.org/history/exhibits/climate/GCM.htm
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Actual and predicted global temperature history
.6
.4
.2
.0
-.2
-.4
-.6
1840
1880
1920
1960
2000
YEAR
T_DICE2007
T_Hadley
T_GISS
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Projected DICE and IPCC: two scenarios
5
4
3
2
1
0
1920
1960
2000
2040
2080
2120
YE A R
T_A2_DICE
T_A2_IPCC
T_B1_DICE
T_B1_IPCC
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Modeling Strategies (III): Impacts
• Central difficulty is evaluation of the impact of climate
change on society
• Two major areas:
–
market economy (agriculture, manufacturing, housing, …)
–
non-market sectors
•human (health, recreation, …)
•non-human (ecosystems, fish, trees, …)
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Summary of Impacts Estimates
Early studies contained a major surprise:
Modest impacts for gradual climate change, market impacts, highincome economies, next 50-100 years:
- Impact about 0 (+ 2) percent of output.
- Further studies confirmed this general result.
BUT, outside of this narrow finding, potential for big problems:
-
many subtle thresholds
abrupt climate change (“inevitable surprises”)
ecological disruptions
stress to small, topical, developing countries
gradual coastal inundation of 1 – 10 meters over 1-5 centuries
OVERALL: “…global mean losses could be 1-5% Gross Domestic Product
(GDP) for 4 ºC of warming.” (IPCC, FAR, April 2007)
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Estimated Damages from Yale Models and IPCC Estimate
11%
Climate damage/global output
10%
RICE-1999
9%
DICE-2007
8%
7%
6%
5%
4%
IPCC estimate
3%
2%
1%
0%
0
1
2
3
4
5
Mean temperature increase (oC)
6
7
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Major problems of impacts analysis
Most impacts analyses impose climate changes on current
social-economic-political structures.
Example: impact of temp/precip/CO2 on structure of
Indian economy in 2005
However, need to consider what society will look like
when climate change occurs.
Example looking backward:
–
2 ˚C increase in 6-7 decades – that was Nazism, period of
Great Depression, Gold Standard, pre-Keynesian macro
– 4 ˚C increase in 15 decades –Ming Dynasty, lighting with
whale oil, invention of telegraph, no cars, many horses….
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Modeling Strategies (IV): Abatement costs
IA models use different strategies:
–
Some use econometric analysis of costs of reductions
–
Some use engineering/mathematical programming
estimates
–
DICE model generally uses “reduced form” estimates of
marginal costs of reduction as function of emissions
reduction rate
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Derivation of mitigation cost function
Start with a reduced-form cost function:
C = Qλμ
(1)
where C = mitigation cost, Q = GDP, μ = emissions control rate,
λ,  are parameters.
Take the derivative w.r.t. emissions and substitute σ = E0 /Q
(2)
dC/dE = MC emissions reductions
= Qλβμ-1[dμ/dE] = λβμ-1/σ
Taking logs:
(3)
ln(MC) = constant + time trend + ( β-1) ln(μ)
We can estimate this function from microeconomic/engineering
studies of the cost of abatement.
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Example from McKinsey Study
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Reduced form equation: C=.0657*miu^1.66*Q
60
50
40
30
20
10
0
0
5
10
15
20
25
30
35
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Further discussion
However, there has been a great deal of controversy about
the McKinsey study. The idea of “negative cost” emissions
reduction raises major conceptual and policy issues.
For the DICE model, we have generally relied on more
micro and engineering studies.
The next set of slides shows estimates based on the IPCC
Fourth Assessment Report survey of mitigation costs.
The bottom line is that the exponent is much higher
(between 2.5 and 3). This has important implications that
we will see later.
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Note that the MC is much more convex than
McKinsey: much more diminishing returns
Source: IPCC, AR4, Mitigation.
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Source for estimates of  (elasticity of cost function)
Source: IPCC, AR4, Mitigation, p. 77.
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Using the IPCC as data for the cost function
Least squares estimate of cost function from IPCC
[Equation is MC = a*(%reduction)^(β-1)
Approach
β-1
SE(β-1)
t(β-1)
Bottom up: A1B
Bottom up: B2
Top down: A1B
Top down: B2
1.8
1.8
3.3
3.4
0.3
0.3
0.4
0.5
5.7
6.4
7.4
6.7
Conclusion is that the cost function is EXTREMELY convex.
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Alternative abatement cost functions: From IPCC
Parameterized as C/Q = aμ2.8 , with backstop price(2005) = $1100/tC
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Alternative abatement cost functions
Parameterized as C/Q = aμ2.8 , with backstop price(2005) = $1100/tC
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Alternative abatement cost functions: IPCC and MK
2000
1600
1200
800
400
0
0
20
40
60
80
100
Parameterized as C/Q = aμ2.8 , with backstop price(2005) = $1100/tC
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Applications of IA Models
How can we use IA models to evaluate alternative
approaches to climate-change policy?
I will illustrate analyzing the economic and climatic
implications of several prominent policies.
For these, I use the recently developed DICE-2007 model.
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1. No controls ("baseline"). No emissions controls.
2. Optimal policy. Emissions and carbon prices set for economic
optimum.
3. Climatic constraints with CO2 concentration constraints.
Concentrations limited to 550 ppm
4. Climatic constraints with temperature constraints. Temperature
limited to 2½ °C
5. Kyoto Protocol. Kyoto Protocol without the U.S.
6. Strengthened Kyoto Protocol. Roughly, the Obama/EU policy
proposals.
7. Geoengineering. Implements a geoengineering option that offsets
radiative forcing at low cost.
Illustrative Policies for DICE-2007
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Per capita GDP: history and projections
Per capita GDP (2000$ PPP)
100
10
US
WE
OHI
Russia
EE/FSU
Japan
China
India
World
1
1960
1980
2000
2020
2040
2060
2080
2100
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CO2-GDP ratios: history
CO2-GDP ratio (tons per constant PPP $)
.7
.6
China
Russia
US
World
Western/Central Europe
.5
.4
.3
.2
.1
.0
80 82 84 86 88 90 92 94 96 98 00 02 04
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IPCC AR4 Model Results: History and Projections
DICE-2007
model
2-sigma
range
DICE
model
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