The payoff matrix (in utility units)
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The state of the environmental world
Good outcome (low damage, many green technologies)
Poor outcome (catastrophic damage, no green technologies)
Climate
Strong policies (high carbon tax, cooperation, R&D) -1% -1%
policyWeak policies (no carbon tax, strife, corruption) 0% -50%
Probability 90% 10%
Optimal policy with learn then act
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The state of the environmental world
Good outcome (low damage, many green technologies)
Poor outcome (catastrophic damage, no green technologies)
Climate
Strong policies (high carbon tax, cooperation, R&D) -1% -1%
policyWeak policies (no carbon tax, strife, corruption) 0% -50%
Probability 90% 10%
Expected loss = 90% x 0 + 10% x -1 = -0.1%
ACT in future
High damages
High carbon tax
Low carbon taxLow damages
This example: Learn then act
LEARNTODAY
What is wrong with this story?
The Monte Carlo approach is “learn then act.”That is, we learn the role of the dice, then we adopt
the best policy for that role.But this assumes that we know the future!
- If you know the future and decide (learn then act)- If you have to make your choice and then live with
the future as it unfolds (act then learn)
In many problems (such as climate change), you must decide NOW and learn about the state of the world LATER: “act then learn”
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Decision Analysis
In reality, we do not know future trajectory or SOW (“state of the world”).
Suppose that through dedicated research, we will learn the exact answer in 50 years.
It means that we must set policy now for both SOW; we can make state-contingent policies after 50 years.
How will that affect our optimal policy?
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Optimal policy with act then learn
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The state of the environmental world
Good outcome (low damage, many green technologies)
Poor outcome (catastrophic damage, no green technologies)
Climate
Strong policies (high carbon tax, cooperation, R&D) -1% -1%
policyWeak policies (no carbon tax, strife, corruption) 0% -50%
Probability 90% 10%
Expected loss depends upon strategy:strong: 90% x -1 + 10% x -1 = -1%weak : 90% x 0 + 10% x -50 = -5%
Conclusions
When you have learning, the structure of decision making is very different; it can increase of decrease early investments.
In cases where there are major catastrophic damages, value of early information is very high. Best investment is sometimes knowledge rather than
mitigation (that’s why we are here!)
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Very extreme distributions
Normal distributions have little weight in the “tails”Fat tailed distributions are ones with big surprisesExample is “Pareto” or power law in tails:
f(x) = ax-(β +1), β = scale parameter.
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Probability of "tail event"
Sigma Normal Pareto 1.5 Pareto 1
2 2.2750% 1.1180% 5.0000%
4 0.0032% 0.3953% 2.5000%
6 0.0000% 0.2152% 1.6667%
10 0.0000% 0.1000% 1.0000%
23 0.0000% 0.0287% 0.4348%
* Sigma = average dispersion (like standard deviation).
Some examples
Height of American women: Normal N(64”,3”).
How surprised would you be to see a 14’ person coming to Econ 331?
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Some examples
Stock market: what is the probability of a 23% change in one day for a normal distribution? Circa 10-230 !!!- Mandelbrot found it was Pareto with β= 1.5.- Finite mean, but infinite variance
Earthquakes: Cauchy distribution β = 1 (see next slide). - Infinite mean, infinite variance
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Distribution of earthquake energy
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1.000
0.100
0.010
0.0019 10 11 12 13 14 15
Energy (E)
Fraction larger than EPower law distribution
Cum
ula
tive
freq
uen
cy
β = 1
Surprise with fat tails
Suppose you were a Japanese planner and used historical earthquakes as your guidelines.
How surprised were you in March 2011? How much more energy in that earthquake that LARGEST in all of Japanese history?
Answer:(9.0/8.5)^10 ^1.5= 5.6 times as large as historical max.
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Distribution of damages from RICE-2010 model
18
0
40
80
120
160
200
240
280
320
0 10 20 30 40 50 60
Damages/ NNP (%)Observations 998
Mean 6.8%Median 4.8%Maximum 66.5%Minimum 0.4%
Damages/ NNP (%) for 2135
Here is another motivation: surprise
Fat tailed distributions are ones that are very surprising if you just look at historical data.
Suppose you were an oil trader in the late 1960s and early 1970s.
You are selling “vols” (volatility options).Let’s rerun history.
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Let’s look at the moving history of oil price changes: 1950- 1965
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-100
0
100
200
300
400
1950 1955 1960 1965 1970 1975 1980
Percentage change in oil prices(3 month, log, annual rate)
Oil price changes: 1950- 1970
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-100
0
100
200
300
400
1950 1955 1960 1965 1970 1975 1980
Percentage change in oil prices(3 month, log, annual rate)
Oil price changes : 1950- 1973:6
22
-100
0
100
200
300
400
1950 1955 1960 1965 1970 1975 1980
Percentage change in oil prices(3 month, log, annual rate)
Oil price changes : 1950- 1974:3
23
-100
0
100
200
300
400
1950 1955 1960 1965 1970 1975 1980
Percentage change in oil prices(3 month, log, annual rate)
Revisit economists’ approach to uncertainty
- Combine structural modeling, subjective probability theory, and Monte Carlo sampling.
- Dynamic system under uncertainty:(1) yt = H(θt , μt)
- Then develop subjective probabilities for major parameters, f(θ). Often, use normal distributions for parameters because so simple:
(2) θ ≈ N (θ, σ)
- This has been criticized by Weitzman and others, who argue that the distributions have much more weight for catastrophic situations.
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Weitzman’s contribution
Weitzman showed that with fat tailed distribution, might have negative infinite utility, and no optimal policy.
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Some technicalia on Weitzman Critique
Weitzman argues that IAMs have ignored the “fat tailed” nature of probability distributions. If these are considered, then may get very different results. (Rev. Econ. Stat, forth. 2009)
Weitzman’s definition of fat tails is unbounded moment generating function:
Note that this is unusual both substantively and because it involves preferences (CRRA parameter, α ).
Combine the CRRA utility with Pareto distribution (β) for consumption.
Dismal Theorem: Have real problems is α is too high or β is too small.
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- c
c=0
E(c)= e f(c)dc=-∞
∞
Go back to earlier example. Here payoffs are c
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The state of the environmental world
Good outcome (low damage, many green technologies)
Poor outcome (catastrophic damage, no green technologies)
Climate
Strong policies (high carbon tax, cooperation, R&D) -1% -1%
policyWeak policies (no carbon tax, strife, corruption) 0% -50%
Probability 90% 10%
1
1
Utility function: ( ) - (1 - ) / (1 )
'( ) - / (1 )
'(1) 1 [so utility is in consumption units]
U c c
U c c
U
Payoffs when act then learn with no policy and fat tails
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Act then learnLoss in expected utility (c units)
α No policy0 -5%
log(c) -7%2 -10%3 -15%4 -23%6 -89%
Payoffs when have policy, act then learn
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Act then learn Loss in expected utility (c units)
α Weak policy Strong policy0 -5.00% -1.00%
1.01 -6.96% -1.01%2 -10.00% -1.01%3 -15.00% -1.02%4 -23.33% -1.02%6 -89.45% -1.03%
Comparing all learning options with catastrophes…
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Act then learn
------ Loss in expected utility (c units) ----
---------- Act then learn ---------Learn then act
α Weak policy Strong policy0 -5.00% -1.00% -0.10%
log(c) -6.96% -1.01% -0.10%2 -10.00% -1.01% -0.10%3 -15.00% -1.02% -0.10%4 -23.33% -1.02% -0.10%6 -89.45% -1.03% -0.10%
Conclusions on fat tails
1. Fat tails are very fun (unless you get caught in a tsunami).
2. Fat tails definitely complicate life and losses.- Particularly with power law (Pareto) with low β.
3. Fat tails are particularly severe if we act stupidly. - Drive 90 mph while drunk, text messaging, on ice
roads.
4. If have good policy options, can avert most problems of fat tails.
5. If have early learning, can do even better.
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