Transcript The problem of fat tails

Treatment of Uncertainty
in Economics (II)
Economics 331b
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The payoff matrix (in utility units)
The state of the environmental world
Good outcome (low
damage, many green
technologies)
Climate
policy
Poor outcome
(catastrophic damage,
no green technologies)
Strong policies (high
carbon tax, cooperation,
R&D)
-1%
-1%
Weak policies (no carbon
tax, strife, corruption)
0%
-50%
90%
10%
Probability
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Optimal policy with learn then act
The state of the environmental world
Good outcome (low
damage, many green
technologies)
Climate
policy
Poor outcome
(catastrophic damage,
no green technologies)
Strong policies (high
carbon tax, cooperation,
R&D)
-1%
-1%
Weak policies (no carbon
tax, strife, corruption)
0%
-50%
90%
10%
Probability
Expected loss = 90% x 0 + 10% x -1 = -0.1%
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This example:
Learn then act
High
damages
High carbon tax
ACT in future
LEARN
TODAY
Low
damages
Low carbon tax
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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Realistic world:
Act then learn
High
damages
ACT
TODAY
?
LEARN
2050
Low
damages
Optimal policy with act then learn
The state of the environmental world
Good outcome (low
damage, many green
technologies)
Climate
policy
Poor outcome
(catastrophic damage,
no green technologies)
Strong policies (high
carbon tax, cooperation,
R&D)
-1%
-1%
Weak policies (no carbon
tax, strife, corruption)
0%
-50%
90%
10%
Probability
Expected loss depends upon strategy:
strong: 90% x -1 + 10% x -1 = -1%
weak : 90% x 0 + 10% x -50 = -5%
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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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The problem of fat tails
Units of dispersion (sample standard deviation)
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Very extreme distributions
Normal distributions have little weight in the “tails”
Fat tailed distributions are ones with big surprises
Example is “Pareto” or power law in tails:
f(x) = ax-(β +1), β = scale parameter.
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%
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0.0000%
0.0287%
0.4348%
* Sigma = average dispersion (like standard deviation).
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Black swans in South Africa
(“Birds of Eden”)
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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
Cumulative frequency
1.000
β=1
0.100
0.010
0.001
Fraction larger than E
Power law distribution
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10
11
12
13
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Energy (E)
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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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Some examples
Climate damages (fat tailed according to Weitzman, but ?)
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Distribution of damages from RICE-2010 model
320
Damages/NNP (%)
Observations 998
280
240
200
160
Mean
Median
Maximum
Minimum
6.8%
4.8%
66.5%
0.4%
40
60
120
80
40
0
0
10
20
30
50
Damages/NNP (%) for 2135
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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
400
Percentage change in oil prices
(3 month, log, annual rate)
300
200
100
0
-100
1950
1955
1960
1965
1970
1975
1980
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Oil price changes: 1950- 1970
400
Percentage change in oil prices
(3 month, log, annual rate)
300
200
100
0
-100
1950
1955
1960
1965
1970
1975
1980
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Oil price changes : 1950- 1973:6
400
Percentage change in oil prices
(3 month, log, annual rate)
300
200
100
0
-100
1950
1955
1960
1965
1970
1975
1980
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Oil price changes : 1950- 1974:3
400
Percentage change in oil prices
(3 month, log, annual rate)
300
200
100
0
-100
1950
1955
1960
1965
1970
1975
1980
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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:
∞
E(c) =

e - c f(c)dc = - ∞
c=0
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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Go back to earlier example. Here payoffs are c
The state of the environmental world
Good outcome (low
damage, many green
technologies)
Climate
policy
Poor outcome
(catastrophic damage,
no green technologies)
Strong policies (high
carbon tax, cooperation,
R&D)
-1%
-1%
Weak policies (no carbon
tax, strife, corruption)
0%
-50%
90%
10%
Probability
Utility function: U (c)  - (1 - c1 ) / (1   )
1
U '(c)  -c
/ (1   )
U '(1)  1 [so utility is in consumption units]
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Payoffs when act then learn with no policy and fat
tails
Act then learn
α
0
log(c)
2
3
4
6
Loss in expected
utility (c units)
No policy
-5%
-7%
-10%
-15%
-23%
-89%
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Payoffs when have policy, act then learn
Act then learn Loss in expected utility (c units)
α
0
1.01
2
3
4
6
Weak policy
-5.00%
-6.96%
-10.00%
-15.00%
-23.33%
-89.45%
Strong policy
-1.00%
-1.01%
-1.01%
-1.02%
-1.02%
-1.03%
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Comparing all learning options with catastrophes…
Act then
learn
------ Loss in expected utility (c units) ---Learn then
---------- Act then learn --------act
α
0
log(c)
2
3
4
6
Weak policy
-5.00%
-6.96%
-10.00%
-15.00%
-23.33%
-89.45%
Strong policy
-1.00%
-1.01%
-1.01%
-1.02%
-1.02%
-1.03%
-0.10%
-0.10%
-0.10%
-0.10%
-0.10%
-0.10%
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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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