1 Economics 331b Treatment of Uncertainty in Economics (II)

The payoff matrix (in utility units) 2

Optimal policy with learn then act 3 Expected loss = 90% x % x -1 = -0.1%

ACT in future High damages High carbon tax Low carbon tax Low damages This example: Learn then act LEARN TODAY

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” 5

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? 6

LEARN 2050 ACT TODAY ? Low damages High damages Realistic world: Act then learn

Optimal policy with act then learn 8 Expected loss depends upon strategy: strong: 90% x % x -1 = -1% weak : 90% x % 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!) 9

10 The problem of fat tails Units of dispersion (sample standard deviation)

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. 11

12 Black swans in South Africa (“Birds of Eden”)

Some examples Height of American women: Normal N (64”,3”). How surprised would you be to see a 14’ person coming to Econ 331? 13

Some examples Stock market: what is the probability of a 23% change in one day for a normal distribution? Circa !!! - Mandelbrot found it was Pareto with β = Finite mean, but infinite variance Earthquakes: Cauchy distribution β = 1 (see next slide). -Infinite mean, infinite variance 14

Distribution of earthquake energy 15 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. 16

Some examples Climate damages (fat tailed according to Weitzman, but ?) 17

Distribution of damages from RICE-2010 model 18

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. 19

Let’s look at the moving history of oil price changes:

Oil price changes:

Oil price changes : :6 22

Oil price changes : :3 23

Revisit economists’ approach to uncertainty - Combine structural modeling, subjective probability theory, and Monte Carlo sampling. - Dynamic system under uncertainty: (1)y t = H( θ t, μ t ) - 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. 24

Weitzman’s contribution Weitzman showed that with fat tailed distribution, might have negative infinite utility, and no optimal policy. 25

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. 26

Go back to earlier example. Here payoffs are c 27

Payoffs when act then learn with no policy and fat tails 28

Payoffs when have policy, act then learn 29

Comparing all learning options with catastrophes… 30

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. 31