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Economics 331b

Treatment of Uncertaintyin Economics (I)

This week

1. Dynamic deterministic systems2. Dynamic stochastic systems3. Optimization (decision making) under

uncertainty4. Uncertainty and learning5. Uncertainty with extreme distributions (“fat

tails”)

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Deterministic dynamic systems (no uncertainty)

Consider a dynamic system:

(1)yt = H(θt , μt )

where yt = endogenous variables

θt = exogenous variables and parameters

μt = control variables

H = function or mapping.

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1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

ExampleAge of Ronald Reagan (years)

Deterministic optimization

Often we have an objective function U(yt ) and want to

optimize, as in

max ∫ U(yt )e-ρt dt

{μ(t)}

Subject to yt = H(θt , μt )

As in the optimal growth (Ramsey) model or life-cycle model of consumption.

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Mankiw, Life Cycle Model, Chapter 17

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Stochastic dynamic systems (with uncertainty)

Same system:

(1)yt = H(θt , μt )

θt = random exogenous variables or parameters

Examples of stochastic dynamic systems? Help?

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Which is stock market and random walk?

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RANDFSP FSPCOMT=year 0 T=year 60

Examples: stock market and random walk

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RANDFSP FSPCOM

Randomwalk

US stock market

How can we model this with modern techniques?

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Methodology is “Monte Carlo” technique, like spinning a bunch of roulette wheels at Monte Carlo

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How can we model this with modern techniques?

Methodology is “Monte Carlo” technique, like spinning a bunch of roulette wheels at Monte Carlo

System is:

(1)yt = H(θt , μt )

So,

• You first you find the probability distribution f(θt ).

• Then you simulate (1) with n draws from f(θt ).

• This then produces a distribution, g(yt ).

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How can we model this with modern techniques?

YEcon Model

An example showing how the results are affected if we make temperature sensitivity a normal random variable N(3, 1.5).

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50 random runs from RICE model for Temp

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Temperature 2100

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Series: Temperature 2100Observations 1000

Mean 3.28Median 3.26Maximum 5.953Minimum 1.267

Temperature degC above 1900

How do we choose?

• We have all these runs, ytI , yt

II , ytIII ,…

• For this, we use expected utility theory.

max ∫ E[U(yt )]e-ρt dt

Subject to yt = H(θt , μt ) and with μt as control variable.

• We usually assume U( . ) shows risk aversion.

• This produces an optimal policy.

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SCC 2015

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0.0 12.5 25.0 37.5 50.0 62.5 75.0 87.5 100.0

Series: SCC 2005$ per ton CO2 Observations 999

Mean 11.07505Median 8.612833Maximum 98.66368Minimum 1.157029Std. Dev. 9.028558

So, not much difference between mean and best guess. So we can ignore uncertainty (to first approximation.)

Or can we?

What is dreadfully wrong with this story?

What is dreadfully wrong with this story?

The next slide will help us think through why it is wrong and how to fix it.

It is an example of – 2 states of the world (good and bad with p=0.9 and

0.1),– and two potential policies (strong and weak),– and payoffs in terms of losses (in % of baseline

utility or income).

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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%