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

This week 1.Dynamic deterministic systems 2.Dynamic stochastic systems 3.Optimization (decision making) under uncertainty 4.Uncertainty and learning 5.Uncertainty with extreme distributions (“fat tails”) 2

Deterministic dynamic systems (no uncertainty) Consider a dynamic system: (1)y t = H(θ t, μ t ) where y t = endogenous variables θ t = exogenous variables and parameters μ t = control variables H = function or mapping. 3

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Deterministic optimization Often we have an objective function U(y t ) and want to optimize, as in max ∫ U(y t )e -ρt dt {μ(t)} Subject to y t = H(θ t, μ t ) As in the optimal growth (Ramsey) model or life-cycle model of consumption. 5

Mankiw, Life Cycle Model, Chapter 17 6

Stochastic dynamic systems (with uncertainty) Same system: (1)y t = H(θ t, μ t ) θ t = random exogenous variables or parameters Examples of stochastic dynamic systems? Help? 7

Which is stock market and random walk? 8 T=year 0T=year 60

Examples: stock market and random walk 9 Random walk US stock market

How can we model this with modern techniques? 10

Methodology is “Monte Carlo” technique, like spinning a bunch of roulette wheels at Monte Carlo 11 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)y t = H(θ t, μ t ) So, You first you find the probability distribution f(θ t ). Then you simulate (1) with n draws from f(θ t ). This then produces a distribution, g(y t ). 12 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). 13

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50 random runs from RICE model for Temp 15

Temperature

How do we choose? We have all these runs, y t I, y t II, y t III,… For this, we use expected utility theory. max ∫ E[U(y t )]e -ρt dt Subject to y t = H(θ t, μ t ) and with μ t as control variable. We usually assume U(. ) shows risk aversion. This produces an optimal policy. 17

SCC 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?

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). 19

The payoff matrix (in utility units) 20