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William D. NordhausYale University

Summer School in Resource and Environmental Economics

Venice International UniversityJune 30 – July 6, 2013

Integrated Assessment Models:Introduction (I) and Uncertainty (II)

Lectures

I. Introduction to Integrated Assessment Models II. Applications to Uncertainty

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Some Critical Issues in Uncertainty in Climate Models

1. What are standard estimates of uncertainty?2. What is the effect of uncertainty compared to a

certainty equivalent?3. What is the state-contingent approach?4. What are the state-contingent policies?5. What is the welfare loss from uncertainty?6. What is the cost of not knowing the state of the

world (i.e., acting before learning)?7. What is the value of early knowledge?8. What is the precautionary principle, and what are

the pros and cons of using it?9. [What are the implications of “fat tailed”

uncertainty?]10.How does uncertainty affect the discount rate?

Dilemmas of uncertainty

• Uncertainty and the potential for catastrophic impacts are a major concern in global warming science and policy today.

• Some major risks include:– Reversal of North Atlantic deepwater circulation– Melting of Greenland and West Antarctic ice sheets– Abrupt climate change– Ocean carbonization

• How can we model risk and uncertainty in IAMs?• I will give an overview and an example.

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Hysteresis loops for Ice Sheets and the “Tipping Point”

5Frank Pattyn, “GRANTISM: Model of Greenland and Antrarctica,” Computers & Geosciences, April 2006, Pages 316-325

General PrinciplesIntertemporal choice. Discounted utility (DU) model,

where U(c) is cardinal utility over alternative bundles.Uncertainty. Very similar model is choice under uncertainty.

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tW U[c(t)]e dt

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EU U[c( )] f( )d

General PrinciplesIntertemporal choice. Discounted utility (DU) model,

where U(c) is cardinal utility over alternative bundles.Uncertainty. Very similar model is choice under uncertainty.

This is expected utility or EU or von Neumann-Morgenstern model: , where U is a cardinal utility, π is a “state of the world,” and f( ) is the probability distribution. In this approach, decision makers wish to maximize EU. (This is like Arrow-Debreu model.)

With two SOW, have

What is U function in EU model? It is the value of consumption or wealth in certain prospects.

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tW U[c(t)]e dt

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EU U[c( )] f( )d

1 1 2 2EU U(c ) U(c )

With linear systems

If the system is linear, then optimal policy is certainty-equivalent; e.g., uncertainty is irrelevant.

Proof left to student. A simple example is the following. Show that the optimal P is invariant to σ(ε).

Max E[U(Y)] = E( a + bY +c Y2 ) = Expected value of UwhereY = kP + εY = income; P = policy; ε = additive uncertainty; σ(ε)

= standard deviation ε; a, b, k, σ are known parameters.

This is Tinbergen-Theil certainty equivalent theorem. See Readings.

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Monte Carlo Approach (Also Learn then Act)

1. Common approach: combines modeling, subjective probability theory, and Monte Carlo sampling.

2. Begin with a structure like DICE model equations. Can represent schematically as follows:(1) yt = H(zt ; θ)

where yt = the endogenous variables (output, emissions, etc.)

zt= exogenous and non-stochastic variables (financial meltdown, land-based emissions, etc.)θ = [θ1, … , θn] = uncertain parameters (including functional forms)

3. We then develop subjective probabilities for major parameters, f(θ).

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Example of temperature distribution

Implementation in DICE model• For DICE model, we examine 100 random runs,

where these are baseline for 8 uncertain parameters.

• Then fit a “response-surface model” that estimates major outcomes as function of uncertain parameters

• These provide information about what the uncertainties are for major variables.

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-2 -1 0 1 2 3 4 5 6 7 8 9

Observations 10,000Mean 3.65Median 3.66Maximum 8.76Std. Dev. 1.49

Temperature, 2100 (deg C over 1900)

Monte Carlo:Baseline run DICE-2013

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A Simple Example

For our example, we will assume that there is one major uncertainty: The damage function is D=YθT2 , θ uncertain.

Assume two states of the world, SOW1 and SOW2.

SOW2 has a damage function that is ten times higher than SOW1, or θ2 = 10 θ2

But we are uncertain about the damages.– State of world 1 (SOW1) has p = 90%– Catastrophic case 2 (SOW2) has p = 10%

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0%

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Dam

age

as %

of i

ncom

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Temperature increase (oC)

Damage % by SOW

SOW2 SOW1

ACT now and in future

High damages

High carbon tax

Low carbon taxLow damages

This example: Learn then act

LEARNTODAY

Assume that we learn about SOW right away

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2015 2020 2025 2030 2035 2040 2045 2050 2055

Pric

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State-dependent carbon prices

Backstop price

Hi Dam: Early Learn

Low Dam: Late Learn

What is wrong with this story?

This 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 30 years.

It means that we must set policy now for both SOW; we can make state-contingent policies after 30 years.

How will that affect our optimal policy?

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LEARN 2045ACT

TODAY?

Low damages

High damages

Realistic world:Act then learn

Optimal policies with 30 years of ignorance

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2015 2020 2025 2030 2035 2040 2045 2050 2055

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Carbon prices with late learning

Hi Dam: Late Learn Low Dam: Late Learn

Optimal policies with late and early learning

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2015 2020 2025 2030 2035 2040 2045 2050 2055

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State-dependent carbon prices

Hi Dam: Early Learn Hi Dam: Late Learn

Low Dam: Late Learn Lo Dam:Early Learn

Compare average early and late learning

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2015 2020 2025 2030 2035 2040 2045 2050 2055

Aver

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n CO

2)Average carbon price in LTA and ATL

Act then learn Learn then act

Lessons

1. Use schema of state-dependence prices and outputs to analyze learning

2. Critical issue of when we learn for policy.3. In the case here, delayed learning leads to

higher average carbon price.- In this extreme case, difference is 10-20%- Generally less in most models and scenarios

4. Note that prices (policies) are close to each other after we learn. Here, no big effect on post-learning policy.

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Aggregate costs and benefits

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Value of learning: $4,710 billion in this catastrophic situation. Economists are earning their wages.

Net world income Trillions of $

Good SOW 2,711.70

Expected value of both SOW

at p(SOW2) = 0.1

Learn then act 2,691.97 Act then learn 2,687.26

Cost of horrible damages Learn then act 19.73 Act then learn 24.44

Value of early knowledge 4.71

Per capita costs and benefits

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Values not huge relative to income.

Net world income Per capita annuity $Good SOW 19,369 Expected value of both SOWat p(SOW2) = 0.1 Learn then act 19,228 Act then learn 19,195

Cost of horrible damages Learn then act 141 Act then learn 175

Value of early knowledge 34

Precautionary principle

1. Act to minimize the damage in the worst case.2. Generally, not a sensible strategy. Too risk

averse.3. However, if you never learn, it is a better

strategy that EU strategy which doesn’t take into account that you do not know the SOW.

4. In catastrophic situation without learning, the precautionary principle may be a better rule than the EU rule.

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Precautionary v. EU in catastrophic situation

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2015 2035 2055 2075 2095

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Learn, then act

Act, then learn

Carbon tax: With Learning

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SOW

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Begin with a utility function, U(C). Examine the expected utility

in year t (say 2100). This is:

E(U) = [ ( )]

In this, are probabilities, and =[ ,..., ] are uncertain

i

ii i i

p U C

p

i

i

θ

θ

parameters, and are uncertain states of the world (SOW). In

general, we apply risk premia to SOWs where '[ ( )] or the

MU of consumption is high. The question is, when do these occur?

i

U C iθ

Macroeconomic risks and climate change

What is relationship between risk and discounting?

Some have argued that we should use a lower discount rate to reflect climate risks. What is the appropriate approach.

(Best reading and highly recommended: Gollier, Pricing the Planet, Princeton U Press, 2012.)

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The macro risk question: theory

Should we be paying a large risk premium on high-climate scenarios? The idea is that we would we invest more than the certainty equivalent to prevent the high-climate outcome.

Earlier studies generally assume yes. These approaches applied a risk-averse utility function to the damages. (This was the approach in Nordhaus-Boyer 2000 and Stern 2007.)

This is a partial-equilibrium approach. The correct approach would be to apply modern risk analysis in the consumption capital asset pricing model.

In this framework, want to pay a risk premium ↔ the high-climate outcome is positively correlated with high MU consumption (or negatively correlated with consumption).

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Consumption and 2100 Temperature

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Temperature change, 1900-2100

DICE model runsDICE response surface runs

log

(per

cap

ita

cons

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ption

, 210

0)

The red dots are 100 runs of DICE model with random draw of uncertain parameters

The blue dots are 10,000 runs of response-surface model.

Key result is that per capita consumption is positively correlated with temperature.

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Implications for Risk Premium

Study suggests that the most important uncertainty in long-run is growth in productivity. High climate damages are associated with high growth rates of TFP.- So good economic news = bad climate news in

baseline.- Think of the $2500 car.

But this also means that the worst climate cases are ones in which the world is rich, which is a situation where we are more likely to be able to afford more costly climate abatement. Leads to higher discount rates (lower prices in state-contingent approach).

This is an open issue. Whether should have positive or negative pricing on climate not settled.

ConclusionsUncertainty does not lead to major change in state-

dependent policy unless there are major non-linearities or learning.

When you have learning, the structure of decision making is very different; it can increase or decrease early investments.

In cases where there are major catastrophic damages, value of early information is very high.

Another important issue is whether should apply higher discounting to climate investments because of risks; open issue.

Best investment is sometimes knowledge rather than mitigation

… that is why we are at summer school!32

Readings: See end of first lecture

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