risk and return in environmental economicsrisk and return in environmental economics robert s....

RISK AND RETURN IN ENVIRONMENTAL

ECONOMICS

Robert S. Pindyck

Massachusetts Institute of Technology

April 2012

Robert Pindyck (MIT) RISK AND RETURN April 2012 1 / 28

Introduction

Environmental policy imposes social costs, yields social return.Like other investments, return is uncertain.

I examine risk/return tradeoff for environmental investments.(Could also apply to other public investments.)

Two related questions:

How to characterize relative riskiness of environmentalinvestments? What are the determinants?How to trade off risk versus expected return as policyobjectives?

Focus on climate change: long time horizon and considerableuncertainty.

Costly abatement would reduce GHG emissions now, and yielduncertain future benefits.


Background

Apart from Stern Review (low discount rate, low abatementcosts, high economic impact), most studies suggest low tomoderate abatement now (or waiting).Dynamically efficient, allows for learning about T and its impact,allows for technological change (e.g., lower abatement costs).If you believe T is within IPCC’s 90% confidence interval, hardto justify stringent abatement now.Might this conclusion change if tail events have high probability?Considerable consensus as to expected temperature change; lessso over confidence intervals, extent of uncertainty.Uncertainty is at the heart of climate change policy:

Uncertainty over underlying physical/ecological processes.Uncertainty over economic impact of climate change, includingpossible adaptation.Uncertainty over technological changes that might reduceimpacts or costs of abatement.


Framework

Consider a policy to reduce expected rate of warming and/orvariance by some amounts.

Compute “willingness to pay” (WTP) for this policy.

WTP is maximum percentage reduction in current and futureconsumption society would give up to achieve that change.(Demand side of policy.)What is the trade-off between reducing expected rate of changeof temperature versus reducing the variance?What combinations of drift reduction and variance reductionyield the same WTP? Calculate “iso-WTP” curves.

Iso-WTP curve is social risk-return indifference curve. For agiven WTP, it describes “demand-side” policy tradeoff betweenrisk and return.


Basic Model

Simple model: temperature follows an arithmetic Brownianmotion (ABM), and reduces GDP growth rate.

Xt = anthropomorphic increase in temperature:

dX = αXdt + σXdz . (1)

Xt reduces real growth rate of consumption, gt :

gt = g0 − γXt , (2)

so process for gt is:

dg = −γαTdt − γσTdz ≡ −αdt − σdz . (3)

Consumption at a future time t is:

Ct = C0e∫ t0 g(s)ds = C0e

g0t− 12 αt2−σ

∫ t0 z(s)ds , (4)

Set C0 = 1.Robert Pindyck (MIT) RISK AND RETURN April 2012 5 / 28

Welfare Measure

CRRA utility. So at t = 0, welfare (under BAU) is:

W0 =1

1− ηE0

∫ ∞

0C

1−ηt e−δtdt . (5)

So we want an expression for E0(C1−ηt ).

Denote F (C , g , 0) = E0(C1−ηt ), for t > 0. Write and solve

Kolmogorov eqn. for F . (See paper.) Get:

E0(C1−ηt )e−δt = e−δt+(1−η)g0t−1

2α(1−η)t2+16σ2(1−η)2t3

. (6)

As t increases, E0(C1−ηt ) first decreases and then increases

without bound, so welfare integral must cover a finite horizon.


Willingness to Pay

Suppose at some cost α and/or σ can be reduced.Let α and σ be drift and volatility of gt under BAU Let α′ andσ′ be new drift and volatility under a policy costing w percent ofconsumption.WTP is maximum w society would accept to achieve this goal.Welfare under the policy is:

W1(α′, σ′) =(1− w)1−η

1− ηE ′0

∫ ∞

0C

1−ηt e−δtdt , (7)

where E ′0 is expectation under α′ and σ′.Under BAU (i.e., no policy), welfare is:

W2 =1

1− ηE0

∫ ∞

0C

1−ηt e−δtdt , (8)

where E0 is expectation under α and σ. Then WTP is value w ∗

that equates W1 and W2.Robert Pindyck (MIT) RISK AND RETURN April 2012 7 / 28

Iso-WTP and Iso-Cost Curves

If we specify a target α′ and σ′, can calculate WTP to achievethis target. Suppose that WTP is w ∗

1 .

We want the tradeoff between drift reduction and variancereduction as targets of policy. That is, want to know whatcombinations of drift reduction and variance reduction yield thesame WTP, w ∗

1 .

Locus of such combinations: an “iso-WTP curve.”

Given costs of reducing α and σ, could plot an iso-cost line. InFigure, costs are linear. Tangency with iso-WTP curve w ∗

1 = .03at point B is the cost-minimizing target combination (α′, σ′).However, actual cost may be > or < than w ∗

1 .


Iso-WTP Curves

Can also find locus of starting values for α and σ that yield thesame WTP for movement to any particular iso-WTP curve.

These are combinations of drift and volatility under BAU thathave the same welfare implications.

In next figure, Curve 1 gives combinations of starting α0 and σ0

such that the movement to point B has a WTP = .03. Sow ∗(A → B) = w ∗(D → B) = w ∗(A → C ) = .03. Butw ∗(D → A) = w ∗(B → C ) = 0, so w ∗(D → C ) = .03. Thusmoving from any point on Curve 1 to any point on Curve 2 hasthe same WTP.


Risk/Return for Incremental Abatement Policy

Consider abatement policy that reduces α by a small amount.

Now (stochastic) welfare is W = 11−η

∫ T0 C

1−ηt e−δtdt and

(stochastic) return from small decrease in α is:

r = −∂W

∂α=

∫ T

0

12t2C

1−ηt e−δtdt

Define ρ0 ≡ δ + (η − 1)g0. Expected return is:

re =∫ T

0

12 t2E0(C

1−ηt )e−δtdt =

∫ T

0

12 t2e−ρ0t− 1

2 α(1−η)t2+ 16 σ2(1−η)2t3

dt

(9)

Variance is V(r) = E0(r2)− (r e)2, so need to find

E0(r2) = E0

(∫ T

0

12t2C

1−ηt e−δtdt

)2


Risk/Return – Incremental Policy

Need expectation of products, G (C , g , 0) = E0(C1−ηi C

1−ηj ).

Write and solve Kolmogorov eqn., etc. (See paper.) Can show:

E0(r2) =∫ T

0

∫ T

0

14 t2s2e−ρ0(t+s)− 1

2 α(1−η)(t2+s2)+ 112 σ2(1−η)2(t+s)3dtds .

(10)

Using eqns. (9) and (10), can find (numerically) expectation,SD, and Sharpe ratio for the cumulative return r .

To illustrate this and WTP, calibrate against numbers in IPCC(2007). (Will show calibration later.)


Sharpe Ratio vs. η. (g0 = .02, δ = 0)


Mean and SD of Return. (g0 = .02, δ = 0)


Willingness to Pay

Now consider policy to move from (α0, σ0) to (α1, σ1).Notation: w1 = WTP, ρ0 = δ + (η − 1)g0 anda(α, σ, t) = −1

2α(1− η)t2 + 16σ2(1− η)2t3.

With no policy intervention, welfare is

W2 =1

1− η

∫ ∞

0e−ρ0t+a(α0,σ0,t)dt . (11)

With intervention, welfare is

W1 =1

1− η

∫ ∞

0(1− w1)1−ηe−ρ0t+a(α1,σ1,t)dt . (12)


Willingness to Pay

Equate W1 and W2 to get WTP:

w1 = 1−[G (α1, σ1)G (α0, σ0)

] 1η−1

, (13)

where G (α0, σ0) =∫ ∞0 e−ρ0t+a(α0,σ0,t)dt, and likewise for

G (α1, σ1).

So given starting values of α and σ we can calculate WTP todecrease α and/or decrease σ.


Iso-WTP Curves

Iso-WTP curves, i.e., combinations of α′ and σ′ for which theWTP is again w1, describe risk-return tradeoff. The “demandside” of policy.

Find (numerically) combinations of α′ and σ′ that satisfy

G (α′, σ′) = G (α1, σ1) . (14)

Can also obtain combinations of α′ and σ′ for which WTPequals some arbitrary number, w . From eqn. (13), findcombinations that satisfy

G (α′, σ′) = (1− w)η−1G (α0, σ0) . (15)


Calibration of α and σ under BAU

22 climate science studies surveyed by IPCC (2007) puts theexpected temperature increase (X ) by 2100 at 2.5◦C to 3.0◦C,with a 17% probability of X ≥ 4.5◦C, and 5% probability ofX ≥ 7◦C.

For 100-year horizon, I take E(X ) = 3◦C. The 5% point is 1.65standard deviations above the mean (X is normally distributed),so 1 SD is 4/1.65 = 2.42.

Thus αX = 3/100 = .03, and σX = 2.42/√

100 = 0.242.

More recent studies imply larger numbers for αX and σX , but Iwill use these earlier (perhaps overly optimistic) estimates.

We need process for the growth rate gt . Can combine expectedfuture consumption with expected loss of GDP for specifictemperature change.


Calibration (Con’t)

IPCC (2007) estimates that loss of GDP from X = 4◦C is “mostlikely” 1% to 5%. Use upper end of range, i.e., 5%.

Then take α′X = .04, i.e., E0(X100|α′X = .04) = 4◦C.

Expected consumption is (see paper):

E0(Ct) = eg0t−(1/2)α′X γt2+(1/6)σ2X γ2t3

= .95eg0t ,

so −12α′X γt2 + 1

6σ2X γ2t3 = ln(.95) .

Substituting t = 100, α′X = .04 and σX = .242:

γ2 − .0205γ + .00000526 = 0

If σX = 0, γ = .000152, so take smaller root: γ = .00026.


Calibration (Con’t)

This is an indirect estimate of γ based on several IAMS.

Bansal and Ochoa obtained direct econometric estimate (147countries, 1950–2007).

B-O estimate implies γ = .001.This is large, but transitory; GDP largely recovers after 10 years.

So I take as “consensus” something in the middle: γ = .0005.

Thus α = αX γ = .0000125 and σ = σX γ = .00012.

Parameter Values. WTP also depends on coef. of riskaversion η, rate of time preference δ, and base growth rate g0.

I set η = 2, δ = 0, and g0 = .02.

If δ ≥ .01, WTP close to zero.

Figures show results for iso-WTP curves.


Iso-WTP Curves (η = 2, g0 = .02, δ = 0)


Calibration: Costs of Reducing α and σ

I assume a linear cost function, i.e., cost of moving from(α0, σ0) to (α′, σ′) is:

Cost = c1(α0 − α′) + c2(σ0 − σ′) (16)

Cost and its components are percentages of consumptionsacrificed annually. So c1(α0 − α′) is percentage of consumptiongiven up each year to reduce α from α0 to α′ < α0.

Estimating c1: U.S. EIA estimated cost of compliance withKyoto Protocol at 1–3% of GDP.

Protocol implies X2100 ≤ 3◦C.Use mid-range cost of 2%; take “X2100 ≤ 3◦C” to meanE(X2100) = 1.5◦C with SD = 1.5◦C.Implies cutting α0 from .000015 to .0000075 at cost .02.So c1 = (.02)/(7.5× 10−6) = 2667.


Costs of Reducing α and σ (Con’t)

Estimating c2: In 2009, public funding in US for climatechange research was about $5 billion.

Make bold assumption that doubling this to $10 billion per yearcould cut σ0 in half, from .00012 to .00006.

$10 billion ≈ 0.067% of GDP.

Implies c2 = .00067/.00006 = 11.17.

Figures show results for iso-WTP and iso-cost curves. At(α′, σ′), slope = −c2/c1 so cost is minimized.

First figure: Cost = WTP = .02, so policy is just feasible.

Second figure: η = 1.5, so smaller reductions in α, σ have WTP= .02, and Cost = .008. So policy has positive social surplus.

If η = 3, would have Cost > WTP, and policy not feasible.


Iso-WTP and Iso-Cost Curves: η = 2


Iso-WTP and Iso-Cost Curves: η = 1.5


Conclusions

I presented a framework for trading off risk vs. expected benefitsas policy targets.

Based on “iso-WTP” curves (social indifference curves) forcombinations of risk and expected returns as policy targets.

Given cost estimates, one can compute the optimal risk-returnmix for policy, and evaluate the policy’s social surplus.

Framework was illustrated using simple ABM process fortemperature, and CRRA utility.

Caveats: Plenty.Costs of reducing α and σ probably convex, not linear.I assumed that we can change α and σ instantly. Unrealistic.I calculated WTP assuming costs are constant percentages ofGDP. Too restrictive.Alternative process for Xt , gradual change in α or σ, cost flowvaries over time — need numerical solutions.


risk and return in environmental economicsrisk and return in environmental economics robert s....

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