Discounting, Equity, Uncertainty

• The problem• The solution

– What will it cost?– What to do?

Optimal emission reduction• Climate change is long-term, global,

uncertain problem• If you do not care about the distant

future, far-away lands, remote probabilities, you do not care about climate change

The Ramsey Rule• The net present value of $1 in t years

time

• Equate marginal utility now to marginal utility then, that is, you are indifferent between getting ε now or 1 later

• Or

rte

01

t

tC CU e U

0

0

0 0

t

t t gttC t

C

e U e C ee CU C C

The Ramsey Rule -2

• That is

• Or

0

0

0 0

t

t t gttC t

C

e U e C ee CU C C

rt t gte e

r g

Time discounting• The Ramsey rule of discount

• That is, we discount the future because– We will be richer and happier then– We are impatient

• Note that the Ramsey rule describes the consumption side of intertemporal trade-offs – on the production side, there is the opportunity cost of capital

• In equilibrium, the two are equal

r g

Alternative discounting• Discounting should

– Reflect the will of the people– Be consistently applied accross public policy– Not ignore the far future

• Can this be done?• There are three contenders• Conventional, exponential discount rates

have that the relative distance between two years does not depend on the time horizon:

• exp(-ρ(t+δ)) / exp(-ρt) = exp(-ρδ)

Hyperbolic Discount Rates• The relative distance between two years does

not depend on the time horizon:• exp(-ρ(t+δ)) / exp(-ρt) = exp(-ρδ)

• This is strange: The difference between year 10 and year 11 is the same as the difference between year 100 and year 101, and between 1000 and 1001

• There is experimental and observational evidence that suggests that people use a lower discount rate when they look further into the future

Hyperbolic Discount Rates -2• There is experimental and observational

evidence that suggests that people use a lower discount rate when they look further into the future, perhaps

• exp(-ρln(t+δ)) / exp(-ρlnt) = exp(-ρln(1+δ/t))• Then, the difference between year 10 and

year 11 is the same as the difference between year 100 and year 110, and between 1000 and 1100

Declining Discount Rates• A similar (but not identical) effect can be

achieved through uncertainty0.04 0.01 0.07 (0.01+0.07)/2 DDR

1 961.5385 990.0990 934.5794 962.3392246 0.039110 675.5642 905.2870 508.3493 706.8181234 0.035350 140.7126 608.0388 33.9478 320.9932921 0.0230

100 19.8000 369.7112 1.1525 185.4318313 0.0170150 2.7861 224.7988 0.0391 112.4189471 0.0147200 0.3920 136.6864 0.0013 68.34385433 0.0135300 0.0078 50.5345 0.0000 25.26724449 0.0123400 0.0002 18.6832 0.0000 9.341583311 0.0118

Declining Discount Rates -2• A similar (but not identical) effect can be

achieved through uncertainty• If we assumed that the discount rate is

uncertain and follows a Gamma distribution, then the certainty-equivalent discount rate equals

• r(t) = α / (1+βt)• where α is a location parameter and β a

spread parameter

Alternative welfare functions• Over an infinite time horizon, a welfare

function cannot simultaneously satisfy Strong Pareto and Anonimity– If one is better off and no one is worse, welfare

improves• Discounted welfare violates Anonimity

– 1 1 2 1 1 1 …– 1 2 1 1 1 1 …

• Koopmans argued that Anonimity is less important

Alternative welfare functions -2• Chichilnisky replaced Anonimity with Non-

Dictatorship and Independence• Welfare is then a weighted sum of net present

welfare and welfare of the last generation• Alvarez-Cuadrado and Van Long dropped

Independence• Welfare is then a weighted sum of net present

welfare and welfare of the worst-off generation

• As long as growth continues, this reduces optimal emission reduction

1( ) ( ( )) d ( )t

t

W U C t e t U C