Hyperbolic Discounting is time consistent: Discounting the far future with uncertain

discount rates2009

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J. Doyne FarmerJohn Geanakoplos

The Environment

• How much should we do today to make the environment better in 200 years or 500 years?

• How to trade off the present vs the future?• Economists all seem to agree we should

exponentially discount the future at some rate.

• Conservatives say 3% per year. Nordhaus.• Liberals say 0.5% per year (Stern report).

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Discounting the future• How does one compare something today with something

tomorrow?• How do we value something for current generations in

comparison with future generations?• Ramsay (1928): For consumption stream (x0,x1,x2,…)

• U(x) = u(x0) + D(1)u(x1) + D(2)u(x2) + … • Ramsay argued for D(t) = 1

– To discount later generations in favor of earlier ones is “ethically indefensible and arises merely from the weakness of the imagination”

– … it is “a polite expression for rapacity and the conquest of reason by passion” (Harrod, 1948)

– Reinterpret as consumption stream by same agent; get discounting.

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Reasons for Discounting

• Impatience: Fisher, Shakespeare• Probability of Death: Rae• Failure of imagination: Bohm-Bawerk

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Exponential discounting

• Standard approach in neoclassical economics is exponential discounting (Samuelson).

• = 1/(1+r0)τ

• Analogous to present value with constant bank interest rate r.– At time you would have– Discount for time is therefore

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D e r

er 1

money now

money latere r

er 1

Value of far future under exponential discounting?

• Under exponential discounting with realistic interest rates, the far future is not worth much

• E.g., with interest rate of 6%, 100 years out the discount factor is 0.0025.

• This is used by some economists to argue that we should put very little effort into coping with phenomena such as global warming that create problems in the far future.

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Copenhagen Consensus(eight leading economists, four Nobel prize winners)

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QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.Concerning global warming:“If we use a large discount rate, they will be judged to be small effects” (Robert Mendolson, criticizing an analysis by Cline using 1.5% discounting)

Bjorn Lomborg

Discounting of far future is very sensitive to the interest rate

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interest rate 10% 5% 1%discount factor 5 x 10-5 7 x 10-3 0.37

100 years into the future:

So how to pick the discount rate?

Market interest rates• We can see what the market interest rates are.• At the moment they are the lowest ever. • 1% per year for under a year, rising to 3% per year or so

in 10 years. Seems to stay thereafter.• But don’t have interest rates for beyond 30 years in

heavily traded markets. Most bonds of 30 year maturity or less.

• Some English consols. Also old railroad bonds. Trade for very low interest rates. Curiosity?

• Must make up rates beyond 30 years.

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Hyperbolic discounting

• Early D(t) goes down exponentially, but for large t, D(t) goes down slowly.

• D(t+1)/D(t) → 1.• The most commonly used functional form

with this property is

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D(t) (1t)

•α = 1, β = ½

•D(t) ≈ 1/√t

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StrotzLaibsonLoewensteinAinslieAinslie-Hernnstein

Animals and Real People Are

Hyperbolic Discounters

E.g. Thaler experiment

• How much money would you need in the future in lieu of $15 today?

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time amount discount interest rate

month $20 D(1) =15

200.751 345%

year $50 D(12) 15

500.9012 120%

10 years $100 D(120) 15

1000.98120 19%

• Fits hyperbolic model with β = 1/2

Rabin Story

• Girl asked to clean her room today vs tomorrow. Much rather do it tomorrow.

• When asked to clean her room in 365 days or 366 days, it doesn’t matter to her. The ratio of what she would pay today to get out of doing it in 365 days to getting out of it in 366 days is barely bigger than 1.

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Iroquois constitution

• Gayanashagowa -- Great Law of Peace --constitution of the Haudenosaunee

• In every deliberation we must consider the impact on the 7th generation … even if it requires having skin as thick as the bark of a pine.

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Even animals use hyperbolic discounting

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Widely viewed as “irrational”, or at least “behavioral”.

Is hyperbolic discounting time consistent?

• Clearly agents could not be Samuelson discounters.

• If D(t)/D(t+1) goes down, is that time consistent behavior?

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Hyperbolic discounting seems irrational

• Only way for D(t+1)/D(t) → 1 seems to be if people think in the future they will become more patient.

• Or in future will become less likely to die in one year.

• Or will develop better imagination about the future as they get older.

• All implausible.

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Hyperbolic discounting is irrational

• If world is certain• People do not think they will grow more

patient, or less likely to die, etc

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Rabin Story

• Girl asked to clean her room today vs tomorrow. Much rather do it tomorrow.

• When asked today to clean her room in 365 days or 366 days, it doesn’t matter to her.

• But if asked today whether she thinks in 365 days if she is asked to clean her room it will matter whether it is then or the next day, she will say it likely will matter

• Sounds time inconsistent, and Rabin and most others agree. They exult in the irrationality.

• But public policy should be rational!

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Solution: One Period Discount is Random!

• Future interest rates are not known today for sure.• People don’t know how urgent their one period

impatience will be.• Death probabilities vary.• An entire industry on Wall Street built to analyze

values when future interest rates unknown.• How does this help if future interest rates on

average are at least as high as today?

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Geometric Random Walk Interest Rate Model

• Called Black-Derman-Toy model• (Ho-Lee model same but with random walk)• Workhorse of finance.• Analyzed to death for t < 30 years• But not for large t

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v = volatility

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D(t) = how much would you pay today for $1 for sure at time t

• Clearly discount factor D(t) must depend on what market expects future one period discounts to be; otherwise there would be arbitrage opportunities.

• Two wrong answers! • D(τ)=e-E₀[r₀]e-E₀[r₁]...e-E₀[rτ-1] < e-r₀τ

• D(τ)=E₀[e-r₀]E₀[e-r₁]...E₀[e-r(τ-1)] ≈ K(1/2)τ

• Both wrong answers lead to more discounting!

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D(t) = how much would you pay today for $1 for sure at time t

• Suppose common knowledge that at any time t, can always make bet at even odds that interest rate will go up or down.

• Then correct answer by no-arbitrage must be average product of one period discounts over all paths to period t.

• D(τ)=E₀[e-r₀e-r₁...e-r(τ-1)]•25

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v = volatility

Theorem

• In geometric random walk, the discount factor D(τ) goes down exponentially at first, at rate faster than r0, but converges to

• D(τ) = Kτ-1/2 as τ → ∞. So hyperbolic discounting is rational.

• D(t+1)/D(t) ≈ √(t+1)/√t• Length of time before entering hyperbolic region is

shorter if vol is higher.• Here K is a constant, or maybe a slowly varying

function like 1/log. Logt/ √t is tiny.•27

r0=4%, v = 50%

•28Farmer and Geanakoplos

Comparison of discount factors x 100 (15% annual volatility, 4% initial rate)

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year rnd. wlk. constant20 46.2 45.660 12.5 9.5

100 5.1 2.0500 0.80 2 x 10-7

1000 0.50 4 x 10-16

Fits Thaler data

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Why is this true?

• Think of one period discount as coming from one year death probability.

• Hazard probability follows random walk.• Conditional on living for 100 years, likely were

following path with very low one year death rates. If one year death probs got bad, you would already be dead.

• Hence conditional probability of living one more year after making it to 100 is very high.

• Familiar idea in economics.•31

Where does 1/√t come from?

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2n

n≈ 22nK/√n

Proportion of 22n possibilities is about K/√n ≈ K/√tt = 2n

Idea of Proof

• Consider the case where volatility v = ∞.• Then have three kinds of paths:• Good paths: Those in which from time 1

onward remain strictly below median.• Mediocre paths: Those in which from time 1

onward hit median but remain below median.• Bad paths: Those in which at some date go

above median. These contribute zero to value

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r0

Good Path: starts down and never hits r0

0

-1

-3

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r0

Another Good Path: same number of ups as downs

0

Good paths• Might as well start all paths at -1 at time 1, and

go for T-1 periods = 2n.• End up at -1 or -3 or -5 etc.• So can count total number of good paths by

adding number that end at -1 plus number that end at -3 plus number that end at -5 etc.

• Total paths that start at -1 and end at -1 have right proportion 1/√T. But need to subtract out non-good paths that start and end at -1, and add good paths that end at -3 plus -5 etc.

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r0

Bad Path

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r0

Reflection Principle

Reflection Principle for paths end -1

• Number of non-good paths that start at -1 and hit or cross 0 and end at -1 is equal to all paths that start at +1 and end at -1.

• But that is equal to number of all paths that start at -1 and end at -3.

• Hence number of good paths that start at -1 and end at -1 is equal to the number of all paths that start at -1 and end at -1 minus the number of all paths that start at -1 and end at -3.

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Reflection Principle for paths end -3

• Number of non-good paths that start at -1 and hit or cross 0 and end at -3 is equal to all paths that start at +1 and end at -3.

• But that is equal to number of all paths that start at -1 and end at -5.

• Hence number of good paths that start at -1 and end at -3 is equal to the number of all paths that start at -1 and end at -3 minus the number of all paths that start at -1 and end at -5.

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Number of good paths equals

• All paths that end at -1 minus all paths that end at -3

• Plus• All paths that end at -3 minus all paths that

end at -5• Plus etc

• Equals all paths that end at -1.•41

End of Proof for v = ∞

• So all good paths has right proportion of all paths.

• Must count total number of mediocre paths that hit 0 but do not cross zero.

• Same technique can be used to show that is exactly equal to number of paths that never hit 0. These paths all get discounted.

• In fact can compute how many paths get discounted k times, for each k.

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Proof for v < ∞

• Key idea is that when interest rate goes down exponentially, discount rate goes up doubly exponentially. So after logT periods of going down, discount factor is essentially 1.

• D = 1/(1+r0(e-v)(1/v)logT)= 1/(1+r0/T)

• Even if one period discount factor is D from then on until T, get virtually no discounting

• DT = (1/(1+r0/T))T = e-r0

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r0

Strip: Everything Below is Good Path

(1/v)logT

Good path starting here

Period TPeriod 0

Proof for v < ∞

• Let N = logT + 1. Note N/√T ≈ 0.• So probability that path starting at 0 goes first to -1 is

½.• For T large probability that path exits strip is ≈ 1.• Then probability the path exits at bottom of strip

before exiting at top is 1/N by gambler’s ruin theorem.• Once out at bottom fraction of good paths that never

enter strip again is at least 1/√T.• So get D(t) ≥ 1/2N√T times discounting while in strip.

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Discounting While in Strip

• Actually spend pretty long in strip before exiting at bottom. If visited every line equally while in strip before exiting at bottom, then too much discounting.

• But can show spend most of time while in strip before exiting at bottom near middle of strip, where discount factor already near 1.

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Discounting While in Strip• Let Π(i) = i/N = prob of exiting at N starting at i.• Let E(i,k) = expected number of times you hit k

starting at i before exiting, with i≤ k ≤ N.• Clearly E(i,N) = Π(i).• Π(i) = E(i,k)(1/2)(1/(N-k)) if i < k < N• E(i,k) = 2 Π(i)(N-k)• Let W(i,k) = expected hits of k that also exit at N.• W(1,k) = (k/N)E(1,k) = 2k(N-k)/N2

• W(1,k)N = 2k(N-k)/N = expected number of hits of k starting at 1 conditional on exiting at N

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Generalizatios

• Asymptotic Behavior of Stochastic Discount Rate

• Geanakoplos-Sudderth-Zeitouni• Instead of binary, let v be any bounded

random variable with zero mean and positive variance. Then

• 1/√t-o(1) < D(t) < 1/√t+o(1)

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End

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Utility today and in the future over certain prizes

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Conclusion from certainty and time consistency

Discount factor Ds(τ) must be the productof one-period discounts Ds+i(1).

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Certainty, time consistency and strict stationarity

• Exponential discounting is time consistent and strictly stationary, I.e. rs+τ = r is constant independent of t.

• Samuelson exponential discounting is the only time consistent, strictly stationary discounting function under certainty.

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Give up strict stationarity

• But then must have Ds(1) declining with time s.

• This is implausible because death hazard is increasing not decreasing with age. People do not get more patient with age; more likely they undertake fewer long term projects. That is point of Rabin story.

• So it would seem that hyperbolic discounting is incompatible with time consistency, i.e. incompatible with rationality.

• But it is compatible with time consistency if we give up certainty as well.

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The discount is not constant• Urgencies vary• Hazards vary• Interest rates vary• The future is uncertain, and uncertainties are typically correlated in

time.• Under these circumstances, on average hyperbolic discounting is

time consistent -- each step uses exponential discounting, but at varying rates. Result is not exponential!

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Key uncertainty

• One period discount Ds+τ(1) is not known at s

• So utility of x depends not just on time s that one considers x, but also the psychological one-period discount rate r at that time, Us,r(x).

• Can rehabilitate weak stationarity by requiring that if x is constant, then utility should depend on r but not on s.

• Can also assume that given r, r one period later is likely to be at least as high, so that on average people get more impatient

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time consistency with certainty to time consistency with

uncertainty

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Discounting certain payoffs with uncertain discount rates

• As long as the payoffs are certain, even if they depend on time, it makes sense to compare consumption today vs consumption at a fixed time in the future. Let us call the resulting certainty discount factors Ds(τ).

• Time consistency requires • Ds(τ) = average[exp(Σi=0

τ-1-rs+i)

• Familiar theorem from interest rate models•58

Weitzman example• Suppose interest rate has probability ½ of being high R per period

forever after, and probability ½ of being low r per period forever after.

• Then Ds(τ) converges to 1/2e-rτ as τ → ∞.

• This is known as convexity if finance. Comes from autocorrelation of interest rates.

• Weitzman argued that for long horizons like environment, should think of very low interest rates.

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Discount factors in geometric random walk

• Where N(τ) = 2τ-1.

• When payoffs xs are certain functions of time

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Theoretical explanation

• Consider high volatility limit• Discount rate tree has a “cliff”: 0 or 1• Discount rate is fraction of paths that do not

cross the cliff.• Random walk with barrier crossing• Scales as • Implies non-integrability!

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t 1/ 2

Who is the better economist?

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pigeon

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12 economists in Copenhagen consensus

r0={.5, 1,100}%, v = 100%

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