How Should the Distant Future be Discounted When

Discount Rates are Uncertain?

Christian Gollier and Martin L. Weitzman∗

November 7, 2009.

Abstract

It is not immediately clear how to discount distant-future events, like climate

change, when the distant-future discount rate itself is uncertain. The so-called “WeitzmanGollier puzzle” is the fact that two seemingly symmetric and equally plausible ways

of dealing with uncertain future discount rates appear to give diametrically opposed

results with the opposite policy implications. We explain how the “Weitzman-Gollier

puzzle” is resolved. When agents optimize their consumption plans and probabilities

are adjusted for risk, the two approaches are identical. What we would wish a reader

to take away from this paper is the bottom-line message that the appropriate long run

discount rate declines over time toward its lowest possible value.

1 Introduction

The concept of discounting is central to economics, since it allows effects occurring at different

future times to be compared by converting each future dollar into a common currency of

equivalent present dollars. Because of this centrality, the choice of an appropriate discount

rate is one of the most critical issues in economics. It is an especially acute issue for projects

involving long time horizons because in such situations the results of cost-benefit analysis

(CBA) can be incredibly sensitive to even tiny changes in the discount rate.

The primary motivator for this paper is the economics of climate change. Answers to

questions thrown up by the climate-change CBA hinge critically on the core issue of how

∗Gollier: Toulouse School of Economics (gollier@cict.fr); Weitzman: Department of Economics, Harvard

University (mweitzman@harvard.edu). The research leading to these results has received funding from the

European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-

2013) Grant Agreement no. 230589.

1
to discount the distant future. There is a high degree of uncertainty about what should

be taken as the appropriate rate of return on capital in the long run, accompanied by much

controversy about its implications for long-run discounting. The investigation of this paper is

focused sharply on the well defined sub-problem of how to perform CBA of small incremental

investments that might alter distant-future events, although the overall ramifications of the

paper are somewhat broader.

The effects of global warming and climate change will be spread out over what might

be called the “distant future” — centuries (or even millennia) from now. The logic of compounding a constant positive interest rate forces us to say that what we might conceptualize

as monumental — even earth-shaking — events do not much matter when they occur in the

distant future. Perhaps yet more disconcerting, when exponential discounting is extended

over very long time periods there is a truly extraordinary dependence of CBA on the choice

of a discount rate. Seemingly insignificant differences in discount rates can make an enormous difference in the present discounted value of distant-future payoffs. In many long-run

situations, including climate change, almost any answer to a CBA question can be defended

by one choice or another of a discount rate.

We think it is important to begin by recognizing that there is no deep reason of principle

that allows us to extrapolate past rates of return on capital into the distant future. The

seeming trendlessness of some past rates of return is a purely empirical reduced-form observation, which is not based on any underlying theory that would confidently allow projecting

the past far into the future. There are a great many fundamental non-extrapolatable factors, just one example of which is the unknown future rate of technological progress. Even

leaving aside the question of how to project future interest rates, additional issues for climate

change involve which interest rate to choose out of a multitude of different average rates of

return that are out there in the past and present real world. Furthermore, there is an ethical

dimension to discounting climate change across many future generations that is difficult to

evaluate and incorporate into standard CBA. A very large number of additional examples of

economic and non-economic features could be given that are currently unknown but would

be highly relevant to determining the distant-future discount rate. The fundamental point

is that there is enormous uncertainty and controversy about choosing an appropriate rate

of return for discounting distant-future events, like climate change. Moreover, the great

uncertainty about discounting the distant future is not just an academic curiosity, but it

has critically important implications for climate change policy. This disturbing ambiguity has given rise to a great deal of controversy and a variety of proposed solutions. Our

purpose here is to focus sharply on clarifying this particularly thorny issue by using a crisp

formulation that abstracts away from all other elements of CBA.

2
In this paper we construct the simplest possible model for analyzing how to discount the

distant future for a CBA decision that must be made now, when future discount rates are

uncertain. We do not defend this model for its realism and immediate applicability to such

long-term issues as CBA of climate change. Rather, we defend our abstract optimizing model

for its ability to isolate, clarify, and hopefully bring some closure to a set of controversial

issues that have bedeviled the discounting of distant-future events like climate change.

The bottom line message we want a reader to take away from this paper: there exists

a rigorous generic argument that the future should be discounted at a declining rate that

approaches asymptotically its lowest possible value.

2 Background: the “Weitzman-Gollier Puzzle”

In this section we focus on a paradoxical issue, whose proposed resolution has featured

prominently in a number of papers on discounting the distant future. We do not mean

to imply that this is the only important issue in long-term discounting, or that our own

resolution of the puzzle is the final word. We are primarily using this alleged paradox as a

way of grabbing a reader’s attention in order to motivate the rigorous analysis that comes

with the next section of the paper.

In the highly stylized model of this paper, time t = 0, 1, 2, ..., is measured in discrete

periods of unit length. To state loosely the issue at hand, a decision must be taken now,

just before time zero (call it time 0−), whether or not to invest a marginal cost δ that will

yield a marginal benefit  at future time t. Right now, at time 0−, it is unknown what will

be the appropriate future rate of return on capital in the economy. There are n possible

future states of the economy, indexed by i = 1, 2, ..., n. As of now (time 0−), future state i is

viewed as having marginal product of capital ri with probability pi > 0, where Σipi = 1. A

decision must be made now (at time 0−, just before the “true” state of the world is revealed

at time t = 0) about whether or not to invest δ now in order to gain payoff  at future time t.

To pose the problem sharply, it is assumed that immediately after the investment decision is

made, at time 0, the true state of the world i is revealed and the marginal product of capital

will thenceforth be ri, from time t = 0 to time t = ∞. The idea that productivity shocks are

permanent seems an appropriate abstraction for analyzing the distant-future discount rate.

Even if we thought of interest rates as being a mean-reverting random variable, our best

statistical estimate of this mean is itself a random variable, which makes the reduced-form

overall stochastic process be non-mean-reverting.

In a pair of articles, Weitzman1 proposed the idea that what should be probability1See Weitzman (1998) and Weitzman (2002).

3
averaged at various times is not future discount rates, but future discount factors. In

other words, one should not apply the average discount rate Σipi ri as if it were a timeindependent constant. Instead one should apply the time-dependent average discount factor

A(t) = Σipi exp(−ri t), whose corresponding time-dependent “effective” discount rate RW (t)

satisfies

exp(−RW (t)t) = Xn

i=1

pi exp(−ri t), (1)

which can be rewritten as

RW (t) = −1

t

ln ÃXn

i=1

pi exp(−ri t)

!

. (2)

Accepting the above logic, it follows that the investment should be made (incurring a marginal cost δ now, at time 0−, in order to yield a marginal benefit  at future time t) if and

only if

 exp ¡−RW (t)t

¢

≥ δ. (3)

It is not difficult to show that RW (t) defined by (2) has the properties

RW (0) = Xn

i=1

pi ri, dRW (t)

dt < 0, RW (∞) = min{ri}. (4)

Unfortunately, Weitzman did not provide a rigorous story to accompany the idea that

what should be probability-averaged at various times is not discount rates, but discount

factors. Instead the argument in his papers was left at the intuitive or heuristic level. A

main purpose of this paper is to provide a rigorous justification for an appropriate version

of (1)-(4). A simple intuition is derived from observing that one can anticipate the future

benefit ε by reducing productive capital by the present value ε exp(−rit) at date 0. This

reduction will be fully compensated at date t by the cash flow ε generated by the project,

so that all cash flows are concentrated at date 0. In expectation, this net present value is

positive if and only if condition (3) is satisfied.

In a series of articles, Gollier2 inverted Weitzman’s logic to produce a seemingly symmetric

discount-rate story with exactly the opposite properties. Gollier reasoned along the following

lines. One can transfer the initial cost of the project by diverting δ from the productive

capital of the economy at date 0. This is offset by investing δ exp(rit) in the productive

capital at date t. This implies that all cash flows are concentrated at date t. In expectation,

2See Gollier (2004, 2009a, 2009b).

4
this net future value is positive if

 ≥ δ exp ¡RG(t)t

¢

, (5)

where the time-dependent internal rate of return RG(t) satisfies the condition

exp(RG(t)t) = Xpi exp(ri t). (6)

This can be rewritten as

RG(t) = 1

t

ln ÃXn

i=1

pi exp(ri t)

!

. (7)

It is not difficult to show that RG(t) defined by (7) has the properties

RG(0) = Xn

i=1

pi ri, dRG(t)

dt > 0, RG(∞) = max{ri}. (8)

Unfortunately, Gollier did not at first provide a rigorous story to accompany his idea of

using the internal rate of return defined by (7). Instead, like Weitzman, his argument was

initially presented at the intuitive or heuristic level.

Aside from sharing the same initial condition RW (0) = RG(0) = Ppi ri, the properties

of RG(t) and RW (t) are diametrically opposed (except for the trivial situation in which there

is but one sure future state of the world, in which case the two are criteria are identical).

While RW (t) declines over time to the smallest value of {ri}, on the contrary RG(t) increases

over time to the largest value of {ri}.

Both the Weitzman discount rate RW (t) and the Gollier discount rate RG(t) have a

superficial plausibility, but with completely opposite conclusions and policy implications. If

the correct discount rate is one of (2) or (7), then of necessity the other one must be wrong

and will give the wrong answers to CBA questions. For want of a better name, this seeming

paradox has been dubbed in the literature the “Weitzman-Gollier puzzle,” and it has featured

prominently in several papers about long-term discounting.3 How might a person resolve

this distressing paradox by choosing between two such seemingly symmetric formulations,

with each one having diametrically opposed implications for distant-future discounting? The

answer can only come from a careful rigorous analysis, which follows in the next section.

3See, e.g., Buchholz and Schumacher (2008) and Freeman (2009).

5
3 Resolving the “Weitzman-Gollier Puzzle”

As Gollier pointed out early on, and elucidated more carefully later,4 neither the formula for

RW (t) given by (2) nor the formula for RG(t) given by (7) are likely correct as they stand

when the evaluation of the safe investment is adjusted for the risk associated to its financing

strategy. Indeed, both formulations contain a germ of truth that can be turned into a

rigorous argument when expressed in units of marginal utility along an optimal consumption

trajectory. Furthermore, and most importantly, the two rigorous formulations give the same

discount rate (as a function of time), thereby resolving the “Weitzman-Gollier puzzle.”

While several generalizations are possible, we focus here on the simplest case, the better

to bring across the main points sharply. Our formal argument proceeds as follows. Break

up time into a series of discrete periods 0, 1, 2, ...t, ... of unit length with corresponding

consumptions C = (C0, C1, ..., Ct, ...). We postulate a very general utility function, in the

spirit of Irving Fisher, of the form V (C). We assume that V is smoothly differentiable, is

strictly concave, has positive first derivatives, and satisfies some kind of generalized Inada

conditions that will guarantee unique interior solutions. Pure time preference is already

built into this general utility function. A special case of V (C) is the Ramsey-Koopmans

form

V (C) = X∞

t=0

exp(−ρt) U(Ct), (9)

where ρ > 0, U0

(0) = ∞ and U0(∞)=0. However, in some ways this special case (9)

obscures rather than illuminates the act of seeing through to the core theoretical structure

driving the model’s results.

There is just one commodity serving as both consumption and investment. The notion

of capital here is intended to be all-inclusive, including human capital, knowledge capital,

and so forth. The underlying production function of this “generalized capital” is linear. In

state of the world i it is of the form

Kt+1 = exp(ri)[Kt − Ct], (10)

where Kt is the capital stock at the beginning of period t. The linearity of the production

possibilities frontier guarantees that the relevant interest rate in state i will be ri — no matter

what is the form of the general Fisherian utility function V .

It is critical in this model to understand the exact timing sequence concerning what

information is available at what time, and when decisions are made. The base-case scenario

is this. At time 0, the inherited capital stock is given as K0 and the state of the world

4See Gollier (2009a, 2009b).

6
is known. It is assumed that the deterministic problem of maximizing V (C) subject to

(10) has a unique interior bounded solution for all i. Denote the deterministic optimal

consumption trajectory in state i as C∗

i = (C∗0i, C∗1i, ..., C∗ti, ...). From the linear production

possibilities frontier (10), reducing consumption at date 0 by one unit in state i would result

in exp(ri t) extra units available for increased consumption at time t (without altering the

rest of the optimal trajectory). Such a marginal change in the consumption plan should

have no effect on intertemporal welfare V . Therefore, the optimal deterministic trajectory

in state i must satisfy the first-order condition

∂V (C∗

i)

∂C0

= ∂V (C∗

i)

∂Ct

exp(ri t) (11)

for all t. This condition, which holds in all states i = 1, ..., n, states that investors are

indifferent about the financing structure of their projects.

Next suppose that an investment opportunity arises at time t = 0−, just before the

“true” state of the world is revealed at time t = 0. This safe investment opportunity

expends marginal cost δ in order to yield a marginal benefit  at future time t. Of course

the representative agent wishes that the investment decision could be made with the precise

information available at time t = 0, just after the “true” state of the world is revealed and

the relevant future marginal product of capital (from time t = 0 to time t = ∞) is known

with certainty. But the essence of the problem of doing CBA with an uncertain future

discount rate is that the investment decision must be made at a time when it is known only

that the uncertain future marginal product of capital will be ri with probability pi.

By the envelope theorem and given the optimality condition (11), the possibility to

reallocate the costs and benefits of the project over time has no effect on its impact on V ,

at the margin. This is the key to resolve the puzzle, as we demonstrate it below.

The investment project raises the expected utility of the representative agent if and only

if



Xn

i=1

pi

∂V (C∗

i)

∂Ct

≥ δ

Xn

i=1

pi

∂V (C∗

i)

∂C0

. (12)

Using the optimality condition (11), this can be rewritten in two equivalent ways. The

"Weitzman approach" consists in eliminating ∂V (C∗

i)/∂Ct from the above inequality. It

yields

ε

Xn

i=1

qW

i exp(−ri t) ≥ δ with qWi ≡

pi

∂V (C∗

i )

∂C0

Pn

i=1 pi

∂V (C∗

i )

∂C0

. (13)

7
This is equivalent to discount the future flow ε at a rate RW

∗ (t) defined as follows:

RW

∗ (t) = −1

t

ln ÃXn

i=1

qW

i exp(−ri t)

!

. (14)

Alternatively, the "Gollier approach" consists in eliminating ∂V (C∗

i)/∂C0 from (12) by using

condition (11). It yields

ε ≥ δ

Xn

i=1

qG

i (t) exp(ri t) with qGi (t) ≡

pi

∂V (C∗

i )

∂Ct

Pn

i=1 pi

∂V (C∗

i )

∂Ct

. (15)

This is equivalent to discount the future flow ε at a rate RG

∗ (t) equaling

RG

∗ (t) = 1

t

ln ÃXn

i=1

qG

i (t) exp(ri t)

!

. (16)

Observe the close links between the definition of (RW

∗ (t), RG∗ (t)) in (14) and (16) and

the definition of (RW (t), RG(t)) in (2) and (7). They are equal up to a risk adjustment

of probabilities. Weitzman converts all cash flows into consumption at time 0 and adjusts

state-contingent net present values with units of marginal utility at time 0. Gollier converts

all cash flows into consumption at time t and adjusts net future values with units of marginal

utility at time t. These risk adjustments are crucial because, although the properties of RW (t)

and RG(t) are very heterogeneous, it is very easy to check from the optimality condition (11)

that

RW

∗ (t) = RG∗ (t)

for all t! This means that the adjustment of the valuation for risk resolves the "WeitzmanGollier puzzle". Let R∗(t) denote this efficiently risk-adjusted discount rate.

It remains to explore the properties of this common risk-adjusted discount rate. Using

the easier formulation R∗ = RW

∗ , it is not difficult to show that

R∗(0) = XqW

i ri, dR∗(t)

dt < 0, R∗(∞) = min{ri}, (17)

so that qualitatively the properties of the efficient discount rate R∗(t) resemble closely those

of RW (t) recommended by Weitzman, with the only quantitative difference being the substitution of “Weitzman-adjusted probabilities” {qW

i } for the unadjusted probabilities {pi}. It

is good to provide an intuition to the result that the term structure of the socially efficient

discount rate be decreasing. In this model in which shocks on capital productivity are permanent, risk on consumption growth are also permanent, as seen from equation (11). This

8
implies that risks are magnified by time, compared to the more standard Brownian motion in

which it is known that the term structure of discount rate is flat. What are the consequences

of this magnification of long term risk on the discount rate for long maturities? Intuitively,

future risk should induce prudent consumers to sacrifice more for this future. This is the

Keynesian notion of precautionary saving. This is translated into using a smaller discount

rate.

Let us illustrate this result by considering the special case of the Ramsey-Koopmans

specification (9) for V . Let us in particular consider the case of the logarithmic utility function U(C) = ln C. In that case, one can rewrite the optimality condition (11) as Cit =

Ci0 exp(ri −ρ)t. Rewriting the intertemporal budget constraint (10) as K0 = ΣtCit exp(−rit)

and eliminating Cit yields that

Ci0 = K0(1 − exp(−ρ))

for all i = 1, ..., n. Observe that the initial consumption is risk free in this special case. As is

well-known, consumption is not affected by a change in the interest rate in the logarithmic

case. It implies that one is neutral to the small risk affecting the net present value of the

project at date 0, which implies in turn that the Weitzman rule RW (t) given in equation (2)

is correct in that case. This can be seen by observing in (13) that qW

i = pi for all i, so that

RW

∗ (t) = R∗(t) = RW (t)! In this case, Weitzman’s rule was qualitatively and quantitatively

correct.

4 Concluding Remarks

We will not bother to go through all of the many caveats that should attach to the results

of this paper. Nor shall we discuss possible extensions to more complicated and realistic

situations.

The bottom-line message that we wish for readers to take away from this paper is the

following. When future discount rates are uncertain but have a permanent component,

then the “effective” discount rate must decline over time toward its lowest possible value.

Empirically, this important feature can have significant ramifications for climate-change CBA

— by weighting the distant future much more heavily than is done by standard exponential

discounting at a constant rate.5

5See, e.g., Newell and Pizer (2003), or Groom, Koundouri, Panopoulou and Pantelidis (2007).

9
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