Discussion Paper

No. 2017-11 | March 13, 2017
Checking Gollier and Weitzman’s solution of the “Weitzman–Gollier puzzle”


In “How should the distant future be discounted when discount rates are uncertain?” (2010) Gollier and Weitzman claimed having solved the Weitzman–Gollier puzzle, concluding from a risk-averse utility maximizing model that Weitzman discounting is qualitatively correct and that when uncertain annual interest rates are highly correlated, long term discount rates are declining functions of time. This paper quantifies a similar model and comes to the opposite conclusion. Weitzman discounting is wrong; there is no puzzle if the correct method is used. Risk-neutral discount rates are growing, rather than declining functions of time under the Weitzman assumptions. Risk-averse discount rates can be declining, but must not be used to discount risky project’s cash flows; risk adjusted rates must be used instead. When long term market yields are a growing function of time, it makes no sense to invest in projects of similar risk but lesser yield, irrespective of one’s degree of risk-aversion.

JEL Classification:

D61, H43


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Cite As

Szabolcs Szekeres (2017). Checking Gollier and Weitzman’s solution of the “Weitzman–Gollier puzzle”. Economics Discussion Papers, No 2017-11, Kiel Institute for the World Economy. http://www.economics-ejournal.org/economics/discussionpapers/2017-11

Comments and Questions

Anonymous - Szekeres, Checking Gollier and Weitzman's solution....
April 25, 2017 - 20:13

Szekeres' paper, Checking Gollier and Weitzman's solution of the Weitzman-Gollier puzzle, is quite worthwhile and should be published. It provides a useful proof of its basic position, along with comments as to the likely applicability of alternatives.

Anonymous - Referee report
April 27, 2017 - 09:48

see attached file

Szabolcs Szekeres - Authors reply to referee
April 28, 2017 - 12:29

I have to thank the anonymous referee for the impressive effort that he has expended in writing his opinion. I find it all the more regrettable, therefore, to have to state that most of his efforts have been misdirected to an issue of no great relevance to the discussion paper's ...[more]

... conclusions, while failing to adequately address its main points.

The discussion paper makes three important points in relation to G&W's bottom line message that "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." The three points that the referee has not adequately addressed are:

1. Under risk neutrality Weitzman discounting is wrong.
2. Risk neutral CERs can be declining, but cannot be used to discount cash flows of projects. These can only be discounted by risk adjusted discount rates, which, under the Weitzman assumptions, will not be declining.
3. The degree of interest rate autocorrelation that is required for the Weitzman postulated phenomenon to be observed is so highly unrealistic that it renders the model, the puzzle, and all related discussion irrelevant to real life policy. In reality, risk neutral CERs neither grow nor decline as a function of time.


Point 1 is a statement about financial arithmetic. It will be made again here (more succinctly than in the discussion paper) because of its importance to the argument of this reply.

Weitzman's implicit definition of expected future value (EFV) adapted to a two-scenario model is as follows. pi are probabilities in two scenarios, ri are the perfectly correlated annual interest rates, and t is time.


a = p1 EXP(t r1) This is the FV in scenario 1 times its probability
b = p2 EXP(t r2) This is the FV in scenario 2 times its probability

Then the EFV of an investment of $1 made at t=0 is:

EFV = a + b

We can normalize this calculation to make EFV = $1 by multiplying both PV=$1 and EFV by

D = (a+b)^(-1)

Then PV = D, EFV = 1, and D is the certainty equivalent discount factor.

Weitzman proposed the following definition of certainty equivalent discount factor of a safe $1 at time t, and therefore the PV of FV=$1:

A = a^(-1)+b^(-1)

Clearly A is not the same as D, which contrary to A, complies with the definition of present value, as D(a+b) = 1.

A (a+b) is not equal to 1. It is this that was considered paradoxical, given that everyone intuitively accepted A as being correct. But A is nor correct, for it is an instance of the error found in the following expression:

(a+b)^n = a^n + b^n

In our case n = -1. The error arises because exponentiation is not distributive over addition. (http://mathmistakes.org/the-distributive-property-of-exponents/)

So there is no puzzle, just the unwitting mistake of an unverified leap of faith. To my knowledge, this has not been shown by anyone else before. In the light of all the literature on this subject, might this not be a result worth publishing?

The referee seems to belittle the import of this conclusion, or to dismiss it as a special case: "the risk-neutral agent would only deploy the ENFV social discount rate if, in the two-state-two-period model presented, the rates of return are greater than the pure time preference rate and all consumption is postponed to the future date. So the result is a special case."

Assuming a zero rate of pure time preference may be special in some sense, but zero time preference is the assumption of Weitzman 1998, of Gollier 2004, and therefore of the Weitzman-Gollier puzzle. As the puzzle is the subject of the discussion paper, it too must assume, as Weitzman and Gollier have, that the pure rate of time preference is zero.

But this is the standard assumption of all CBAs as well, for the very good reasons that CBAs are about comparing alternatives, and time preference is irrelevant to that comparison because it equally affects the future values of project returns and those of the alternative market returns, with which discounting compares them. So it is not a special case at all.

Is the referee suggesting that computing EPVs breaks down for some data combinations? Possibly postponing all consumption to the future is a consequence of risk neutrality, not a deficiency of the decision model. We all know that linear indifference curves result in corner solutions. But what makes that a special case? If the investor's pure time preference is higher than the highest possible interest rate, then he will not invest. If some of the possible interest rates are higher than his rate of time preference, he will have to calculate. For the example of the discussion paper, if the pure rate of time preference is set to 3%, half way between the low and high scenario interest rates, then the investor will postpone all consumption, and his risk neutral market CERs will be exactly the same as if his pure rate of time preference were zero.

I would respectfully ask the referee to state in what sense does the discussion paper fail treat adequately the Weitzman-Gollier puzzle? Anything beyond that, however, lies outside the scope of the discussion paper.

As for other papers attempting to show that

(a+b)^(-1) = a^(-1) + b^(-1)

all I can say is that either they have redefined the problem in a way that it is no longer the original Weitzman-Gollier puzzle, or they must be wrong.


The discussion paper uses a model like that suggested by G&W and uses it to calculate risk averse CERs, following the method proposed by G&W, which yields correct results. This assertion rests not only on the mathematical derivations shown, but is also buttressed by the fact that the same numerical results are obtained by purely numerical methods that use no formulas, just maximize the utility function assumed. Anyone can readily test this using Excel, so there can be no doubt about their correctness. For the purposes of Point 2, the use of the G&W method is just illustrative, however, and its conclusions are not dependent on it. As footnote 8 of the discussion paper states, "All conclusions remain valid even if the assumption of clairvoyance made in Gollier and Weitzman (2010) is dropped. In such case the investor will choose an optimal time 0 consumption and will be uncertain about consumption at time t, which will be scenario dependent. He will not be on the optimal consumption path in either scenario, and the CER calculation formulas dependent on such optimality cannot be used, but optimal expected consumption and CERs can still be calculated using numerical methods."

How risk averse CERs are calculated is therefore immaterial to the main message of Point 2, the most important assertion of which is that risk averse CERs cannot be used to discount cash flows of risky projects. As risk averse investors trade yield for safety, they are willing to take the lower yields of certain returns. But this is true not just for market yields, but also for project yields. So just as market returns have a risk averse CER, so do project returns. The right way to decide about a project is to compare its CER to that of the market. Discounting uncertain project cashflows with risk averse CERs, as G&W appear to imply, is a colossal mistake. There is a discount rate that can be directly used in such cases, however. It is the Risk Adjusted Discount Rate (RAR), which is defined as the IRR derived from the project's risk averse EPV and the expected values of its monetary flows. Section A.12 of the paper shows the magnitude of the error involved in discounting project cash flows with risk averse CERs, and Section A.14 shows the congruence between risk averse CERs and RARs: both give the same EPV when derived from the same cash flow. The former discounts certainty equivalent payoffs, the latter discount their expected values.

As the discussion paper states "Investment projects with risk equal to that of the market must have (growing) expected monetary returns equal to those of the market for them to have sufficiently high (possibly declining) risk-averse CERs to be acceptable to risk averse investors. This is true for any degree of risk-aversion. In other words, these projects must have IRRs equal to growing risk-neutral CERs, even for risk averse investors!" So the fact that that risk averse CERs are declining functions of time is immaterial, as is how they are computed. The G&W assertion that the cash flow of projects can be discounted at rates that are declining functions of time is never right under the assumptions of the Weitzman 1998 model.

The referee states that "the paper makes claims that are too strong considering the broader literature in this area and the absence of theoretical results."

My response to this is that the scope of this paper is the same as that of G&W, which is limited to that of the Weitzman 1998 model. The discussion paper never claimed that its conclusions extended beyond that scope. For instance, they may not necessarily pertain to the peculiar conditions of a Lucas economy, which is not what the Weitzman assumptions correspond to.

But within the scope of the model on which the discussion paper is based, the claim that discount rates directly applicable to cash flows are never declining is not too strong. It has been proven. I would be most interested in reading about a specific example in the frame of the Weitzman model for which this conclusion is not true.

And what is the absence of theoretical results? The derivation of D above is mathematical. What else is needed? That RARs are higher than risk averse CERs has been explained. I thought that this was self-evident from how RARs relate to CERs, a relationship amply illustrated with numerical examples in the paper. But I would gladly provide formal proof if it is felt to be needed.

A final remark on Point 2: to my knowledge nobody has yet shown that regardless of the degree of risk aversion, the discount rate applicable to projects with risks equal to that of the market is the risk neutral CER. I expected this claim to be startling. Unless the referee has found this claim elsewhere in the literature, or can prove it wrong, he might have remarked on its novelty.


Granted, the degree of correlation needed for the phenomenon of accelerating compound factors that forms the sine qua non basis of the Weitzman model is not given great prominence in the paper, but it underpins a fundamental part of the conclusion, namely that despite the fact that the correct conclusion of Weitzman’s risk averse model is that discount rates are a growing function of time, the discussion paper does not endorse growing discount rates because of the highly unrealistic basis of the Weitzman model itself.


The referee dismissed Point 1 as a special case. The case is not particularly special, however, as standard CBAs ignore time preference for good reason, but even if it were, it is the case in which Weitzman discounting is defined; it is the case in which a determination about its correctness must be made; and, therefore, it is the case that he discussion paper had no choice but to adopt.

The referee spent considerable effort showing an alternative fashion of computing risk averse CERs. I did not engage in a discussion regarding that part of his opinion for two reasons. Reason 1: I am persuaded that my calculations are right because they are confirmed by simple numerical analyses that are based only on the formula of the utility function, so I don't find it useful to investigate the results of alternative formulations. Reason 2: calculating CERs was done only as an illustration. The main message of Point 2 is not to show how risk averse CERs are calculated, but to point out that project yields cannot be discounted by risk averse CERs, however calculated; and that when project risks are identical to market risks, then RARs coincide with risk neutral CERs, even for risk averse investors, regardless of their degree of risk aversion. This is just a fancy way of saying what every economist's common sense should be telling him: "When long term market yields are a growing function of time, it makes no sense to invest in projects of similar risk but lesser yield, irrespective of one’s degree of risk-aversion."

Point 3 simply means that in reality all of this should be moot: the Weitzman-Gollier puzzle and all the literature generated by it, this discussion paper and the referee's response to it included, for it is all premised on a fantastically unrealistic assumption about interest rate autocorrelation that is simply unreal. Absent that, as Gollier pointed out, the term structure of discount rates should be flat.

It all should be moot, therefore, but it is not. The reason why it is not, is the strange motivation of policy makers in various countries who are willing to allocate resources on the basis of a model that only exists in the imagination of some economists, to the great loss of their countries. (See Section A.11 of the paper).

I find it regrettable, therefore, that the referee recommended rejecting this paper without due consideration of its main messages.

But I take solace from the fact that this discussion is taking place on Economics e-journal. Elsewhere, this would be the end of the story. The editor agrees with his referee, and the paper is trashed. No editor can delve into the complexities of all papers, so he must rely on his referees, and sometimes, as in this case, be misled by them. This could very well happen here as well, but here I can respond, others can see the response, make up their minds, and opine if they so wish.

It is a tribute to the founders of Economics e-journal that the discussion paper remains available, as do these arguments. Regardless of how fervently some would like to see declining discount rates justified by the Weitzman thesis, sooner or later it will be recognized that Weitzman discounting is wrong, and that it is wrong to discount project cash flows with risk averse CERs.

Szabolcs Szekeres - Author's reply - Detailed response to specific comments
May 01, 2017 - 22:42

In my first response to the Referee’s comments, I highlighted the aspects of the paper to which the Referee failed to pay adequate attention. In this response, I address some of his specific comments.

The fundamental problem with the comments received is that the Referee misunderstood the objectives of ...[more]

... the paper, even though they are clearly stated in its introduction. They are to answer the following questions:

1. Did G&W solve the Weitzman-Gollier puzzle?
2. Under what conditions do the bottom line recommendations of G&W hold?

Instead of reviewing whether the paper has accomplished these objectives, the Referee sets up an alternative set of objectives for the paper: “However, these critiques must now take into account the fact that the theoretical model used as motivation does not support their claims, even though some other model might.”

I utterly reject this point. The discussion paper is not a survey article about declining discount rates. Its only subject is whether they can be derived from the premises of the Weitzman 1998 model.

Having set up alternative objectives for the paper, he then finds fault with it for not living up to them: “other models of risk-neutral agents have better explained the Gollier-Weitzman puzzle (Freeman 2010, e-journal)”, which “solves the puzzle by separating out risk aversion from preferences for consumption smoothing and shows that the Gollier-Weitzman puzzle is simply a manifestation of an old problem in Finance.” Regardless of this claim, Freeman 2010 is not a paper about the original Weitzman-Gollier puzzle, for there is no preference for consumption smoothing in either Weitzman 1998 nor in Gollier 2004.

The Referee refers to “social discount rate,” which is a concept of CBA that has nothing to do with the financial arithmetic of the Weitzman 1998 model, and speaks of a “social planner,” whereas in the Weitzman model there is only an individual investor. There is no pure rate of time preference in the model, and no consumption smoothing. The investor is simply out to maximize the expected present value of his monetary returns. Where does the following statement come from: “when the social planner is more impatient, the risk-neutral social discount rate is that of Weitzman?” Clearly not from the Weitzman 1998 model, which has no room for impatience.

The Referee must have some other paper in mind. Adding assumptions to the Weitzman 1998 model changes the nature of the model and at the same time the subject: such papers are no longer dealing with the original Weitzman-Gollier puzzle. This is why such papers were not cited (“missing out some important references in the literature”). Such papers deal with something other than the original puzzle, and therefore are irrelevant to the submitted paper. Neither is any paper cited that explicitly or implicitly computes ENPVs by probability weighing discount factors, as that is precisely what the submitted paper proves to be wrong.

The Referee does not seem to even agree with the validity of the subject of the paper “(it’s not really a puzzle if a model is underspecified, but anyway),” and is obsessed with time preference “The basic problem with the model is that a risk neutral planner does not need to smooth. Consumption will either all happen in the future, or in the present depending on the weather returns are higher than impatience or not. This is what determines the social discount rate in this model.”

This reveals confusion about (1) which stage of the optimization is being discussed when in the paper, and (2) how the discount rate is defined. (Investor’s market CER, not planner’s social discount rate, but anyway). G&W set up a model for risk averse investors in which investors must proceed as follows:

1. First, optimize their consumption path. The pure rate of time preference or “impatience,” if any, will affect the solution of this optimization.
2. Second, decide about making a small investment that yields a safe return. It is in this second optimization that the discount rate is defined: it is the safe rate of return that leaves utility unaltered. The pure rate of time preference or “impatience,” if any, do not affect this choice, as they would affect project and market returns alike, and therefore the result obtained would be the same as if these values were ignored or set to zero.

In the risk averse case, the first optimization is required to establish the benchmark utility level against which the second optimization can take place. The decision is much simpler for risk neutral decision makers. They will either invest all or none of their endowment. Those who choose not to invest, cease to be the subjects of the Weitzman 1998 model.

Contrary to what the Referee appears to state, the fact that the risk neutral investor postpones all his consumption has no bearing on his discount rate. Neither does the amount of his consumption, for unlike in risk neutrality, his marginal utility is constant. But his discount rate can be computed none the less, just like in the risk averse case, by finding the return a small safe investment such that there is no loss in diverting the required investment in it from the market. Given that risk neutrality means being indifferent to risk, the required return will be the expected monetary return of the market. It has nothing to do with the investor’s level of consumption or time preference.

Raising these questions indicates that the Referee would like the paper to deal with subjects that are irrelevant to its objectives. He seems to want it to follow the rest of the literature engendered by the puzzle, virtually all of which abandoned the original Weitzman 1998 model, because it was unable to explain the puzzle on its own terms. Most papers changed the objective function of the investor. In contrast, the submitted paper sticks to the original model, and actually solves the puzzle.

The Referee fails to acknowledge that the paper solves the puzzle on its own terms. Solving a puzzle or paradox, which is a logical contradiction between some elements, can only be accomplished in one of two ways: either by showing that there is no need for the conflicting elements to be in harmony, or by showing that one of the elements is wrong. In this case, there is no disagreement on what the expected FV of an investment is, and there no disagreement on the need for specifying the inverse function of compounding. Therefore, the only item left that could be wrong is Weitzman’s specification of that inverse function, which is indeed what both Section 5.2 of the paper and my previous reply prove, as logically required.

Returning to the paper’s objectives, the Referee fails to address the claim that G&W do not solve the puzzle, but states that “one of the clever things about G&W is that they solve the ‘puzzle’ without specifying the preferences, particularly the nature of pure time preference and the precise extent of risk aversion.” Other than the statement about G&W having solved the puzzle, I agree with the Referee: G&W say nothing specific about parameters. All they care about is to derive a morphological similarity between the expression for Weitzman’s risk neutral CER and one they derive under risk aversion from their very general welfare function, a similarity from which they conclude that the former must be right, and therefore declining discount rates are justified. This solution is rather shaky, as is the derived recommendation. As the paper points out, “the same could just as easily have been concluded about RG(t) recommended by Gollier.” In other words, the morphological similarity could have just as easily supported the opposite view. But whatever morphological similarity there may be between a risk averse expression with Weitzman’s risk neutral expression proves nothing, and hence does not constitute a solution to the puzzle.

The second objective of the paper is to determine when, on the basis of the Weitzman model, are declining discount rates justified. This question must be answered in two parts: the case of risk neutrality and the case of risk aversion.

The fact that under risk neutrality Weitzman discounting is mathematically wrong is proven in Section 5.2 of the paper. The Referee does not dispute it, but considers it to be limited to a special case, but as discussed in the previous response, that happens to be the only relevant case, for that is the case of the Weitzman 1998 model. For this reason, it is puzzling that he writes “The numerical example in Table 3 [showing that as the degree of risk aversion declines to zero, all discount rates converge to the correct risk neutral ones] […] is at least a possibility: the example is reproducible at least. However, the problem is that this does not constitute a proof.” Of course, it is not a proof, it is just an illustration. The proof is the mathematical one already cited. The paper states “The above results are illustrative examples, which sensitivity analyses show to be robust. Independently of the illustrative results, however, the following generalizations are universally valid.” It is the assertions made following this statement that the Referee should challenge, if he disagrees with them, rather than suggesting that they all depend on illustrative examples.

For the risk averse case the paper never disputes the possibility that CERs can be declining functions of time. In fact it illustrates their behavior under varying assumptions about the degree of risk aversion. But the paper explains that such rates cannot be used to discount cash flows. Those can only be discounted by RARs (risk adjusted rates). The paper explains why for projects with risks similar to the market’s, discounts rates applicable to cash flows cannot be declining in the Weitzman model. This concept is well explained and illustrated with examples, the numerical values of which make no difference. The concept should be self-evident, so no formal proof is provided. It would be easy to present one, however, if needed.

I will not respond to the Referee’s comments regarding his alternative derivation of risk averse CERs for the reasons already mentioned in the previous reply. CERs only serve illustrative purposes in the paper. None the less, their correctness has been verified through numerical methods, so the expressions used to compute them must be correct and consistent with the postulated utility function from which they were derived. Otherwise, analytically and numerically calculated sets of numbers would not have exactly matched in all cases. But even more importantly, as already stated in the previous reply, the paper’s conclusions are independent from the underlying model assumed. If the Referee is worried about special cases, he should really be worried about the assumptions of the G&W model: clairvoyance an instant after a small investment decision has been made, interest rates unchanging forever thence, but shrouded in uncertainty just an instant before (what’s so special about that instant?). If the paper’s conclusions depended on that model, they would be on shaky grounds indeed. Fortunately, they are independent from it. This is why the Referees views on computing CERs do not affect the conclusions of the paper.

On minor points

• “The paper has a selective reading of the literature.” Only literature genuinely germane to the puzzle proper is cited. Those papers that tried solving something else are not. Neither is cited the literature on the term structure if interest rates in financial markets, because the term structure of interest rates in the Weitzman 1998 model is defined by the model itself, so how interest rates behave in real markets is not germane to the paper.

• “The function V (:) needs to be introduced.” This comment is unclear. V() in the paper is the same welfare function that G&W define in their expression (9). G&W leave the utility function within it unspecified, but other than supplying that, there is no change to it in the paper. It is not clear what the reviewer means by “Nor [is it clear] why [its value] should vary depending upon the state of the world.” The investor has a fixed endowment to invest. In one state of the world, he will be able to do so at low interest rates, in the other, at high ones. Why would his welfare not be higher in the latter case than in the former? The rest of this comment is not clear either.

• Comment about the use of double quotes to represent first derivatives. The Referee is right. This is an error that will be corrected.

• Re minor comment on Section 4: the Referee states earlier that “G&W choose the discounted utilitarian form of preferences to make their point,” and observes that the parameters of the functions are not given. He then comments that “as written, this paper seems to suggest that only DU is considered by G&W.” I fail to see why the absence of specific parameters would affect the nature of the function.

• “On Page 7 it is stated that Weitzman (1998) only considered 2 time periods. This is incorrect. That paper has a limiting result, and also characterises a declining forward rate rather than the average rate of discount. So the 2-period statement (irrespective of the zero minus epsilon moment) is inaccurate. This inaccuracy reduces the credibility of the research by making it seem out of touch with the general literature.” I disagree with this statement. Expression (5) in Weitzman 1998, which defines A, the certainty equivalent discount factor, only has a single time variable t. The future safe sum from which A is derived is due at time t, a single period. Add the present, and that’s two periods. Declining rates are derived by continuously varying the value of t, but for each discount rate calculation, there are only two periods. The model would only be multi-period if the discount rates were all computed simultaneously and were interdependent. But then the Referee speculates “Perhaps the meaning is that in effect only 2 time periods are treated in Weitzman (1998), but if so this needs to be explained.” He guessed right, considering that the paper is about the Weitzman 1998 model. On page 18 it reads: “two-period long-term Weitzman model.”

• “The paper also suffers from being poorly organised. It was difficult to navigate.” This comment is not informative enough. The introductory section gives a roadmap explaining where everything is in the paper. Granted, Section 5 is long, but it is segmented into sub-sections with descriptive titles. The sequence of the presentation seemed logical to me, but everything can be improved upon. The statement by itself is of little help, however, without specific suggestions.

Szabolcs Szekeres - Authors's reply - Analysis of the Referee's derivation of risk averse CERs
May 08, 2017 - 10:13

The Referee created a puzzle: "In conclusion, there is a chance that the proof below contains a mistake. If so I look forward to the corrections. However, if the proof withstands scrutiny, I think it shows that the paper is not of sufficient quality to be published." The puzzle is ...[more]

... this: his proof is correct, but what he has proven is that the paper's method of CER calculation is right. Is that an argument for or against publication of the paper?

This final note reviews the Referee's analysis of the paper's CER calculation, as it would not have been just to ignore it, even though the paper's conclusions do not depend on this question.

This note also formally contests the Referee's statement that "the results in the two state two period model are not as general as claimed in the paper."

Because formulas are involved in the argument, the text format of this reply is unsuitable to make the case. To read it, please download the attached document.

Szabolcs Szekeres - Run your own tests on the correctness and Relevance of Weitzman discounting
April 29, 2017 - 22:37

The attached Excel workbook was designed to allow you to set up your own numerical example with which to test the effects of standard and Weitzman discounting.

Bonus: this workbook contains a macro that runs Monte Carlo simulations. You can copy it and use it for your own work. ...[more]

... It can handle correlated random numbers.

By changing the degree of auto-correlation of interest rates, you can check the relevance of the Weitzman-Gollier puzzle.