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Discounting the future, yet again

February 23rd, 2007

Felix Salmon gnashes his teeth at yet another incorrect report on discounting and the Stern review, by David Leonhardt in the New York Times.

Using his discount rate and other assumptions, a dollar of economic damage prevented a century from now is roughly as valuable as 7 cents spent reducing emissions today. (In fact, it’s less than that, because Stern adds another discount rate, called delta, on top of eta.)
Leonhardt says that “spending a dollar on carbon reduction today to avoid a dollar’s worth of economic damage in 2107 doesn’t make sense” – but this is a straw man, since Stern never comes close to saying that we should do such a thing. Leonhardt also spends a lot of time on the academic qualifications of Stern’s opponents, but neglects to mention that Stern himself, a former chief economist of the World Bank, is actually a real expert on discount rates, and understands them much better than most economists do.

Salmon is right, both about the Leonhardt piece and, unfortunately, about the limited understanding of discounting issues on the part of economists in general.

Leonhardt’s error follows a column by Hal Varian which, while not strictly wrong, was ambiguous enough to lend itself to this misreading. And the same error has been made by lots of Internet commentators who have enough economics that I would have expected them to know better.

But even economists who avoid the obvious error of confusing the pure rate of time preference with the money discount rate, as Leonhardt has done, have been badly confused about this question, being led astray by a presumption that the money discount rate has to be fairly high. There are a number of reasons for this.

First, standard practice in benefit-cost analysis is to use high discount rates, often as high as 8 or 10 per cent, and this seems to work reasonably well (by no means perfectly) in terms of selecting good projects and rejecting bad ones. But this is a paradigm case of “being right for the wrong reasons”. In a typical project evaluation, the project’s proponents (in the case of infrastructure, usually engineers) have a lot of influence over the projections on which cash flows are based, and they tend to be biased upwards (mostly by ignoring things that might go wrong). By contrast, economists usually get to choose the discount rate, and they almost always go for a high rate. If an economically correct discount rate is applied to cash flow estimates with a pervasive optimism bias, too many projects will pass the test. But using a high discount rate offsets, on average, the bias in cash flow projections. This is far from a perfect approach, but it’s hard to implement a better one. However, it leads you badly astray in the case of climate change, where the big risks of unforeseen bad outcomes arise with the do-nothing option.

Second, there’s a belief that market rates of discount are high and that we should follow the market. The problem here is that the premise as false at least for the obvious choice of market rate, namely the rate of interest on high grade bonds, which has averaged about 1 or two per cent. This is much lower than the rate of return to equity, which seems to be what most economists have in mind (at this point we need to start thinking about the equity premium). But, on the face of it, the bond rate is the appropriate rate for discounting riskless flows of either cash or utility. The best way to deal with risk is not to use a risk-adjusted cash rate to convert risky cash flows into certainty equivalents using expected utility or some more general model.

Third, it’s commonly assumed that individuals display high rates of time preference, and therefore so should society. In fact, individual behaviour on this score is inconsistent, with some decisions implying unreasonably high rates of time preference and others low or negative rates. More importantly, the putative fact that individuals have high rates of time preferences implies almost nothing about the social rate of time preference, for a couple of reasons. First, individuals are mortal whereas (except for a small probability of nuclear annihilation and similar disasters) society is not. To quote Richard Tol and his co-authors in 2006 paper in Environmental Science and Policy

The PRTP is the ‘utility discount rate’,which reflects our time preference for utility. Estimates of utility discount rates for individuals are almost always positive – an estimate of 1.5% is considered plausible for the UK for instance (HMTreasury, 2003) – for the simple reason that humans prefer good things to come earlier rather than later. Given the inevitability of death for individuals, a preference for benefits to accrue earlier rather than later is entirely sensible. At the social level, however, the arguments are more nuanced, and indeed whether or not the PRTP should be equal zero has been debated by philosophers and economists for decades. Cline (2004), for example, proposes to use a zero PRTP in evaluating climate change policies. Reasonable ethical considerations suggest using a zero PRTP—a positive PRTP involves placing a lower weight on the welfare of future generations, which is impartial and contrary to intergenerational equity. However, there are also persuasive arguments for employing a very small positive PRTP.�

We can sharpen this up a bit by observing that the average annual mortality probability for adults is around 1.5 per cent, suggesting that this factor alone is sufficient to explain positive time preference.

A more fundamental problem is that individual time preference is relevant to optimal individual consumption profiles, but not to the equitable distribution of resources between different age cohorts. I doubt that many gen X-ers (certainly not the esteemed Paul Watson) will agree that, having been born earlier, baby boomers like myself are entitled to a higher weight in social welfare calculations. But there is no other coherent basis for using a positive social rate of time preference. You can’t discriminate between generations without discriminating between people who are alive at the same time.

You can read my general summary of the issues here

Note: I’ve been a bit mischievous in a couple of places above. The phrase “right for the wrong reason” is quoted by Leonhardt, and comes from Marty Weitzman’s excellent review of Stern, where he observes that Stern tends to overweight known risks as a way of dealing with unknown ones. I’m just making the point that this kind of offsetting bias is widely prevalent, and is incorporated in the unexamined assumptions of most economists on discounting.

Also, while I’ve quoted Richard Tol in support of Stern’s position (as indeed Stern did) he’s strongly criticised Stern and (on my blog) has repeatedly denounced the idea that a zero social rate of time preference could ever be appropriate. He has also claimed that the correct rate for both individuals and society is between 2 and 4 per cent, whereas the quote above suggests 1.5 per cent for individuals and either zero or “a very small positive rate” for society. I’ll leave it to Tol to reconcile these positions; I’m happy to endorse the passage I’ve quoted.

* Discounting and the social cost of carbon: a closer look at uncertainty by Jiehan Guo, Cameron J. Hepburn, Richard S.J. Tol and David Anthoff, Environmental Science & Policy, 9(3): 205-216

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  1. Richard Tol
    March 6th, 2007 at 17:27 | #1

    John H: My objection to money discount rates is not emotional. It is simply that there is no single future market — there are many, with different public/private mixtures, different risks, different time horizons and what have you. If you work out why the money discount rates are what they are, you can work out what the money discount rate for investing in greenhouse gas emission reduction would be.

  2. March 6th, 2007 at 20:09 | #2

    I think I can simplify that list — the relevant market must be private. I think the time problem is a false one… we’re talking about an average annual return.

    Regarding risk we should pick the realistic expected return for an average investor — which could be taken as the average stock market increase (7% inflation adjusted for the US since 1802) or an average of managed fund returns or average real economic growth as a percentage of savings/investment. This represents the reality of what $1 will turn into for future generations. Damn reality. :)

    Of course the discount rate should be higher than this return. Even if somebody is indifferent between the value of $1 now or $1.07 next year they might still have a real preference for the $1 now because they know that their big expense (that holiday to majorca) is happening in 6 months and they have to pay off their little brothers uni debt in 8 months.

    Of course, it must be a short trip and a small debt if $1 can pay for it all.

  3. Richard Tol
    March 6th, 2007 at 21:48 | #3

    John H: I disagree. Climate policy is an investment in a global, public good, and a particularly uncertain one. This cannot be compared to a private investment; or at least not without further ado.

  4. March 6th, 2007 at 23:17 | #4

    It doesn’t matter what issue we’re measuring… all that matters is what a dollar today is worth in the future.

  5. Majorajam
    March 7th, 2007 at 03:44 | #5

    Richard, pls remove your head from your colon. By God’s grace I’m not your student so don’t think you can cover your intellectually clumsy tracks by playing teacher. The question regarded public policy right- not time preference? So my critique of your inane illustration of the impossibility of stated preference and of your deference to the wisdom of philosopher king opinion by way of Tol approved integrated assessment model stands.

    And what of the approval process? Let’s see if I can’t assemble a partial list of considerations you’ve dismissed with a mere smattering of syllables, and often less: US real long term bond yields, public opinion, drinking water, infectious disease, the plausibility of global environmental treaties, HM Treasury’s preferred PRTP, medical technology- as I said, a partial list. It would be remarkable to see what you can do in a calendar year.

    Fyi, Weitzman discounting “calibrated to reproduce observed short-term discount rates”, is fatally ambiguous, a fact of which I’m sure you are well aware seeing as you teach the stuff and don’t mind telling people so. But I’m less interested in that than by the fact that your “contrast with Stern” conclusion, (a likewise ambiguous statement but here I’ll guess as to your meaning), is contradicted by the man himself, but hey, who’s he to say?

    With B = .5 in (7), the relevant interest rate for a century from now becomes r(100) = 1.7%, which is close to Stern’s r = 1.4% or Cline’s earlier r = 1.5%.

    You’re no end of fun Tol. Keep ‘em coming.

  6. Majorajam
    March 7th, 2007 at 03:55 | #6

    JH, ‘expected value’, by which I assume you mean expected return does not sufficiently factor in the ‘probability’, by which I assume you mean uncertainty of investments. To illustrate, consider this example: the expected return on an investment with 100% probability of a 5% return is 5%. The expected return on an investment with 50% probability of a 0% return and 50% probability of a 10% return is also 5%. So here you have two investments with the same expected return, but with evidently differing risk profiles. These differences will affect investor preferences and hence are fundamental to the determination of discount rates.

  7. Richard Tol
    March 7th, 2007 at 06:34 | #7

    Majorajam: r(100) = 1.7% => r(10) > 1.7%, at least according to Weitzman

  8. Majorajam
    March 7th, 2007 at 07:47 | #8

    Yes, and I live in Queens. You put that together for yourself Einstein? What, do you got a team of monkeys working on this?

  9. March 7th, 2007 at 09:54 | #9

    Majorajam — I already responded to that point. While people are indeed risk averse, their revealed preference for an expected return of 10% (with risk) over 5% (without risk) shows that the higher return more than compensates them for their risk aversion. As you yourself point out, there is a very active market for risk.

    The simple reality is that the average return on investment over the next 100 years is likely to come to more than 0.1%.

    And please don’t mention that Stern used a discount rate higher than 0.1%, because that was not to measure time value of money, but the decreasing marginal utility from money. They are different issues and including one does not absolve somebody of the responsibility of considering the other.

  10. Majorajam
    March 7th, 2007 at 10:24 | #10

    John, the Ramsey formulation used by Stern to derive the discount rate r, is r = p * ng where p is the PRTP, g is forecasted economic growth and n represents risk aversion. Stern used the value .1% for p, 1 for n and 1.3% for g. Since the cost of global warming scales with the economy, the most critical parameters there are the taste parameters of which PRTP is the favorite fixation of one Richard Tol. The bottom line is the money discount rate Stern uses is indeed 1.4%. You can argue that all you like, but you will be wrong.

    As regards people’s “revealed preference for an expected return of 10% (with risk) over 5% (without risk)”, you should know there is no such thing. The only preferences that can be implied of investors is the price of risk, (i.e. the return to taking on uncertainty), and that is on average and over a long period (as measurements of shorter periods tend to be dominated by changing expectations and external shocks).

    More broadly, it may help you to understand that there is more to utility than return, or even the concave utility of wealth. There is utility in risk reduction even if it leaves you less wealthy on average. This explains things like insurance.

    Hope it helps.

  11. March 7th, 2007 at 10:44 | #11

    I understand risk aversion (ie “more to utilty than return”).

    Where Stern uses “g” it would have been better to use “g*(GDP/I)” where I is investment. If people chose to spend now instead of investing that doesn’t mean their expected return from investment is 0… what it means is that their time value for that money exceeds the expected investment return.

    I also disagree with 0.1% for PRTP as it implies an intergenerational utility maximising framework while is inequitable, unrealistic and rightly ignored by most economists… but that’s a different story.

    The returns on investments are determined by the quality of the ideas… not the risk profile of the investors. If I find a great investment that returns me 20% a year with relatively low risk, that doesn’t mean I’m amazingly risk averse and demanding huge returns for my low risk. It means that the market return on that investment is high. Yay for me. In eco-jargon… there is consumer surplus in the market for risk.

  12. Majorajam
    March 7th, 2007 at 11:24 | #12

    JH, try as though I may it seems I am unable to get through to you. It would help if you made more of an effort.

    Firstly, “The returns on investments are determined by the quality of the ideas… not the risk profile of the investors.”, is flatly wrong. I covered this earlier:

    in particular, investors bid up and down the price of risky investments- e.g. shares- based on [their risk and return] preferences… The higher the aggregate risk aversion, the fewer dollars/investors are interested to buy a risky asset- e.g. a share- the lower the share price, and- all else equal- the higher the earnings yield/return to your investment

    What this means is that the price of the investment that you enter into is determined by aggregate expectations regarding its cash flows, and aggregate risk aversion. Given that the cost basis of an investment is a rather critical determinant of its return, your statement is false. That said, if you know better than the aggregate what the earnings potential is and can identify mispriced assets, or even what the future holds and therefore what the risk is, most definitely yay for you (although that only means that you will be able to beat the market, not that the market doesn’t matter).

    Secondly, the g Stern uses is an allusion to the per capita growth potential of the economy which is the well spring for the value of all assets. As such it directly relates to the expected returns investors are able to achieve on average (and as such is a starting point from which to evaluate investments). Where you are going with I over G except after C I can’t say I grasp.

  13. March 7th, 2007 at 19:04 | #13

    I’m sorry if I’ve been unclear. I’ll try again:

    1. Returns on investment. The demand for loanable funds is a downward sloping curve with “i” on the y-axis (interest rate) and “I” on the x-axis (investment). For any i* the market will have I* worth of investment that is profitable. Now imagine that the people in our market get smarter and are able to find more profitable ideas. The demand curve shifts to the right, so that the same i* will lead to more investment (say I^).

    However, the supply of loanable funds is not (as implicitly assumed above) perfectly inelastic. The supply of loanable funds is an upward sloping curve with the same axis. When the demand curve shifts out, that is the same as a shift along the supply curve, up and right. So an increase in the number of good ideas leads to a higher interest rate (say i^).

    Note that the interest rate (or broadly speaking, the return on investments) has increased without any change in the risk preferences of the demanders or suppliers of loanable funds. The more good ideas that exist in a market, the higher the return will be. Ultimately (assuming constant risk profiles) the driver of returns is the new ideas in an economy… and that leads me to growth theory which is totally off-topic.

    In economies with identical risk preferences, the appropriate time value of money is higher in the economy that has higher expected returns. The people who were happy to invest at i* are now getting i^ and so they receive an additional benefit of (i*-i^).

    The summary of this point is that the market for risk is not a zero-sum game. There is real economic welfare to be gained in that market and using a risk-free return ignores the legitimate economic gain from risk.

    2. Stern — my point about g * (GNP/I) was that the growth in an economy comes from the return on investments, not return on all income (and indeed only on new net investment, not total investment — but lets not add unnecessary complexity at this point).

    Using the simplified closed eoconomy identity Y = C + I + G then it is only “I” that leads to the growth. An an exmaple consider:

    100 (Y) = 60 + 20 + 20, which lead to real economic growth of 1%.

    The correct expected return on money is the return you can get from your investment = 1% * (100/20) = 5%. While it is theoretically possible to reach the “golden” level of capital where further investment will lead to lower growth — in reality that never happens (because of the constantly changing knowledge and preferences that require regeneration of capital). With the above example, at the margin a person has a choice of taking $1 out of consumption and putting it into investment and getting 5%. The fact that they choose not to shows that their time preference for that money exceeds 5%!

    I hope you consider that more of an effort.

  14. March 7th, 2007 at 19:12 | #14

    Richard, can you please explain to me how my time preference for money changes because you (the universal “you”) want to use my money to prevent global warming? Does that mean I wouldn’t have been able to invest? Or does it mean the holiday I planned for this year has been banned by parliament? Why wouldn’t I have the same time preference?

  15. Richard Tol
    March 7th, 2007 at 19:57 | #15

    John H: You (again, universal) should have the same time preference, but because risk etc are different, you would not have the same required return on investment.

  16. March 7th, 2007 at 20:30 | #16

    The risk of what is different? Are you saying my expected return on my savings/investments are lower? Why?

    Are you predicting a significant economic slowdown because of global warming (or our actions to prevent it)? I didn’t think even Stern was suggesting that.

    This seems a little like double counting the costs of global warming.

    And the extra capital depreciation caused by global warming will be in part offset by higher returns on new replacement capital or capital moved into new areas with new-found comparative advantages. To borrow from Stern’s example — think of the German or Japanese economy post-WW2 and the “catch-up” that economies have as long as they haven’t destroyed their knowledge base.

  17. Majorajam
    March 8th, 2007 at 02:02 | #17

    John,

    I pointed out now on four occasions where you’ve made erroneous statements. In each case, rather than engaging on those points, disagreeing or trying to understand the error, you’ve simply gone on to make new erroneous statements. This is not encouraging, and does not qualify as making an effort.

    Granted, you did write a lot in your post, even going so far as to create new and revolutionary macroeconomic identities. There is something missing however there which perhaps you can explain to me- what happens to the existing capital stock in your formulation? Or technological advance for that matter? You have explained growth in income entirely as a function of annual net investment, so presumably there is a good reason why the return on investment made in a prior year goes to zero and why labor-augmenting technological progress comes to a screeching halt. Perhaps you can explain it in the context of “people in the market”s spontaneous brain growth (presumably akin to natural male enhancement)?

  18. Richard Tol
    March 8th, 2007 at 03:32 | #18

    John H:

    I think the question is, why should risks be the same?

    Investing in climate policy is unlike any other investment. Both costs and benefits are very uncertain, and it may be that the real benefit is in fact a reduction of the uncertainty. Besides, the benefits of climate policy will fall on the children of people in faraway lands. This makes climate policy a different prospect than buying shares in WalMart. Climate policy is still an investment, however, and should be treated as such.

    BTW, I’m not predicting much economic slowdown because of climate change; but Stern is (well, he does in some chapters but not in others).

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