Discounting and climate mitigation
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    If we want to reduce the negative impact of future global warming, we will have to defray some costs now. In return , we get some benefits to future generations, who would otherwise have to endure an adverse climate.
   Let us for a moment assume that the costs to be defrayed now are equal to the future benefits, i.e. they make out the same amount of money. In that case, should we spend that amount of money ourselves, to the detriment of our descendants, or should we invest the money so that we preserve some value for them? This is an ethical question  - should we think of ourselves, or of those who come after us - and there is no objectively correct answer to that. However, usually we would say that if the benefit to them is not much bigger than the costs for us, we would rather think of ourselves first. But on the other hand, the costs to us now could be quite modest, and the benefits to them could be so huge that it would be prudent to think of their wellbeing first. So the question is: how much bigger should the future benefits be relative to the costs today, before we think that we should indeed defray the costs? We might claim for instance that if the benefits in a hundred years from now are 10 times greater than the costs today, then it would be warranted to invest today. That is, we may use a discount factor of 10.
    A discount factor of 10 over 100 years corresponds approximately to a yearly discount rate of 2.5 %, because (1.025)^100 is approximately 10.
    There is no objective procedure to set the proper value of the yearly discount rate. It may be different for different assets. Imagine for instance that in a hundred years from now people will be much richer in terms of money and material goods. If they will be ten times richer than we are now, then one money unit is just as precious for us today as ten money units will be to them. On the other hand, there may be much less nature in a hundred years than there is today, and the few intact nature reserves may be much more precious to them than it is to us now. In that case we might argue that the discount factor for nature values should be less than 1. So a fair discount rate may be different for different assets.
   It may also be discussed if there should be a discount rate at all. Those who think so will posit that if there were no discount rate, then it would always be ethically correct to postpone some benefit to future generations, rather than use the money now. The number of future generations that might harvest the benefit is nearly infinite, so how could it be justified that the benefit should go just to us out of so many possible generations? If you think like that, ethical considerations could always lead to the conclusion that we should not use any money now, and we would remain poor. So, most economists are not in favour of a discount rate of zero; but some economists think differently and claim that the discount rate should be zero.
    It should be mentioned that there exist situations where the only sustainable procedure is not to discount future benefits.  This is the case when utilising slow-growing organisms such as broad-leaved trees for timber production, or products from large whales. The stocks of such organisms grow at most by a few percent per year, and after allowing for the costs of capture, felling etc., they may even yield less than 1 % per year. Here, a totally different way of thinking is required. We can consider the stock of trees or whales as some sort of "principal", where it is economically sound to harvest the interest, but where we should abstain from touching the principal. So here the proper rate of harvesting is set by properties of the living beings that we utilise, not by any features of human society.
    Most of us believe that future generations will be more prosperous than we are, and therefore better able to pay the costs of environmental damage. The ability to meet these costs will therefore, in a first approximation, increase at the same rate as the growth in wealth. Wealth, in turn, can be measured as the growth in purchasing power, or the growth in consumption of goods, or in some other way. If this growth rate for instance is 1 % per year, then we can use this as a discount rate. We calculate the present value of the future damage, and compare this with the present-day costs of preventing the damage. If the latter are the larger, then we pass on the bill to our descendants. If not, we should prevent the future damage by appropriate action now. This way of thinking is called the presciptive approach - the discount rate will be set according to some considerations of what will happen in the future.
    But many economists think in a different way. According to them, we should not prescribe a discount rate based on guesses on what will happen in the future. Rather, we should observe what choices people actually make today. If an investment in climate mitigation will yield a return of 2 % per year over a long time scale, and an investor can get a return of 5 % per year on alternative investments, then the investor will not choose to invest in climate mitigation. The sum of the choices made by many private investors will be the best indication of what investments are sensible. Actual market behaviour will reveal what investments are ethically sound, according to such economists. All you have to do is to describe what happens now, and then use a discount rate corresponding to the returns that people actually obtain. This is the descriptive approach.

Descriptive or prescriptive discounting
    So, when deciding on what is a proper discount rate, we have the two approaches, the descriptive and the prescriptive. These designations are somewhat misleading, because both approaches are based on empirical observations. In the descriptive approach, you observe what rates of return are obtained by people who invest money now. In the prescriptive apporach, you observe the rate at which the total economy - e.g. GDP - grows now.
    In the descriptive approach, the arguments are that society must base its decisions on the rate of return on invested capital. The opportunity to avoid environmental damage at some time in the future should only be utilised, it is argued, if the discounted net benefit from this is at least as large as the benefit from some other investment, say in production of goods. In principle, what is done is to find the best possible allocation of sparse economic resources. Thus, the alternative investments that are compared must all be investments that benefit society as such. However, this approach may be distorted in such a way that the investment in environment is compared to financial investments, say in bonds. And if the rate of return is largest with bonds, it is claimed that money should be spent on bonds, not on the environment.  
    In developed countries, the rate of return on capital in financial investments is typically 4 - 6 %, and in developing countries it is often 10 - 12 % or even higher. Therefore, it is claimed that the discount rate should be placed at these levels, depending on the country.
    If instead we use the prescriptive approach, we arrive at discount rates that are much smaller. Future rates of growth are typically assumed to be the same as in the recent past. For the USA, there are various estimates. For instance, the GDP per capita (in constant prices and PPP corrected), has grown on average by 2.2 % annually from 1950 to 2000. This rate overestimates the growth in per capita consumption, however. Economists argue that a better estimate is the rate of growth of total factor productivity, which was 1.3 % p.a. from 1960 to 1989 (W. D. Nordhaus), or the real treasury bill rate, which was on average 1.1 % from 1949 to 2003 (W. R. Cline). If we choose an average rate of 1.2 %, then it can be adjusted slightly upwards, because the wealth of our descendants will be composed of goods that are not all as necessary for a good life as those we use today. They should therefore, it is argued, be able to pay a relatively larger share of their wealth than we are. For this reason the appropriate rate was set at 1.5 % by W. R. Cline, in his "challenge paper" presented for the Copenhagen Consensus.  
    These rates will, of course, be different for different countries. At present, national wealth grows much faster in certain countries in eastern Asia and Latin America than in the western countries. To obtain estimates of expected future growth rates, Lomborg has studied the data presented by IPCC (his reference IPCC 2000b). By taking for instance the figures of the B1 scenario, which is approximately an average scenario, for the whole period 2000 to 2001, we obtain a growth rate in GDP per capita which is 1.2 % for the OECD countries, but 2.3 % for the whole world. Thus, the proper discount rate may have to be twice as large if we consider the whole world as if we consider only the rich world.

Which is more correct - the descriptive or the prescriptive approach?
    A main problem with the descriptive approach is that it gives a high discount rate. This will lead to absurd results in the long term. To illustrate this, consider that the present value of all real estate in Denmark is worth $238 billion (as explained by Dubgaard in chapter 10 in Sceptical Questions and Sustainable Answers). If this were also the value of all Danish properties in 500 years from now, we could discount 500 years back with 5 % p.a. to find its present value, which would amount to $6, i.e. the equivalent of half a barbecued chicken with potato fritters.
    In relation to climate change, a time horizon of 500 years is relevant. Surface water that has been warmed up due to recent temperature increases will sink to the ocean bottom, and rise again later, taking about 1,000 years to complete the circuit.  In other words, the full effect of the temperature rise will be seen only when all up-welling of deep sea water consists of water that has already been warmed up. So, the time to obtain the full effect of the heating that happens now could well be 1,000 years.
    So on a time scale that is relevant for climate issues (500 years), descriptive discounting means that even the existence of all real estate in a developed nation with more than 5 million inhabitants simply does not matter at all. Such an absurd result demonstrates that the method does not work on such time scales.
    The point is that our wealth, for example our ability to afford to build dikes along sea shores to avoid flooding, increases by only 1 to 2 % per year. So the increase of 5 % per year is an increase in "shadow prices" that are irrelevant in the real, material world.
    How come, then, that investors today are able to obtain rates of return on capital that vastly exceed the rate of increase in wealth? How do we explain the gap between the rate of return on capital and the consumer discount rate? This question has also been raised recently by the French economist Thomas Piketty (in his book from 2014: Capital in the twenty-first century). He states that the annual rate of return on capital (before tax) is typically 4 - 5 %, whereas the growth rate of the economy is typically 1 - 1.5 %. In his view, this will lead to growing inequality, where a minority of the population accumulates wealth, at the expense of the rest of the population. On a very long term, this will mean a labile political situation. If he is right, it will be very wrong to operate with discount rates as high as 4 % or more, because this will mean endorsing an unstable political situation.
    But how can one explain in the first place that the return on capital can be higher than economic growth, at least for a period of many decades ?
    One explanation is that the increase in wealth depends mainly on the rate of technological innovation and is rather independent on the rates of return og invested capital.
    Another explanation may be found if we focus on the rate of re-investment. Consider a typical investor who earns 5 % in return on his capital each year. On average, he will re-invest only 1 % out of these 5. He will maybe pay nearly 2 % in tax, and the remaining 2 % will go on consumption. As tax may be considered "public consumption", he will altogether use 4 % on consumption. This money does not disappear - it returns into the general circulation of money - but on the other hand, it does not contribute to the growth of the economy. Only that which is re-invested contributes to general growth, and therefore we see a rate of growth in the general wealth of about 1 %. Many economists will use the wording that the savings rate is 1%, that is, they will postulate that the savings rate is equal to the rate of growth of the general economy. In saying so, they disregard the possibility mentioned above that the rate of growth may be more dependent on technological advance than on the rate of saving and reinvestment.
    A third explanation is that the difference between the actual rate of return on investments and the risk free discount rate is explained by the uncertainty on risks, which implies that the actual rate of return includes a premium for risk-taking.
   If we earmark money for an "AIDS fund", which is to be used within 10 years, then it might be fair to expect a yield of 5 % (out of which, tax will have to be paid). But if we earmark money for a "climate fund", which is to be used in 100 years´ time, then during all that time the accumulating amount will not be reduced by spending for tax or consumption. It is therefore unlikely that it could sustain a yield of 5 % for so long. The amount would be so large that the bank would hardly be able to pay it out when the time came. On the other hand, this would surely be possible if the rate of interest were only 1 %, because that would correspond the increase in wealth in society in general.
    In general we cannot expect that the usual rates of return on capital - say 5 % - can be sustained for more than about 30 years. No investments are available that extend for longer. If we have to save money for longer periods, such as in life insurances, we can obtain less than 2 % rate of interest. In the guidelines for preparing economic analysis issued by the American Environmental Protection Agency (EPA), it is recommended that intra-generational discounting use a discount rate of 2 to 3 %, which is reckoned to be the market interest rate after tax. But for the inter-generational discounting (i.e. over time spans of more than one human generation) they recommend a discount rate of only 0.5 %.
    In summary, the descriptive approach to discounting is relevant only on time scales of under 30 years. On longer time scales - as in the case of climate changes - this approach is definitely wrong, and should not be used.

The concept of  discount rates in the Stern Review.
     The Stern Review is a report made by the economist sir Nicholas Stern for the British government in 2006. In this report (chapter 2 and appendix to chapter 2), the discount rate is seen as composed of two parts. This appears from the formula used by Stern, a formula which goes back to Ramsey (1928):
    s = eta x G + delta
s is the social discount rate, also termed the `social rate of time preference´. Eta is a parameter indicating the price elasticity (see below). G is the growth rate of the total economy, i.e. the expected future growth rate of consumption of material goods, and delta is the `pure rate of time preference´.
   The first part is eta x G.
   G - the growth rate - will be different in different future scenarios. For instance, G may be different in a society where we engage in strong reductions in CO2 emission very soon, and another society where we will cut CO2 emissions only later. It is dependent on what `path´ we follow. In the situations discussed by Stern, G is on average about 1.3 % per year.
   Eta is the elasticity (the elasticity of marginal utility with respect to consumption). It indicates how much weight should be given to the consumption of the poor relative to the rich. If eta = 1, then it means that if we become twice as rich as we are now, then we would value an increment in material welfare half as much as we do now. If eta = 2, then we would value an increment in material welfare only one fourth as much as we do now. Many economists claim that eta should be larger than 1, which would increase the total discount rate, s. However, there is a problem because eta represents three different concepts (risk, inequality and allocation over time) in one single parameter, and different economists may focus on different concepts. Actually, Ramseys formula does not adequately keep these concepts apart, and therefore the parameter eta gives rise to considerable confusion (Beckerman & Hepburn (2007), World Economics 8(1): 187-210). Here is a point where the science of economics has not yet been able to describe the situation adequately, and further clarification is needed.
   The second part is delta, the `pure rate of time preference´. It is a figure expressed as percent per year, and it indicates how impatient we are - how much we would like to have the goods now rather than wait until some time in the future. This makes much sense when a private person considers whether to invest money or to use the money now. For instance, this may be a 50 year old man. Would this man be willing to wait twenty years to receive the returns of an investment? By that time he will be 70 years old, and maybe too old to enjoy an adventurous journey to exotic places. He may even be dead. How sensible would it be for him to wait twenty years? The money might be much more valuable to him now. So delta should be larger than zero for him. But if money is invested by society and the benefits will accrue in a hundred years from now, then such considerations are irrelevant - none of those who decide to invest the money, will be alive to see the benefits.
    As pointed out by Stern and coauthors (Dietz et al. (2007): World Economics 8(1): 121-168), it would be unethical to let delta have a value much higher than 0. If , for instance, delta  = 2%,  someone born in 2008 would have around half the ethical weight of someone born in 1973,  and if  delta = 3 %, individuals existing at the end of the century would be worth roughly one tenth of individuals living now, irrespective of their relative income. The only argument that delta should be larger than zero, says Stern, is that there is a theoretical chance that society will not exist in a hundred years or more. The globe may  be hit by a large meteorite, or society may crash so that all bank accounts are cancelled. Because of the risk that such disasters could happen, it might not necessarily be wisest to invest the money, and therefore delta should be slightly larger than zero. Stern sets it at 0.1 % per year. If this is taken as the chance that our society might disappear or crash, this is a very high rate. Actually, however, there was probably also another argument for setting the rate above zero. That has to do with some technicalities concerning the optimal savings rate.
   The total social discount rate applied by Stern is eta x G + delta =  approximately 1 x 1.3% + 0.1 % = 1.4 %. However, this is only approximately. The rate is different for each run of the computer model. To assess risks and uncertainty with each setting of parameters in the computer model, a thousand model runs are made in order to study the spread of the results. In each of these thousand runs, a discount rate is employed that varies with the "realized" rate of economic growth, after climate damages have been subtracted. Because the discount rate is made to depend on economic growth, model runs with  higher damages have lower discount rates. As a low discount rate means relatively great weight on losses in the distant future, those runs with higher damages are weighted more heavily in the aggregation of the thousand runs. Thereby, the possible risk of bad outcomes is given special weight. This effect is to some degree intended, because the modellers want that the importance of avoiding high risks should somehow enter into the calculations.

Nordhaus´ concept of discounting, and the discrepancy between prescriptive and descriptive approaches
    One of the world´s leading scientists dealing with the economics of climate change, William D. Nordhaus, has written an article in which he criticises the use of discounting in the Stern Review (Nordhaus (2007): Journal of Economic Literature 45 (3): 686-702). Here he argues strongly for the descriptive rather than the prescriptive approach, and he argues that the low discount rate used in the Stern Review leads to absurd results. Nordhaus himself uses the values G = 1.5% per year, eta = 2 and delta = 2.5% per year. This gives a total value of the social discount rate of 1.5% x 2 + 2.5 % = 5.5 %. In this way he bridges the gap between the observable rate of growth of GDP (1.5%) and the observable rate of return on invested capital (5.5%). However, one may argue that this represents a misunderstanding of the concept of eta (the elasticity parameter). Actually, economists today have not been able to give a full and satisfactory explanation of the discrepancy between  the observed rate of growth and the observed rate of return on investments.  It does not clarify anything to postulate that this discrepancy is "explained" by having high values of eta and delta. It remains a postulate that eta should be e.g. 2 or some other figure - this is just a figure coming out of the blue that in itself explains nothing. So this criticism - and much other criticism of the Stern Review - goes back to the more fundamental problem that different economists attach different meanings to "eta" and use it for different purposes.
    In the same issue of Journal of Economic Literature, there is an interesting paper by M. L. Weitzman, pp. 703-724. Here it is stated that if eta = 1 and delta = 0 (approximately the values used by Stern), then the growth rate and the rate of return are equal, which implies that all returns are reinvested, i.e. all capital is saved, and nothing is used. This, of course, is impossible in the real world, and therefore these parameters have been given impossible values. The problem here is that eta is taken to express what proportion of the money earned is saved, whereas Stern uses eta to express what weight is put on equity between different groups of people. Thus, eta is used to express two completely different issues, which leads to absurd results concerning the savings rate.
   As said, in the article by Nordhaus he uses a social discount rate of 5.5%, approximately equal to the observed rate of return. As stated above, such a high rate of discount is untenable over long time periods, and leads to absurd results, such as the value of owning all of Denmark in 500 years from now being equal to the value of having a single fried chicken today. Nordhaus, on the other hand, says that using a low rate of discount over long time periods in the way that it is done in the Stern Review, leads to absurd results. So, no matter what solution one prefers, the results are absurd in one way or the other.
   It seems that no matter what values economists choose to use, the results will, on a very long time scale, be absurd. The implication of this is that the state of the art of economics is not yet good enough to cope with the problems we are facing. This is more or less what is said in the above article by Weitzman. On pp. 714 he says that the biggest disconnect between the reasonings behind the prescriptive approach and the descriptive approach are the asset-return puzzles. "These puzzles very strongly suggest that something fundamental is amiss in the paradigm framework for pricing assets and deriving the rates of return that we are relying upon to produce discount rates . . . ". In conclusion, the different opinions, with various economists accusing other economists of using flawed discount rates, may arise from basic flaws in the economic paradigms.
   (The economic model developed by Nordhaus, the DICE model, is criticized here on Lomborg-errors).
Uncertainty of future growth rates
    At least since the Second World War, the yearly rate of growth in the economies of the western developed countries has been fairly constant, and one could reasonably assume that this will continue. However, in some countries where the standard of living lags behind that of the west we are now seeing very high growth rates - most notably in China, but also in other countries of east and south Asia. It is likely that growth rates in these countries will remain high only as long as there is an obvious scope for growth. When their standard of living some time in the future approaches that of the western countries, (or when resource and environmental constraints prevent it from ever doing so) we may expect the rate of growth to slow down. This has already happened quite markedly in Japan. The present rate of growth in the total world economy will therefore probably not continue. This in turn means that, if we discount with time horizons as distant as 100 years, we cannot base the discount rate solely on present high growth rates.
    Another reason for reducing the discount rate is the uncertainty about the future. This may be illustrated by the principles followed by life insurance companies. The company is obliged to pay a certain sum to its customer at a certain time in the distant future. Here, it is not enough to be 50 % sure that the amount will actually be available. The company must be, say, 90 or 95 % sure. So if the most likely rate of return on that time scale is 2 %, then the company will have to make an "uncertainty deduction", which has the same effect as yielding a rate of return that is somewhat lower than 2 %.  Due to this principle, the larger the uncertainty about the future, the lower must be the discount rate.

Some other reservations
    Discounting requires that all benefits be expressed quantitatively. This does not necessarily mean that they be expressed in terms of money, but in practice they are expressed just so. This means for instance that human lives have to be converted into sums of money. Such analyses have been carried out by putting a monetary value on each human life lost equal to two years´ income of the person concerned. This means that a life-year lost in Africa is worth only 1.4 % of a life year lost in USA. In many situations, this is ethically unacceptable, and we will have to refrain from expressing values in economic terms.
    Likewise, there may be situations where we do not accept that the life of other organisms is expressed in economic terms. For instance, we may find it unacceptable that a particular species is exterminated, even if this may pay off economically, because we assign a value to the existence of that species that is independent of its economic value.
    If we use discounting of money sums to find the optimal path of development, then we will maximise the amount of money in the world, and only that. We will probably not maximise the sum of all human happiness. Two examples will illustrate this:
    First, human happiness depends not only on the average situation, but also on the predictability of the situation. If for instance certain very destructive weather extremes - droughts, floods, hurricanes - appear at irregular, unpredictable intervals, then there will be a high level of general uncertainty. This will increase the level of worry and unhappiness.
    Second, if climatic zones are shifted, it may become necessary to transfer large human populations from one region to another, or from one country to another. In economic terms, such transfers may be quite feasible. But the social costs may be very high, for instance if people in the recipient country do not accept the newcomers.
    On these and many other grounds, we should not accept that all the problems of the world are solved only with respect to optimisation of the economy.

Policies for climate mitigation
    In his "challenge paper" at the Copenhagen Consensus (3), W. R. Cline describes optimal rates of carbon taxation with different discount rates. Under one set of assumptions, for instance, he finds that the optimal reduction in CO2 emissions in the middle of the twenty first century will be a 45 percent "cutback" with a discount rate of 0, 20 percent cutback with a discount rate of 1.5 percent, 6 percent cutback with a discount rate of 3 percent, and 4 percent cutback with a discount rate of 4.5 percent. Clearly, a high rate of discount means that it only very small reductions in CO2 emissions will pay.
    In the same paper, he describes three possible policies, i.e. strategies for coping with CO2 emissions:
    a) The Kyoto protocol, where emissions are cut only in the developed countries
    b) Optimal carbon tax, a policy where emissions are taxed in all countries, but where revenues are kept within each country.
    c) A so-called value-at-risk approach, which aims to avoid `maximum´ damage.
    Cline finds that the Kyoto protocol, where rises in emissions in The Third World are not reduced, will have little effect on future warming. Therefore, for the rich countries, the benefits will not outweigh the costs. But for the two other policies, he finds, benefits may outweigh costs (measured in constant prices) on a time scale of 300 years. This is so for discount rates from zero up to a level of 1.5 %, even though the computer model does not allow for technological adaptation to non-carbon energy sources. But with rates of 3 % or 4.5 %, the conclusion is reversed - costs outweigh benefits.
   In the computer model that Cline has used in his paper for Copenhagen Consensus, the abatement costs, when expressed as percentage of GWP (gross world product, i.e. the GDP of the whole world) are as follows: A 10 % cut in emissions in the year 2045 would cost 0.03 % of GWP. A 30 % cut would cost 0.32 % of GWP, and a 50 % cut would cost 0.97 %. A 75 % cut would be expensive - it would cost 2.3 % of GWP. It should be noted, however, that these "costs" could be tax revenues which do not disappear out of the economy.

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