time preference. Key words. Discount, logistic growth, limits, present value. Introduction. The rate of discount has been the subject of considerable discussion.
Note
Discounting in a World of Limited Growth 1 THOMAS STERNER Department of Economics, University of Gothenburg, Viktoriag 30, 41125 GOteborg, Sweden
Abstract. This paper explores the consequences for discounting of assuming limits to growth. One of the main determinants of the discount rate is the rate of economic growth. If growth rates decline in the future then the discount rate should not be constant but also decline over time. In fact, we would then need not a single discount rate but rather a variable discount schedule. This would imply higher present values for the distant future. The paper analyses how discount rates would vary with different assumptions about the patterns of growth and the pure rate of time preference. Key words. Discount, logistic growth, limits, present value
Introduction The rate of discount has been the subject of considerable discussion. In this paper we will discuss the role of economic growth and of the pure rate of time preference or "myopia" which Pigou (1932) also referred to as a "defective telescopic faculty." A m o n g the works that have inspired later research we note in particular Arrow and Lind (1970), Baumol (1968), and Marglin (1963). The literature is by now so large and well known that a complete survey would be rather pointless in this context. We will limit ourselves to referring the reader to others such as the collection of papers in Lind (1982) which cover many of the economic and philosophical arguments. More recent studies that are particularly relevant include Kula (1988) and Norgaard and H o w a r t h ' s (1991) work on intergenerational transfers. Price (1991) and Hanley (1992) again review m a n y of the arguments concentrating on the issue of whether environmentalists should prefer high or low discount rates. Much of this recent work addresses fundamental issues of intergenerational equity and ethics, overlapping generations, intertemporal efficiency, issues of the productivity of natural capital, uncertainty, risk and risk aversion. However, in this paper, we will not deal with these issues but assume that discounting is possible and appropriate in order to focus on the relationship between e c o n o m i c growth and the rate of discount. Growth is one of the main factors determining the rate of discount as shown by the so called " R a m s e y rule" f o r the optimal rate of social time preference (1), see for instance, Dasgupta and Heal (1979), Chapter 10. r, = p, = 6 + TIg
(1)
where r t is the rate of interest from the production side (determined by the marginal productivity of capital), Pt is the interest rate from the utility side Environmental and Resource Economics 4: 527-534, 1994. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.
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which is determined by 8, the pure rate of time preference and the rate of growth in consumption g = (dC/dt)/C. "q = -dZU/dC z * C/(dU/dC) is the elasticity of marginal utility. 2 The first part of (1) is an equilibrium condition equating the marginal productivity of capital to consumers' marginal rate of time preference. The second part, which is of particular interest in this paper, equates these both to the sum of the pure rate of time preference and the product of consumption growth and a factor that expresses the rate at which consumption impacts on utility. This part is derived by assuming an intertemporal welfare function W = I e4tU(Ct) dt" The slope of this function between any two arbitrary points in time t and t + E is the marginal rate of time preference from the consumption side and gives us the right hand side of (1) by derivation, see Dasgupta and Heal (1979) for details. Naturally (1) is derived under a number of simplifying assumptions about the utility and welfare functions as well as about technology. It is also a starting point which can be modified by considerations of taxation, funding, social and private perceptions of risk, risk aversion and risk spreading, relative productivity differentials (and price changes) between sectors and other considerations raised in the literature. Since the purpose of this paper is to discuss the effects of various assumptions about growth, we will work with the simple case in which (1) holds. THE LIMITS TO GROWTH
When discussing growth, economists and ecologists have different perspectives. We will start by considering the ecologists' frame of reference which is formed by looking at natural biological growth in various ecosystems. Even in these settings we find exponential growth as a common model - but only under special conditions or limited periods of time when there is an abundance of space and resources. For instance, a bacteria culture will grow at a roughly exponential rate as long as there is enough space and a suitable substrate. The logic behind exponential growth in biological systems is that the number of individuals born in any period depends on the number in the previous generation. Sooner or later, however, other factors come in to check growth: these may be predation, disease or lack of space, food or some other vital input. The combination of underlying tendencies for exponential growth with limiting factors can give a number of well-known growth patterns such as the so-called logistic curve or various oscillatory (or even chaotic) paths. AN = rN(K-N)/K
(2)
(2) is a logistic difference equation for the increase in population, AN, as a function of population N and carrying capacity K. When the latter is very large, growth is just equal to rN (exponential), when N approaches the carrying
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capacity growth falls to zero. The growth rate for continuous time, may be given by a formula such as (3) 3 (3)
dN/dt = ~ N - 7N 2
The difference in long run paths between exponential and logistic growth is illustrated in Figure 1. Ecologists generally argue that human population must, at some stage, follow such a path (if, indeed, we do not follow a more drastic one). One might argue that the limits to growth argument only applies to population but not to economic growth. Substitution and technical progress should allow at least some sectors which need small (or decreasing) amounts of physical resources, to go on growing for ever. On the other hand, there are some sectors that do require certain minimum amounts of energy and material. The actual feasibility of long-term economic growth depends on a whole set of factors including the character and rate of technical progress, economies of scale, the availability of natural resources, and the elasticity of substitution between such natural resources and reproducible capital. There is a considerable and rather well known literature on this subject and we shall merely assume, for the sake of our argument, that, in the long run, the average rate of economic growth must decrease either to zero or some fixed (low) rate. The rationale for this may, for instance, be the existence of some essential natural capital which implies limits to the substitutability of natural and reproducible capital. DISCOUNTING UNDER VARIOUS ASSUMPTIONS ABOUT GROWTH
From (1) it is clear that a slow-down in growth should imply decreasing rates of discount. If we apply a logistic growth formula to income Y and substi-
INCOME
]
Exponential
TIME Fig. 1. Exponential vs logistic growth.
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tute the equivalent of (3) into (1) we obtain a modified discount schedule (4). r = 5 + q(fJ - yY) = 5 + rlY(Yo - Y)
(4)
where [3 is the inherent growth rate (which in principle corresponds to a hypothetical initial rate when Y = 0) and Yo is the final equilibrium income level at which the economy is assumed to settle down. 4 Thus it is clear that discounting is jointly determined by preferences and technology. We will take the preferences as given and examine the consequences of alternative growth paths for the discount rate. For expositional purposes, we first assume a zero rate of time preference 5 and concentrate on economic growth) In the next section this assumption will be relaxed and a positive rate of time preference introduced. The "traditional" discount factor is simply the inverse of exponential growth. It therefore follows that, if instead o f exponential growth, we have logistic growth (with zero time preference) we can then calculate analogous discount factors inversely proportional to these new income levels. In Table I, the first two columns show exponential growth (at 3%) and the corresponding discount rate. The latter is inversely proportional to the former so that the present value of £1.00 in year 40 is equal to 1/3.26 = £0.307, or about 31 pennies. Column 3 shows a logistic growth which starts off with a growth rate of 3% but then levels off. The carrying capacity or sustainable income level has been chosen at ten times the size of the present global economy.6 This is to show that even if we believe in considerable scope for technical progress and growth, discounting the future should still be adjusted for the eventual levelling-off of income levels. Column 4 shows a "logistic discount schedule" which is inversely proportionate to the levels of income attained with logistic growth. As will be appreciated this schedule does not directly correspond to any single, ordinary (exponential) discount rate but rather to a gradually declining rate which falls as growth levels off. After 40 years of logistic growth (col 3) the economy has grown to 2.66 times its present size instead of 3.26 (in col 1) and so the discount factor is 0.376 and the present value of a pound is now 37.6p instead of 30.7p, For most projects the time horizon is less than 30 years and thus our distinction makes little difference. This is because we count on growth for another couple of decades. For longer periods, however, growth is here expected to cease. After 250 years of exponential growth the economy would be 1619 times as big and the appropriate discount factor would be the inverse of this number (the present value of a pound would only be 0.06p). With logistic growth the size of the economy would already have levelled off at ten times its current size and the present value of £1 would still be 10p. If the final carrying capacity of the economy had been chosen at some other level than ten the present
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value of £1 at any particular future date would be different (and proportional to the inverse of the carrying capacity chosen).
Combining Logistic Growth and Pure Time Preference From column 4 we see that after year 40 the discount factor hardly changes at all since the size of the economy has levelled off. In fact, the present value of a £1 never falls below 10p. While this constancy may appeal to some it would generally appear to be a problem to most people. Surely, if we can postpone a major cost to humanity from year 2100 to year 5000 this would be an advantage? The answer depends on the other factor in (1), namely, the pure rate of time preference. The correct procedure is, therefore, to combine a discount schedule based on logistic growth with an (exponential) time preference as in (1). Columns 5 and 6 illustrate the result of such a combination (see notes to Table I for formulae). Alternatively they can be taken to illustrate the effect of a certain (low) rate of perpetual growth due to technical progress. The pure time preference (or equilibrium growth rate) has been chosen at 1 and 0.1% rates. The present value of £1 is now either 36p or 25p after 40 years and a £1 cost that is 1000 years away is now valued at 3.68p or 0.0005p for a time preference of 0.1% or 1% respectively. In both cases it continues to decline although at a much slower rate than in the purely exponential case of column 1. Figure 2 illustrates the difference between logistic and exponential discounting: The curve that declines fastest is the ordinary exponential discount. By way of contrast the logistic discount schedule levels off (at a level Present
Value
L o..qist/c discount
Time
Fig. 2. Different discount schedules.
Discounting and Growth
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proportionate to the inverse of the carrying capacity) and the combined logistic growth cum pure time preference schedule is intermediate. It does fall to zero but more slowly than the pure exponential discount rate. The consequences of how we choose to carry out our discounting only become drastic in the long run. We will, therefore, focus on a hypothetical long run environmental problem (such as global warming or nuclear wastes) and assume it implies a constant lump sum cost for each and every year during the next Millennium (from 2000 to 2999 AD). Imagine we were trying to assess total costs: How would we then value the first century's costs as a fraction of the total costs throughout the entire thousand year period? The final three rows of Table I answer this question by showing the relative weight of an annual cost of £1 for the next century compared to the same annual cost for the next ten centuries. 7 We value the first century at 95% of the total cost and the last 9 centuries at 5% of the total with ordinary discounting based on exponential growth. If instead we use the pure logistic discount schedule proposed, with a carrying capacity of 10 we would instead value the first century at 30% leaving 70% of our concern for the last 900 years. Combining discounting based on logistic growth with a pure rate of time preference, (or allowing growth to fall to 1% or 0.1% instead of zero) as in the last two columns does of course produce intermediary results. With a lower pure rate of time preference (0.1%) the share of the first century is still relatively low at 40% but already with a pure time preference of 1% the logic of exponential growth increases this share to 87%.
Conclusion In this paper, we have shown that if economic growth is expected to level off in the long run then discount rates should be adjusted. In fact, the single discount rate should be substituted for a nonlinear discount schedule which decreases with the inverse of the growth rate. We show how it will depend on three parameters: initial growth, carrying capacity and the pure rate of time preference. This way of discounting will typically give higher values for very tong run problems. Considering the increased importance of such longterm environmental problems further analysis of these discount schedules should be an important task for future research.
Notes 1 An earlier version of this paper was presented at the conference "ECOLOGICAL ECONOMICS", Stockholm,August 1992. I am grateful for valuable comments from Partha Dasgupta, Gunnar K0hlin, Karl-G~SranM~iler, Mike Young and two anonymous referees. 2 Dasgupta and Heal call this the elasticity of marginal felicity to make it clear that they do not necessarily make appeal to classical utilitarianism. For simplicity we have chosen to use the more common term utility throughout this paper.
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3 This is sometimes written dN/dt = ~ + ~N - yN ~ but we will here assume that growth is zero when population is zero. 4 Yo is given by g = 0 = 13 - YYo so that Iio = 13/% 5 In our numerical example, we also assume for simplicity that the elasticity of utility, rh is constant and equal to one. The principles would be the same with any other value. The effect of exponential discounting with a growth of 3%, in our example, could also be used to illustrate the effect of 2% growth and a marginal elasticity of utility of 1.5. Furthermore, we assume that growth does actually decline to zero although the essence of the argument would be the same if it decreased to some other arbitrary (but low) level. 6 Young (1992) proposes this limit, pointing out that it allows for a doubling of World population and an average living standard such as currently found in industrialized countries. 7 Without discounting this would naturally be 10%.
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