Discounting: A Review of the Basic Economics

Transcription

1 Disounting: A Review of the Basi Eonomis Geoffrey Heal I review the justifiations given for disounting future benefits relative to present, and distinguish between the pure rate of time preferene, or utility disount rate, and the onsumption disount rate, also sometimes known as the soial rate of disount. I disuss when to hoose one or the other, and how to hoose a disount rate in a real-world projet. I. FUNDAMENTAL DILEMMAS OF DYNAMIC WELFARE ECONOMICS Environmental eonomis provides a good entry point to a disussion of disounting, as intertemporal issues play a partiularly entral role: the time sale of many environmental proesses is radially longer than onventional eonomi time sales. Global warming and loss of biodiversity provide perfet illustrations. Global warming may have its main impats on human soieties one hundred or more years hene, and likewise the osts of loss of speies diversity in terms of simplifiation of eosystems and loss of geneti variability are likely to be felt most strongly by generations quite remote from us. This is not to say that there will be no short-run impats from these phenomena. There will be, but they are likely to be dwarfed by onsequenes that will beome apparent only over very long periods. So to assess and evaluate properly these hanges we need to look relatively far into the future possibly a entury at least. More generally, environmental assets suh as watersheds, speies diversity, rangelands, marine eosystems, and limate regimes are assets that are in priniple very longlived. They have funtioned as they do today for millennia, and if well managed will ontinue to do so equally far into the future. In this, they are rather different from the assets that humans onstrut and that we are used to valuing. These typially have life spans measured in years or deades. So to appreiate fully the ontributions that environmental assets an make to human welfare we need a very long view. This does not sit easily with the eonomist s standard pratie of disounting the future at a real rate of at least 3 perent or 4 perent. Paul Garret Professor of Publi Poliy and Corporate Responsibility, Columbia Business Shool, and Professor, Shool of International and Publi Affairs, Columbia University, gmh1@olumbia.edu, This Artile draws heavily on my entry Intertemporal Welfare Eonomis and the Environment, in Karl-Göran Mäler and Jeffrey R. Vinent, eds, 3 The Handbook of Environmental Eonomis: Eonomywide and International Environmental Issues 1106 (Elsevier 2005). 59

2 60 The University of Chiago Law Review [74:59 After all, if we disount at 3 perent per year, then $100 one hundred years from now is worth only $5 now; thus events so far into the future will be of little onsequene in a ost-benefit analysis. In an obvious and intuitive sense, disounting seems to tip the sales against the future. 1 Is this indeed the ase? What are the arguments for disounting over suh a long period? What are the alternatives, if any? The debate to whih these questions give rise is not a new one. It runs right bak to the origins of dynami welfare eonomis. Frank Ramsey, who wrote a paper on optimal growth from whih we an still learn today, ommented that disount[ing] later enjoyments... is ethially indefensible and arises merely from the weakness of the imagination. 2 And his ontemporary Roy Harrod was equally outspoken, remarking that disounting is a polite expression for rapaity and the onquest of reason by passion. 3 They are saying, rather learly, that the right disount rate is zero. More reently, The Eonomist, normally a repository of mainstream thinking on eonomis, was driven to remark: There is... something awkward about disounting benefits that arise a entury hene. For, even at a modest disount rate, no investment will look worthwhile. 4 Yet Ramsey s pointed remark, though intuitively appealing, misses some deep tehnial points relating to the ranking of alternative onsumption streams over time. These are important issues oneptually, and also omplex and indeed treaherous from a tehnial perspetive. In working through these issues, we need to start by understanding our options for ranking intertemporal utility or onsumption streams. Consider an eonomi model in whih eah generation lives for a finite number of periods and generations may overlap in time. Many people though not all would agree with the idea that we ought to give equal weight to the welfare levels at all points in time. It is after all diffiult to make a really strong ase for treating some generations better or worse than others. Of ourse if we are utilitarians, we may plae less weight on a marginal inrement of onsumption of rih generations than of poor ones, but this is not an intertemporal judgment. It is an interpersonal one. It arises from diminishing mar- 1 See Geoffrey Heal, Nature and the Marketplae: Capturing the Value of Eosystem Servies (Island Press 2000) (desribing the effets of applying disounting to environmental values). 2 F.P. Ramsey, A Mathematial Theory of Saving, 38 Eon J 543, 543 (1928) (examining the issue of how muh a ountry should save over time). 3 R.F. Harrod, Towards a Dynami Eonomis: Some Reent Developments of Eonomi Theory and Their Appliation to Poliy 40 (Mamillan 1948) (laiming that a government doing what is best for its subjets will not disount). 4 What Prie Posterity, Eonomist 73 (Mar 23, 1991), ited in Geoffrey Heal, Valuing the Future: Eonomi Theory and Sustainability (Columbia 1998) (explaining that many have questioned the wisdom of disounting).

3 2007] Disounting: A Review of the Basi Eonomis 61 ginal utility and not from differential treatment of generations. Suppose then that we want to treat all generations equally, and that we are utilitarians at least in the limited sense that we assume at eah date t welfare is a onave inreasing funtion of onsumption, u( t ). This assumption embodies the lassial one of diminishing marginal utility. We want to maximize the total welfare, the sum of welfare over all periods t T = t = 1 u( This is easy when we have a finite number T of periods: we just give equal weight to the utility of eah, as we are doing here. But eonomists have typially wanted to work with infinite horizons. This is mainly beause of a relutane to speify a date beyond whih nothing matters, as one does when one hooses a terminal date T that is finite. Also influential is a onern about the impat of end effets, by whih I mean sensitivity of the ranking of paths to the preise end date speified. So we tend to look instead at the sum of utilities from the present and forever on, the infinite sum t= t= 1 u( In many interesting and relevant ases, this sum will be infinite. In fat, it will be infinite for all onsumption sequenes on whih utility levels are bounded away from zero, and on many others as well. This means that we annot now represent hoosing the best onsumption sequene as finding the highest value of an objetive funtion, whih is the standard way of finding a best hoie. Finding the best path is not in this ase a onventional maximization problem. The only way of making sure that the infinite horizon sum is a finite number, so that this is a standard maximization problem, is to treat generations unequally, and in partiular to give little weight to most of them, whih of ourse is what disounting does. So there is apparently a pratial reason for disounting. It is a way of ensuring that we have a well-defined preferene order over the set of alternative infinitely long-lived onsumption sequenes between whih we must hoose, and that the best aording to this ordering is the solution to a maximization problem. Is it the only way to solve the problem or are there others? Ramsey was learly aware of the diffiulty bak in the 1920s and had an ingenious alternative. His approah ensures that we give equal weight to all generations, but depends on some speial assumptions and is not t t ) )..

4 62 The University of Chiago Law Review [74:59 a general resolution of the issues. 5 Others have developed more general approahes. For example, Carl Christian von Weizsäker 6 and others have suggested the overtaking approah, whih is now widely used. This ranks as best the onsumption sequene, if any, whose umulative utility sum eventually exeeds that on any other path. Formally this 1 means that path t ranks above path t if and only if there exists a time T 1 suh that whenever T > T 1, then t = T u ( ) > t = T u ( t t t = 1 t = 1 This is a nie idea: we ompare the umulative welfare levels along alternative paths and see if there is one that eventually dominates. The idea behind this approah is that we an avoid disounting future utility levels and thus treat all generations equally. It is not ompletely suessful in this. 7 And, one again, it does not allow the best hoie to be found as the solution to a onventional maximization problem. But for most purposes it is a good solution. The onlusion of this disussion is that we must be ingenious if we are to evaluate onsumption sequenes over time in a manner that gives equal weight to all generations, if at the same time we insist on working with infinite horizons and a roughly utilitarian framework. Perhaps in this ase we should waive the requirement of infinite horizons? In fat, as we will see below, this onession buys surprisingly little. A seond problem, less fundamental but nonetheless demanding and only reently moving to the enter of the stage, is dynami onsisteny. A hoie of a onsumption path is dynamially onsistent if it has the following property: if at some date during the exeution of the hosen path we stop and ask what path we would now hoose, given what we have done to date, then (provided that no parameters have hanged) the answer is that we ontinue with the original hoie. In other words, we see no reason to revise our hoie merely beause of the passage of time. Not all algorithms for making hoies over time 1 ). 5 Ramsey made some rather speial assumptions that ensured that this sum onverged in his ase. He assumed that utility levels are bounded above and then sought to minimize the total shortfall over time of atual utility levels from their maximum level: t = [ b u ( t )]. t = 1 See Ramsey, 38 Eon J at 545 (ited in note 2). Here b is the upper bound of the utility funtion: think of it as Bliss. Id. 6 See generally C.C. von Weizsäker, Lemmas for a Theory of Approximate Optimal Growth, 34 Rev Eon Stud 143 (1967) (disussing the best utility path to take for preferred levels of eonomi growth). 7 See Heal, Valuing the Future at (ited in note 4).

5 2007] Disounting: A Review of the Basi Eonomis 63 are dynamially onsistent. Indeed, relatively few are. If we feel the need for dynamially onsistent hoies over time, then this again onstrains how we an approah this area. Finally, a third issue is that a path of onsumption over time should generally be intertemporally effiient. This means that it is a path with the property that no variation will make some generation better off and none worse off. Obviously, if suh a variation were possible, we would normally want to take advantage of it. Finding a way to make hoies that attain or ome lose to these three desiderata equal treatment of all generations, onsistent hoies, and effiient hoies is diffiult. It is the subjet matter of dynami welfare eonomis. As noted, it is partiularly relevant in the environmental ontext beause of the unusually long time horizons implied by the unfolding of anthropogenially indued modifiations to the biosphere. II. DISCOUNTING UTILITY OR CONSUMPTION? As a vehile for illustrating the ideas surrounding disounting, onsider an intertemporal optimization problem in whih we seek to maximize the integral of welfare over an infinite horizon, where welfare depends on the flow of onsumption and on a state variable (the state of the environment perhaps) and where there is a tradeoff between onsumption and the state variable: dt ds Max u (,s ) e subjet to + f ( s ) 0 t t =. (1) dt Here is onsumption of a final good, and s is a stok that ould be a apital stok or an environmental state variable. This is a standard optimal growth model. Output f(s) an be onsumed or invested. We are disounting future utilities at the rate d. If d > 0, this means that future generations are being given less weight than the present generation just beause they are in the future: futurity alone ondemns them to a disount relative to us. A very important distintion in this type of utilitarian model is between the rate d at whih utility is disounted, also known as the soial rate of time preferene or pure rate of time preferene, and the rate at whih onsumption is disounted. The latter rate is sometimes alled the soial disount rate. The distintion is simple. Within this framework, one an ask the following question: Suppose the eonomy follows an optimal time path, one that solves the above problem, and we onsider adding an inrement of onsumption at some date t. What is the value of this inre-

6 64 The University of Chiago Law Review [74:59 ment in terms of its ontribution to the objetive funtion, and how does this value hange as the date t is hanged? 8 This ontribution to the objetive funtion is the value that should be assigned to an inrement of onsumption. The rate at whih this ontribution hanges over time is the onsumption rate of disount. It is the rate at whih the weight of an inrement of onsumption hanges over time. It is the rate at whih inremental units of onsumption are impliitly disounted along the path. Clearly, the value of an inrement of onsumption at date t is the marginal utility of onsumption at t disounted to the present 9 u (,s t t )e dt and the rate at whih this value hanges with t is, 1 u (,s )e t t dt whih we an easily ompute to be du (t,st )e dt dt, d η, d/dt η,s ds/dt s where η, = u,/u > 0 is the elastiity of the marginal utility of onsumption with respet to the level of onsumption and η,s = su,s /u is the elastiity of the same quantity with respet to the level of the stok of the asset. These measure how quikly the marginal utility of onsumption hanges in response to hanges in the levels of onsumption and of the state variable. This expression is the onsumption disount rate. Consider for simpliity the ase in whih the utility funtion is additively separable, so that the ross derivative is zero and u,s = 0. Then the onsumption rate of disount is d/dt d + η,. 8 For a mathematial explanation of the distintion between the utility disount rate and the onsumption disount rate, see Geoffrey Heal, Depletion and Disounting: A Classial Issue in the Eonomis of Exhaustible Resoures, in Robert W. MKelvey, ed, 32 Proeedings of Symposia in Applied Mathematis: Environmental and Natural Resoure Mathematis 33, (Amerian Mathematial Soiety 1985). 9 u and u s are the first partial derivatives of the funtion u with respet to its arguments and s. u,, et., are likewise the seond partial derivatives, using obvious notation.

7 2007] Disounting: A Review of the Basi Eonomis 65 For a linear utility funtion, or for variations in the level of onsumption small enough that a linear approximation to the utility funtion suffies, this redues to the utility rate of disount: the two onepts are the same. In this ase, we have a positive onsumption rate of disount if and only if we have a positive utility rate of disount. As Partha Dasgupta, Karl-Göran Mäler, and Sott Barrett note, the onsumption disount rate is in general not onstant over time, as the terms d/dt ds/dt η, η,s s are time varying. 10 More generally, the two disount rates the onsumption disount rate and the utility disount rate differ. And, if onsumption falls over time so that d/dt < 0 then the onsumption disount rate may be negative, so that the weight given to a inrement of onsumption atually rises over time (a point also emphasized by Dasgupta, Mäler, and Barrett). In fat, if an eonomy is following an optimal path in the utilitarian sense, then the first-order onditions for optimality give us further information about the onsumption disount rate. Intuitively, the onsumption disount rate will differ from the utility disount rate beause welfare levels hange over time. If, for example, future generations are riher than the present generation, then within a utilitarian framework the value of a marginal unit of onsumption to them will be less than to us, and this will be refleted in the onsumption disount rate. This is the genesis of the term d/dt η, in the expression above, as this term shows how marginal utility is falling beause onsumption is rising. If onsumption were to be falling rather than rising over time, this effet would go into reverse and future inrements of onsumption would be more highly valued than present ones. The disount rate ould be negative. One obvious and general proposition is the following. If the utility funtion has an elastiity of marginal utility of onsumption that is 10 See Partha Dasgupta, Karl-Göran Mäler, and Sott Barrett, Intergenerational Equity, Soial Disount Rates, and Global Warming, in Paul R. Portney and John P. Weyant, eds, Disounting and Intergenerational Equity 51, 53 (Resoures for the Future 1999).

8 66 The University of Chiago Law Review [74:59 bounded, and if the utilitarian-optimal path tends in the limit to a stationary solution, then at this stationary solution the onsumption disount rate is equal to the utility disount rate. Another way of saying this is that the soial disount rate equals the pure rate of time preferene. This is immediate from the expression d/dt d + η, for the onsumption disount rate: if d/dt is zero at a stationary solution, then this expression is just the utility disount rate. So, in general, the utility and onsumption disount rates onverge in the limit along utilitarian-optimal paths, or indeed any paths that have limits. Consider some partiular ases in more detail, firstly the pure depletion model of Harold Hotelling (this orresponds to the model (1) above when f(s) = 0 for all s). The first-order ondition is that d/dt = d η so that along an optimal path the onsumption disount rate is always zero, whatever the utility disount rate. 11 In fat this is obvious. The first-order ondition for optimality in the Hotelling problem is just that the marginal ontribution of an inrement of onsumption to the objetive should be the same at all times, whih is preisely that the onsumption disount rate be zero. On an optimal path in the Hotelling model, whatever the utility disount rate, and however uneven the distribution of onsumption between generations, the onsumption disount rate is zero. This shows that a zero onsumption disount rate does not imply any degree of equality of onsumption over time. Note that in this ase there is no stationary solution to the first-order onditions for optimality so that the onsumption and utility disount rates do not onverge. For the model speified above, the onsumption rate of disount along a utilitarian optimal path is u s + f u whih is generally positive and again goes to the utility disount rate in the limit. In this ase, the first order onditions admit a stationary 11 See Harold Hotelling, The Eonomis of Exhaustible Resoures, 39 J Polit Eon 137, (1931) (defining an exhaustible resoure as an absolutely irreplaeable asset, that is, an asset that does not replenish at all over time).

9 2007] Disounting: A Review of the Basi Eonomis 67 solution so that the utility and onsumption disount rates pure rate of time preferene and soial rate of disount onverge in the limit. A similar onvergene holds for models with apital aumulation and prodution. 12 In summary, the onsumption and utility disount rates, or soial disount and time preferene rates, are not independent onepts: when there is a stationary solution to the utilitarian problem, they are equal asymptotially along a utilitarian optimal path, and are always linked by the first-order onditions. Furthermore, the fat that the onsumption disount rate is zero does not imply any degree of intergenerational equity, as the Hotelling example shows learly. Having a zero onsumption disount rate is not a solution to the ethial problem that led Ramsey and Harrod to dery the disounting of future utilities. The entral point is that we annot avoid the need to hoose a utility disount rate by fousing instead on onsumption disount rates. Nor an we justify a positive utility disount rate by invoking an argument that the onsumption disount rate may be quite different. The onsumption disount rate is driven by the utility disount rate, the form of the utility funtion, and the tehnology of the eonomy. Whih of these two rates should be used for ost-benefit analysis? The answer should be lear from the analysis so far. The utility disount rate is a general equilibrium onept used in models of the evolution of an entire eonomy over time. If we have a planning model of the eonomy as a whole and are using this to assess how muh to devote to preventing limate hange, the general equilibrium onept is appropriate: we should use the utility disount rate. If, however, we are evaluating a small projet that will have no eonomy-wide impliations say the onservation of a regional forest or fish stok then this is a partial equilibrium exerise, and the onsumption disount rate is appropriate. I will return to this issue in the onluding setion. How muh are the diffiulties that arise in disussing disount rates artifats of the use of an infinite horizon? Surprisingly, the answer is very little. Certainly one of the fundamental diffiulties is treating all generations equally when the horizon is infinite. However, we have a partial solution to this, in the form of the overtaking riterion. This gives us an infinite-horizon utilitarian approah with a zero disount rate, that is, equality of generations. Perhaps more important than this is the fat that problems with large finite horizons do not really look that different from infinitehorizon problems. By this, I mean that the solution to a finite-horizon 12 See Heal, Valuing the Future at (ited in note 4) (exploring sustainability of both renewable and nonrenewable resoures in eonomies with apital aumulation and prodution).

10 68 The University of Chiago Law Review [74:59 problem with a very long horizon looks for most of the long horizon like the solution to the infinite-horizon problem. In fat, for suffiiently long horizons the differene between the finite- and infinitehorizon solutions is lose to zero for muh of the time. This idea lies behind the Hammond-Mirrlees onept of an agreeable plan, 13 and is disussed in Heal. 14 Working with infinite horizons is an analytial onveniene and puts some entral issues into a sharp perspetive, but does not distort the issues. None of the disussion of onsumption versus utility disount rates, for example, is dependent on the hoie of time horizon. III. EMPIRICAL EVIDENCE There is interesting evidene that in making hoies over time people use a framework different in ertain salient respets from the standard disounted utilitarian approah. Of ourse, even if we have a lear piture of how individuals form their judgments about the relative weights of the present and future, this does not neessarily have normative impliations. We might still feel that relative to some appropriate set of ethial standards they give too little (or too muh) weight to the future, and so are an imperfet guide to soial poliy. 15 However, in a demorati soiety, individual attitudes towards the present-future tradeoff presumably have some informative value about the appropriate soial tradeoff and have at least an element of normative signifiane. There is a growing body of empirial evidene suggesting that the disount rate that people apply to future projets depends on, and delines with, the futurity of the projet. 16 Over relatively short periods up to perhaps five years, people use disount rates that are higher even than many ommerial rates in the region of 15 perent or in 13 See generally Peter J. Hammond and James A. Mirrlees, Agreeable Plans, in James A. Mirrlees and N.H. Stern, eds, Models of Eonomi Growth: Proeedings of a Conferene Held by the International Eonomi Assoiation at Jerusalem 283 (Mamillan 1973). 14 See Heal, Valuing the Future at 114 n 12 (ited in note 4) ( The zero disount rate utilitarian path that approahes the [highest environmentally sustainable utility level] is probably an agreeable plan, in the sense of Hammond and Mirrlees. ). 15 See generally Geir B. Asheim, Ethial Preferenes in the Presene of Resoure Constraints, 23 Nordi J Polit Eon 55 (1996) (analyzing the suitability of different ethial preferenes in reahing ethially aeptable results in a apital aumulation and resoure depletion model). 16 See, for example, Maureen L. Cropper, Sema K. Aydede, and Paul R. Portney, Preferenes for Life Saving Programs: How the Publi Disounts Time and Age, 8 J Risk & Unertainty 243, (1994) (desribing empirial researh suggesting that the disount rates individuals apply derease as a funtion of time delay); George Loewenstein and Drazen Prele, Anomalies in Intertemporal Choie: Evidene and an Interpretation, in George Loewenstein and Jon Elster, eds, Choie Over Time 119, (Russell Sage 1992); George Loewenstein and Rihard H. Thaler, Anomalies: Intertemporal Choie, 3 J Eon Perspetives 181, (1989); Rihard Thaler, Some Empirial Evidene on Dynami Inonsisteny, 8 Eon Letters 201, 205 (1981).

11 2007] Disounting: A Review of the Basi Eonomis 69 some ases substantially more. For projets extending about ten years, the implied disount rates are loser to standard rates perhaps 10 perent. As the horizon extends, the implied disount rates drop to in the region of 5 perent for thirty to fifty years and down to of the order of 2 perent for one hundred years. The empirial evidene also indiates that the disount rate used by individuals, and the way in whih it hanges over time, depends on the magnitude of the hange in inome involved. A. Logarithmi Disounting and the Weber-Fehner Law This empirially identified behavior has been termed hyperboli disounting 17 and is onsistent with a very general set of results from the natural sienes, whih find that human responses to a hange in a stimulus are nonlinear and are inversely proportional to the existing level of the stimulus. As an example, the human response to a hange in the intensity of a sound is inversely proportional to the initial sound level. The louder the sound initially, the less we respond to a given inrease. The same is true of responses to an inrease in light intensity. These are illustrations of the Weber-Fehner law, whih is formalized by the statement that human response to a hange in a stimulus is inversely proportional to the preexisting stimulus. In other words, for a given hange, the response is less, the greater the existing value. In symbols, or, integrating, dr K = ds s r = K log where r is a response, s a stimulus and K a onstant. The empirial results on disounting ited above suggest that something similar is happening in human responses to hanges in the futurity of an event: a given hange in futurity (for example, postponement by one year) leads to a smaller response in terms of the derease in weighting the further the event already is in the future. This is quite natural. Postponement by one year from next year to the s 17 See Loewenstein and Prele, Anomalies in Intertemporal Choie at (ited in note 16) (noting that a hyperboli disount rate mathes empirial findings); George Ainslie and Nik Haslam, Hyperboli Disounting, in Loewenstein and Elster, eds, Choie Over Time 57, (ited in note 16) (arguing that only hyperboli disount rates an explain the pervasiveness of selfdefeating behavior in human beings). See also David Laibson, Life-Cyle Consumption and Hyperboli Disount Funtions, 42 Eur Eon Rev 861, (1998) (using a hyperboli disounting model to explore various types of eonomially irrational human behavior).

12 70 The University of Chiago Law Review [74:59 year after is learly quite a different phenomenon from postponement from fifty to fifty-one years hene. The former obviously represents a major hange; the latter, a small one. If we aept that the human reation to postponement of a payoff or ost by a given period of time is indeed inversely proportional to its initial distane in the future, then this suggests that the Weber-Fehner law an be applied to responses to distane in time as well as to sound and light intensity. The result is that the disount rate is inversely proportional to distane into the future. Put differently, we reat to proportional rather than absolute inreases in the time distane. Denote the disount fator at date t by Δ(t), so that this represents the weight plaed on benefits at date t relative to a weight of unity at time zero. In this ase the disount rate q(t), whih is minus the rate of hange of this weight over time, 18 is q(t) = (dδ(t)/dt) / Δ(t). We an formalize the idea that a given inrease in the number of years into the future has an impat on the weight given to the event, whih is inversely proportional to the initial distane in the future as: 1 dδ K q( t) = = Δ dt t or K t K Δ ( t) = e log = t for K a positive onstant. Suh a formulation has several attrative properties. 19 A disount fator K log t Δ( t) = e has an interesting interpretation: the replaement of t by log(t) implies that we are measuring time differently, by equal proportional inrements rather than by equal absolute inrements. We reat in the same way to a given perentage inrease in the number of years hene of an event, rather than to a given absolute inrease in its number of years hene. We shall all this logarithmi disounting. This approah is quite onsistent with that of aoustis, where, in response to the Weber-Fehner law, sound intensity is measured in deibels, whih re- 18 The inlusion of a minus sign is required beause it is onventional to report the disount rate as a positive number, whereas the rate of hange of the disount fator is normally negative. 19 The disount rate q goes to zero in the limit, the disount fator () t also goes to zero and the integral 1 Δ ( t ) dt = 1 e K log t is finite for K positive and greater than unity. The lower limit of integration in this example is one, not zero, as t K is ill-defined for t = 0 and K > 0. dt = 1 t K dt

13 2007] Disounting: A Review of the Basi Eonomis 71 spond to the logarithm of the energy ontent of the sound waves, and not to energy ontent itself. In general, nononstant disount rates an be interpreted as a nonlinear transformation of the time axis. IV. CHOICE OF A DISCOUNT RATE Eonomists are often asked by international agenies, government offiials, or members of environmental organizations how in pratie one should hoose the disount rate to be used in projet evaluation, or whether one should move away from disounting and use another approah. It should be lear from the disussion above that there are no simple answers to suh questions. However, it is possible to give some general riteria that may be useful. One an start with trying to larify whether disounting utility or onsumption is at stake. Do we need a pure rate of time preferene (utility disount rate) or a onsumption disount rate (soial disount rate)? The former is appropriate when we are dealing with deisions that will affet the entire growth path of an eonomy or a region. To rephrase, utility disounting is appropriate when we are working with a general equilibrium model and general equilibrium onsequenes will follow from the hoies under onsideration. By ontrast, disounting onsumption is appropriate when we are working in a partial equilibrium ontext and the underlying growth path and resoure alloation of the eonomy an be taken as given. In suh situations we are onsidering hanges that will amount to marginal alterations of the initial situation. Some examples an help to larify. Alterations in eonomi poliy designed to redue greenhouse gas emissions in an industrial ountry are probably in the first ategory. They ould be suffiiently far-reahing to alter the general equilibrium of the eonomy. So ould a deision about the onstrution of a large dam in a small developing ountry. But with a purely loal deision, suh as the onservation of a loal fishery or forest, it is learly appropriate to view this in a partial equilibrium framework. We are onsidering marginal alterations about the eonomy s initial pattern of resoure alloation. As noted above, hoosing utility and onsumption disount rates involve different issues, and in general the former is a neessary but not suffiient ondition for the latter: the onsumption disount rate depends on, but is not fully determined by, the utility disount rate. This makes it appropriate to start with a disussion of the hoie of the utility disount rate. This, of ourse, presumes the hoie of a utilitarian framework, or one like it, by whih I mean a Ramsey-esque or overtaking approah. All of these require the seletion of either a single utility disount rate, whih may be zero, or a shedule of timevarying disount rates as implied, for example, by the hoie of a onstant logarithmi disount rate. The only approah that lies outside

14 72 The University of Chiago Law Review [74:59 this general utilitarian-style framework is the Rawlsian, whih I believe ultimately has less than the others to reommend it, partiularly in the intertemporal framework. 20 Martin Weitzman reently onduted a survey of 2,160 professional eonomists, seeking their opinion on the appropriate hoie of disount rate for long-term environmental problems suh as global warming. 21 The modal rate reommended by this group was 2 perent, the median 3 perent, and the mean 4 perent. 22 Unfortunately, it was not lear from the survey whether the hoie referred to a utility or a onsumption disount rate, although it is perhaps reasonable to assume that respondents took the question to refer to utility disount rates as there was no information about growth rates and the other fators neessary to selet a onsumption disount rate. 23 These responses give us a lear indiation that the majority of the eonomis profession is aware of some of the issues disussed above. In general, the hoie of a disount rate for projets with a life of a deade or less would be onsiderably above these rates. Clearly eonomists are seleting lower rates in these responses in reognition of the impat of normal rates over time horizons that are very long by onventional eonomi standards. This does not prove that any partiular answer or approah is the orret one, but it does provide a degree of reassurane that there is a general reognition of these problems. The positions of Ramsey, Harrod, von Weizsäker, and indeed of most eonomi theorists and philosophers who have written on this, is that the utility disount rate should be zero. Suh a position is an ethial rather than an eonomi judgment, and there is no obvious ethial reason why future people should be onsidered less valuable than present people. Whatever one s ultimate aim, the first move has to be a hoie of a long-term utility disount rate. We do not need to know risk premia or the riskiness of any projets that may be undertaken. Nor do we 20 See Heal, Valuing the Future at 58 59, 76 (ited in note 4) (explaining the Rawlsian framework and rejeting it in the intertemporal ontext beause it does not allow sarifie by the present generation beause it is likely to be the poorest generation in a world with eonomi growth); P.S. Dasgupta and G.M. Heal, Eonomi Theory and Exhaustible Resoures 270 (Cambridge 1979) (arguing that we still do not know how individuals will hoose when faed with unertainty). 21 See Martin L. Weitzman, Gamma Disounting, 91 Am Eon Rev 260, (2001). See also generally Martin L. Weitzman, Why the Far-Distant Future Should Be Disounted at Its Lowest Possible Rate, 36 J Envir Eon & Mgmt 201 (1998) (arguing that unertainty about interest rates in the distant future suggests that the lowest possible disount rate should be used in ost-benefit analyses of long-term projets). 22 See Weitzman, Gamma Disounting, 91 Am Eon Rev at 268 (ited in note 21). 23 See id at 266 (reproduing the survey, whih asked: Taking all relevant onsiderations into aount, what real interest rate do you think should be used to disount over time the (expeted) benefits and (expeted) osts of projets being proposed to mitigate the possible effets of global limate hange? ).

15 2007] Disounting: A Review of the Basi Eonomis 73 need to be aware of the nature of distortions in the eonomy and the impat of these on the relationship between market and shadow pries. Nor, finally, do we need to know the rate of return on apital investments or the ost of money. At an equilibrium or an optimum these are a funtion of the utility disount rate, and not the other way around. At issue are the relative weights to be plaed on welfare levels ourring at different dates: should these weights deline with futurity and, if so, aording to what pattern? This judgment about intertemporal distribution is at the heart of the hoie of a utility disount rate. This judgment has to be made first. Given suh a judgment, we an work out the onsumption disount rate from the formulae above. Reall that the onsumption disount rate, or soial rate of disount, is the utility disount rate plus terms that measure the rate of hange of marginal utility. These terms depend on the urvature of the utility funtion its elastiity and on the rates of hange of onsumption: d/dt ds/dt d + η, + η,s. s For a onventional Ramsey-type model without stok externalities (so that η,s = 0 and the third term is zero) the first-order onditions for optimality imply that d/dt d + η, = f () s where f is the marginal produt of apital s. This states that the onsumption disount rate d/dt d + η, is equal to the marginal produt of apital f. Note that here both the left and right hand sides soial disount rate and rate of return are endogenous to the solution of the model and are driven by the utility disount rate and the preferenes and tehnology. This makes an important point, namely that in a general equilibrium framework we should not use the historial return on apital as a utility disount rate (in ontrast with the positions of Weitzman 24 and William Nordhaus 25 ). 24 See Martin L. Weitzman, Just Keep Disounting, But..., in Portney and Weyant, eds, Disounting and Intergenerational Equity 23, 25 (ited in note 10) (reognizing that long-term produtivity has grown over time at a more or less trendless rate and arguing that this will ontinue into the future). 25 See William D. Nordhaus, Disounting and Publi Poliies That Affet the Distant Future, in Portney and Weyant, eds, Disounting and Intergenerational Equity 145, 150 (ited in note 10) (using observed returns on apital when disounting).

16 74 The University of Chiago Law Review [74:59 In a partial equilibrium situation with no stok effets on welfare, however, we an take the soial rate of disount to be the return on apital. With a stok-dependent utility funtion, the soial disount rate is d + η d/dt, + η,s ds/dt s and from the first-order onditions 26 on an optimal path d/dt d + η, = f. k Hene, in a model with both natural and produed apital the soial rate of disount or utility disount rate is ds/dt f + k η,s s. For a separable utility funtion, η,s = 0, we again have equality between the utility disount rate and the return on apital, but not in general. In general, we expet that the elastiity of marginal utility with respet to the stok variable is not zero, as the utility of many goods will be affeted by the state of the environment. If a better environment enhanes the values of other goods environmental stoks and other goods are omplements then η = su < 0,s,s /u so that the utility or soial disount rate is greater or less than the return on apital when the environmental stok is falling or rising respetively. If environmental stoks and other goods are substitutes, these inequalities are reversed. In suh ases, we annot ompute the utility disount rate from the return on apital without information about preferenes, about omplementarities or substitutabilities between environmental stoks and other goods, and about movements in environmental stoks. In summary, we are working either with a general equilibrium or a partial equilibrium framework. In the former ase we need a utility disount rate. This reflets an ethial judgment and is not obtained from eonomi data suh as the return on apital, the risk premium, et. If we are in a partial equilibrium situation, we use the onsumption or soial disount rate. We an use the return on apital as a start- 26 See Heal, Valuing the Future at (ited in note 4).

17 2007] Disounting: A Review of the Basi Eonomis 75 ing point for suh alulations. However, we have to modify this by the term ds/dt η,s, s and the net result ould be zero or negative if the environmental stok is hanging. The result is also time varying. All of these omments on the onsumption disount rate are premised on the assumption of a first-best eonomy, for example an eonomy with no distortions and with a fully optimal alloation of resoures. To understand this point, and to begin to understand what would happen if we were to drop this assumption, reall that there is a standard duality between shadow pries arising from intertemporal optimization problems and ompetitive market pries. The shadow pries ould also emerge as market-learing pries in a ompetitive eonomy with a omplete set of markets. A omplete set of markets in the present ontext means a omplete set of futures markets, from the present to the infinite future. Given suh a set of markets, a ompetitive eonomy would attain an intertemporal equilibrium at whih pries were idential to the shadow pries and would follow the path desribed by the onditions for optimality. An alternative to the assumption of a omplete set of futures markets would be the assumption of fully rational expetations. This amounts to the same thing. The only differene is in the pakaging. Under these admittedly rather strenuous assumptions, we an think of f k as the market return on apital and then talk about how to adjust this to alulate the soial rate of disount to be used in a market eonomy. If the assumption of a fully first-best eonomy is not met, it beomes muh harder to haraterize the fators that determine the onsumption rate of disount. The assumption of a fully first-best eonomy may fail in several ways. One, as noted, is the lak of futures markets, or, equivalently, imperfetions in apital markets. There is an equivalene here beause futures markets are used for moving onsumption and inome over time, and that is also what apital markets do. We an therefore think of futures markets as devies for borrowing and lending. Another possible soure of departure from first-best is the presene of wedges between borrowing and lending rates: this ould be a onsequene of taxes on inome or of apital rationing another aspet of apital market imperfetions. In addition, there are reasons for differenes from first-best that are more speifially related to the environmental nature of the problems under onsideration. For example, the environmental stoks onsidered in the model above are often publi goods forests, stoks of biodiversity, limate regimes and of ourse the investor in these will often have diffiulty

18 76 The University of Chiago Law Review [74:59 in appropriating the returns that the investment generates for soiety as a whole. Likewise, there may be external effets driving a wedge between the private and soial returns to investment. Given differenes between borrowing and lending rates, between private and soial returns, and other deviations from the first-best framework impliitly assumed above, what is the right hoie of a onsumption disount rate? Unfortunately, this is an extremely omplex subjet, and one with no easy generalizations. Ideally we would modify the model to reflet the preise departures from first-best that are relevant in a partiular situation, and then use this revised model to derive onlusions about the onsumption disount rate, using the same mathematial methods as above. As a rude illustration of the model above, suppose that apital market imperfetions made it impossible for agents in this model to save or dissave at more than a speifi rate, say ε > 0. This would then impose a onstraint that the differene between output and onsumption (whih is savings) must be less than this amount, so that in graphial terms the system would be fored to stay within an ε-neighborhood of the urve = f(s). This onstraint would, in turn, have a shadow prie that would interat with other shadow pries and affet the onsumption rate of disount. In pratie there are too many different possible departures from first-best for it to be pratial to model eah partiular ase. Some general points, however, are obvious. We should try to orret returns for the differenes between private and soial osts, and should impute to investments in publi goods the full soial benefits resulting. There is one less-obvious general point that is robust. Above, we noted that in a first-best situation with no stok externalities the onsumption disount rate would equal the return on investment. The reason is that a small inrease in onsumption will lead to a small (equal) derease in investment, and the returns to the two must be equal on the margin on an optimal path. Suppose that there is a wedge between these two returns: whih is the more appropriate as a disount rate, if either? In the presene of taxes on inome, the return on investment is typially greater than the onsumption rate of disount. The former is before tax and the latter after tax. In onsidering the rate at whih the valuation of onsumption hanges over time, we should note that a hange in the time pattern of onsumption will alter the time path of investment and of apital and output, whih will have further impliations for future onsumption. A redution in future onsumption will make soiety immediately worse off, but may make soiety better off by inreasing output and onsumption at a later date. With a first-best alloation of resoures all of this is refleted in shadow or market pries. With a seond-best, it is not. The rate of hange of the marginal utility of onsumption, whih is the

19 2007] Disounting: A Review of the Basi Eonomis 77 onsumption rate of disount and is also the rate of hange of the shadow prie of investment, does not reflet hanges in the level and valuation of output if there is a differene between the onsumption disount rate and the return on investment. Some ommentators have suggested that the appropriate response is to use a weighted average of the rate of hange of the marginal utility of onsumption and the return on apital as a onsumption disount rate. 27 A better approah is to assess the impat of onsumption hanges on the level of investment and to value these hanges in investment by the shadow prie of apital. This shadow prie of apital at date t reflets the full ontribution that extra apital available at date t makes to onsumption from t onwards. 28 We then value the sequene of onsumption and investment hanges resulting from a hange in poliy in terms of onsumption (the shadow prie of apital makes this onversion for hanges in apital) and disount them to the present at the onsumption rate of disount. We are using the onsumption rate of disount as a disount rate, but the return on investment is being used in alulating the shadow pries of apital. Robert Lind provides an exellent introdutory survey of these issues, 29 and David Bradford s paper is a readable and authoritative original soure See, for example, Robert H. Haveman, The Opportunity Cost of Displaed Private Spending and the Soial Disount Rate, 5 Water Resoures Rsrh 947, (1969). 28 See Kenneth J. Arrow, The Rate of Disount on Publi Investments with Imperfet Capital Markets, in Robert C. Lind, et al, Disounting for Time and Risk in Energy Poliy 115, 118 (Resoures for the Future 1982) (outlining how to mathematially predit onsumer behavior for disounting purposes); Robert C. Lind, A Primer on the Major Issues Relating to the Disount Rate for Evaluating National Energy Options, in Robert C. Lind, et al, Disounting for Time and Risk in Energy Poliy 21, 23 (using the onept of the shadow prie of private apital to aount for the opportunity ost of finaning publi investments); David F. Bradford, Constraints on Government Investment Opportunities and the Choie of Disount Rate, 65 Am Eon Rev 887, 889 (1975) (explaining onsumption patterns in relation to government investments); Kenneth J. Arrow and Mordeai Kurz, Publi Investment, the Rate of Return and Optimal Fisal Poliy (Resoures for the Future 1970) (disussing the hypothesis that onsumption is a funtion of wealth and the interest rate); Kenneth J. Arrow, Disounting and Publi Investment Criteria, in Allen V. Kneese and Stephen C. Smith, eds, Water Researh: Eonomi Analysis, Water Management, Evaluation Problems, Water Realloation, Politial and Administrative Problems, Hydrology and Engineering, Researh Programs and Needs 13, (Resoures for the Future 1966) (same). 29 See generally Lind, A Primer on the Major Issues (ited in note 28). 30 See generally Bradford, 65 Am Eon Rev 887 (ited in note 28).