Abstract
In this paper I offer a fairly complete account of the idea of social discount rates as applied to public policy analysis. I show that those rates are neither ethical primitives nor observables as market rates of return on investment, but that they ought instead to be derived from economic forecasts and society's conception of distributive justice concerning the allocation of goods and services across personal identities, time, and events. However, I also show that if future uncertainties are large, the formulation of intergenerational well-being we economists have grown used to could lead to ethical paradoxes even if the uncertainties are thin-tailed. Various modelling avenues that offer a way out of the dilemma are discussed. None is entirely satisfactory.
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Notes
For the analysis involving multiple consumption goods, see Sterner and Persson (2008).
This rules out the influence on an individual's felicity of habitual consumption or the average consumption of the person's peer group. The implications of habitual consumption on social rates of discount have been studied by Ryder and Heal (1973); the influence of peer group by Layard (1980, 2005) and Arrow and Dasgupta (2007), among others.
Expression (1) has the structure of “utilitarianism”, though not necessarily its classical interpretation (see below). Some ethicists have proposed an ethical theory they call “prioritarianism”, which says that an increase in the well-being of a rich person (i.e., someone who enjoys a high consumption level) should be assigned less social value than the same increase in the well-being of a poor person (someone whose consumption level is low). I have not understood why such an ad hoc ethical principle should be awarded a name. I would have thought the utilitarian who is averse to inequality in consumption has it right: he assigns a lower social value to an increase in the consumption level of a rich person than to the same increase in the consumption level of a poor person.
We write U′(C) = dU(C)/dC and U"(C) = dU′(C)/dC.
If we wished to study the intra-generational distribution of consumption as well, the simplest move would be to disaggregate each generation by imagining that there are N people at each date (i = 1,2,...,N), as in expression (1), and assuming that people have the same felicity function, U. Intergenerational well-being at t = 0 would then be \(W_0 = _{t = 0} \sum ^\infty \left[ {{{V_t } \mathord{\left/ {\vphantom {{V_t } {\left( {1 + \delta } \right)^t }}} \right. \kern-\nulldelimiterspace} {\left( {1 + \delta } \right)^t }}} \right] = _{t = 0} \sum ^\infty \left[ {_i \sum \left\{ {{{U\left( {C_{it} } \right)} \mathord{\left/ {\vphantom {{U\left( {C_{it} } \right)} {\left( {1 + \delta } \right)^t }}} \right. \kern-\nulldelimiterspace} {\left( {1 + \delta } \right)^t }}} \right\}} \right].\)
In work under preparation, I have tried to construct a framework that builds an intergenerational welfare economics admitting the idea of selfhood. The model I have constructed permits someone to discount his own future felicities in any way he likes (that’s the demand of his “self”), but requires of him to give a weight to the lifetime well-being of each of his children that equals the weight he gives to his own lifetime well-being. The model would seem to reconcile the widespread finding from consumption behaviour that people do discount their future felicities at a non-negligible positive rate (see below) and the philosophical injunction that many people would seem to adhere to, namely, that they should not discriminate against their children’s futures (see below).
Roughly, the “independence” assumption amounts to the requirement that the marginal rate of societal indifference between felicities in any two periods is independent of the felicities in all other periods.
So,
$${{C_{t + 1} } \mathord{\left/ {\vphantom {{C_{t + 1} } {C_t }}} \right. \kern-\nulldelimiterspace} {C_t }} = 1 + g\left( {C_t } \right).$$(F1)Formally, \(\eta = {{ - CU\prime \prime \left( C \right)} \mathord{\left/ {\vphantom {{ - CU\prime \prime \left( C \right)} {U\prime \left( C \right) > {\text{ }}0}}} \right. \kern-\nulldelimiterspace} {U\prime \left( C \right) > {\text{ }}0}}\)
Arrow (1965) observed that the simplest U that is bounded at both ends is one for which η is an increasing function of C and is less than 1 at low values of C and greater than 1 at high values of C.
Proof: Because the pair of variations ΔC t +1 and ΔC t leave the numerical value of expression (2) unaltered,
$${{\text{U}}\prime {\left( {{\text{C}}_{{\text{t}}} } \right)}\Delta {\text{C}}_{{\text{t}}} } \mathord{\left/ {\vphantom {{{\text{U}}\prime {\left( {{\text{C}}_{{\text{t}}} } \right)}\Delta {\text{C}}_{{\text{t}}} } {{\left( {{\text{1 + }}\delta } \right)}^{{\text{t}}} }}} \right. \kern-\nulldelimiterspace} {{\left( {{\text{1 + }}\delta } \right)}^{{\text{t}}} } + {{\text{U}}\prime {\left( {{\text{C}}_{{{\text{t}} + {\text{1}}}} } \right)}\Delta {\text{C}}_{{{\text{t}} + {\text{1}}}} } \mathord{\left/ {\vphantom {{{\text{U}}\prime {\left( {{\text{C}}_{{{\text{t}} + {\text{1}}}} } \right)}\Delta {\text{C}}_{{{\text{t}} + {\text{1}}}} } {{\left( {{\text{1 + }}\delta } \right)}^{{{\text{t}} + {\text{1}}}} }}} \right. \kern-\nulldelimiterspace} {{\left( {{\text{1 + }}\delta } \right)}^{{{\text{t}} + {\text{1}}}} } = {\text{0}}{\text{.}}$$(F2)By definition,
$$\rho _t = {{ - \Delta C_{t + 1} } \mathord{\left/ {\vphantom {{ - \Delta C_{t + 1} } {\Delta C_t - 1}}} \right. \kern-\nulldelimiterspace} {\Delta C_t - 1}},$$(F3)where ΔC t +1 and ΔC t satisfy equation (F2). Now use equations (3), (F1)–(F3) to obtain equation (4) in the text.
I have friends in the US who find illustrations involving negative economic growth to be unrealistic. In fact a number of countries in sub-Saharan Africa suffered from negative growth during the period 1970–2000. What discount rates should government project evaluators there have chosen in 1970 if they had an approximately correct forecast of the shape of things to come?
See Dasgupta et al. (1999). This parallels the well-known fact that if the external disbenefits arising from anyone's use of a commodity are large enough, the commodity's shadow price will be negative even when its market price is positive.
I am grateful to William Cline for correspondence on this way of studying how η should be chosen.
Quite obviously, I am making outrageous assumptions regarding aggregation of capital. In this I am no different from contemporary growth economists.
Proof: If ρ t is less than r, society would be advised to save a bit more at t. But to save a bit more at t is to consume a bit less at t, and this tilts consumption more toward the remaining future, which in turn raises ρ t . Alternatively, if ρ t exceeds r, society would be well advised to save a bit less at t. But to save a bit less at t is to consume a bit more at t, and that tilts consumption more toward t, which in turn lowers ρ t . It follows that along the optimum C t , ρ t = r.
This is the same as approximation (4a), with r = ρ t .
In Section 3.4 I suggest that in a deterministic world δ should be set equal to zero.
The rigorous argument would have us check that the saving rate in equation (8) satisfies the transversality condition, namely, that the present discounted value of wealth (in well-being units) tends to zero as t tends to infinity. Readers can check that it does.
Proof: Re-write equation (5) as
$$K_{t + 1} - K_t = rK_t - \left( {1 + r} \right)C_t ,$$which says that a consumption level of C t at the beginning of period t is equivalent to the consumption level (1+r)C t at the end of that period. So saving out of output at the end of t is (rK t -(1+r)C t ). Therefore the ratio of saving to output is (rK t -(1+r)C t )/rK t , which, as is easily confirmed, equals the normalized saving-wealth ratio.
This result is very old. It dates back to Ramsey (1928). In defense of his choice of η = 1,Stern (2008) complains that the 97% saving rate I have just obtained is a feature of a very artificial model. Of course it is. But “non”-artificial models, such as those Stern used in his computer runs, don’t reveal which parameter is doing what work in generating his findings. How is one to test the robustness of ethical assumptions if not by putting them to work in stark, artificial models?
Proof: a marginal additional unit of capital at t = 0 yields a small change in consumption, ΔC t , equal to (1-s)(s(1+r))t. At the consumption discount rate ρ, the present value of that small change, from 0 to ∞, is the expression for P k . (Note that, because s > (1+r)-1, the present value exists.) Equation (10) is due to Marglin (1963).
Ramsey (1928: 261) famously wrote that to discount future well-being is “ethically indefensible and arises merely from the weakness of the imagination”. That is, of course, not an argument; merely an expression of one's beliefs. Broome (1992) contains a summary of the arguments that support Ramsey's position.
Possible extinction of the human race offers a reason for δ > 0, but that is a different reason for positive time discounting. We discuss that in Section 4. We should also bear in mind that infinite-horizon deterministic models are mathematical artifacts: we know Humanity will not survive forever.
Barrett (2003) contains an interesting discussion of those obligations.
Dasgupta and Maskin (2005) have offered an explanation for hyperbolic discounting (preference reversal, more generally), among starlings and pigeons, that is based on selection pressure over evolutionary time. The authors assume that the decision maker has to choose between two options: (i) a reward, V (> 0), that will appear at an uncertain date (the expected date being T), and (ii) a reward, V* (> 0), that too will appear at an uncertain date (the expected date being T*). Assuming V* > V and T* > T, the authors show that, under quite general circumstances concerning the distributions of the uncertain arrival times, a risk neutral decision maker would display preference reversal (from V to V*) if neither reward appeared for a while. In the present paper, I am studying social ethics, not private preferences. The viewpoint I am adopting here is that individual behaviour based on hyperbolic discounting is a constraint the social evaluator must take into account when he evaluates public policies, but that the evaluative criterion for social choice should be intergenerational well-being (expression (2)).
The subsequent, asset-pricing literature (e.g., Brock 1982) has explored models that are more general than the one studied by Levhari and Srinivasan (1969). I use the Levhari-Srinivasan formulation to illustrate my points because of its simplicity and because its findings are directly comparable to those discussed in textbooks on asset pricing (e.g., Cochrane 2005), where asset prices are taken to be exogenous stochastic variables.
It is easy to show that,
$$1 + \overline r = \exp \left( {\mu {{ + \sigma ^2 } \mathord{\left/ {\vphantom {{ + \sigma ^2 } 2}} \right. \kern-\nulldelimiterspace} 2}} \right),$$(F4)$$and\quad \operatorname{var} \left( {1 + \tilde r} \right) = \operatorname{var} \left( {\tilde r} \right) = \left( {\exp \left( {\sigma ^2 } \right) - 1} \right)\exp \left( {2\mu + \sigma ^2 } \right).$$(F5)Notice that s** = s* (equation (8)) if σ = 0.
That is, \(\tilde s** = {{\left[ {s** - \left( {1 + \overline r } \right)^{ - 1} } \right]} \mathord{\left/ {\vphantom {{\left[ {s** - \left( {1 + \overline r } \right)^{ - 1} } \right]} {\left[ {1 - \left( {1 + \overline r } \right)^{ - 1} } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {1 - \left( {1 + \overline r } \right)^{ - 1} } \right]}}\). See footnote 24.
Alternatively, we could call it the “certainty-equivalent rate”.
Approximation (12a) summarized a related finding that the optimum saving rate increases with increasing risk.
See Lenton et al. (2007).
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The ideas I apply here were presented in my Plenary Lecture to the World Congress of Environmental and Resource Economists, held in Monterey, California, June 2002, and were explored in Dasgupta (2001: Ch. 11). For discussions and correspondence over the years, I am very grateful to Kenneth Arrow, Geir Asheim, and Karl-Göran Mäler. While revising the paper I have benefited greatly from the comments of William Cline, William Nordhaus, W. Kip Viscusi, and Martin Weitzman.
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Dasgupta, P. Discounting climate change. J Risk Uncertain 37, 141–169 (2008). https://doi.org/10.1007/s11166-008-9049-6
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DOI: https://doi.org/10.1007/s11166-008-9049-6
Keywords
- Utilitarianism
- Prioritarianism
- Intergenerational well-being
- Social discount rates
- Uncertainty
- Inequality aversion
- Risk aversion
- Rate of time preference
- Hyperbolic discounting
- Rate of return on investment
- Precautionary principle
- Elasticity of marginal felicity
- Risk-free discount rates
- Thin-tailed distributions