Partha Dasgupta1 Faculty of Economics, University of Cambridge, Cambridge CB3 9DD, United Kingdom
T
he units that are subject to selection pressure in evolutionary biology are ‘‘strategies,’’ which are conditional actions (for example, do P if X occurs; otherwise do Q). In contrast, the units in economics (they are called ‘‘people’’) select strategies from available menus so as to further their projects and purposes. If evolutionary dynamics involves mindless competition among strategies, economic dynamics involves the ways in which people revise their beliefs (and so, their strategies) in light of what they know and what they observe. However, even though the two are different, a remarkable body of work that began with a publication by John Nash (1) has shown that evolutionary and economic dynamics are embedded in a common analytical framework, namely, multiagent decision theory, commonly known as the theory of games (2, 3). A necessary preliminary in any game theoretic exercise is the analysis of a decision unit (DU) that inhabits an environment in which there are no other DUs. The problem before the analyst is to predict what the DU would choose under a given set of circumstances. The common tool to use there is optimization theory (4, 5). A classic problem is to analyze how the DU trades off present against future consumptions, which is the concern of Arrow and Levin (6) in this issue of PNAS. For example, squirrels collect food for winter. Because that requires energy they face tradeoffs between current and future consumption. The question arises: what collection size would maximize inclusive fitness? Similarly, people save a portion of their wealth, not only for their own future consumption but also for their offspring and, by recursion, the offspring of their offspring, the offspring of their offspring’s offspring, and so on, down the generations. If the DU’s objective is to maximize the sum of the expected present values of his dynasty’s utilities and if a person’s utility is an increasing function of his consumption level, how much of his wealth should the DU consume? We assume that the DU has inherited a certain amount of wealth from his parents. Consider first the case where he faces no future uncertainty. Even on intuitive grounds it is clear that the DU’s optimal consumption-to-wealth ratio would depend on four factors: www.pnas.org兾cgi兾doi兾10.1073兾pnas.0906908106
(i) the numbers of his descendents, (ii) the extent to which people prefer early consumption to delayed consumption because of an impatience to consume, (iii) the functional dependence of utility on consumption, and (iv) the rate of return on saving (e.g., market interest rates). The precise way in which those four factors play their part has been known for a long time (7–10). From that work it can be shown, for example, that if the rate of return on saving was suddenly to increase (e.g., because of a technological innovation), or if the rate of growth in the size of his dynasty was suddenly to increase (e.g., owing to an
Risk aversion alone does not lead to the Precautionary Principle. innovation in health care that lowered the infant mortality rate), the DU would reduce his current consumption (i.e., increase his current saving). The reason he would respond in that manner is that an increase in the rate of return would make a further bit of saving yield more consumption in the future than it would if there had been no increase; likewise, an increase in the rate of growth in the dynasty’s size would reduce the consumptions of each future member unless current saving was raised. But what if, say, the rate of return on saving is uncertain? Should that uncertainty be a reason for saving more, as a loose use of the Precautionary Principle might suggest? Economists have studied that question, too (11–15). Consider an environment where the rate of return is known to be independently and identically distributed over time. Assume that the DU is risk-averse, implying that the utility of additional consumption declines with increasing consumption. It can be shown that the response even such a DU would make to an increase in uncertainty in the rate of return on saving is ambiguous. Whether the increase would lead him to raise his saving-to-wealth ratio depends on the rate at which the utility of additional consumption declines to zero. It is only when the utility of additional consumption declines relatively fast to zero that an increase in uncertainty would lead
the DU to raise his saving rate (i.e., decrease his consumption). So the analysis tells us that risk aversion alone does not lead to the Precautionary Principle; something stronger is required. We may call that something, ‘‘prudence.’’ Economists have provided a precise definition of prudence (13). Imagine now that the rate of return on saving is independently distributed not only over time, but also across members of each cohort of the DU’s dynasty. Then members of no given cohort will inherit the same wealth. Imagine next that every one of the DU’s descendents will face the same probability distribution. The Central Limit Theorem then says that the asymptotic distribution of wealth among the DU’s descendents is log-normal. A parallel problem is to analyze how the DU trades off present against future consumptions if the numbers of offspring he and his descendents will have are exogenous random variables; where by ‘‘exogenous’’ I mean that the numbers, although uncertain, will be unaffected by his descendents’ actions. Notice that whatever consumption level the DU chooses for himself, his offspring’s consumption will be low if he happens to be fecund, high if he is not, other things remaining the same. How much of his wealth should he consume? In their interesting article, Arrow and Levin (6) study that problem. They assume that the rate of return on saving (the parameter ␥ ⫺ 1 in their notation) is known, but that every member of the DU’s dynasty will produce an exogenous but random number of offspring (the parameter n in their notation). The random variables are assumed to be identically and independently distributed across cohorts and across members of every cohort. It should be noted that although ␥ and n have similar roles in the two problems, the roles are not exactly the same. Arrow and Levin deploy a neat mathematical trick to show that, nevertheless, the effect of uncertainty in n on the optimal saving rule for the DU is similar to the case where the numbers of offspring are known for sure but the Author contributions: P.D. wrote the paper. The author declares no conflict of interest. See companion paper on page 13702. 1E-mail:
[email protected].
PNAS 兩 August 18, 2009 兩 vol. 106 兩 no. 33 兩 13643–13644
COMMENTARY
Saving for an uncertain future
rate of return on saving is random. They also show that by the Central Limit Theorem the asymptotic distribution of wealth among the DU’s descendents is log-normal. Extensions suggest themselves. Because the DU and his descendents would be genetically related, the distribution of n should be expected to be correlated across generations. We should imagine, too, that the DU and his descendents are able to influence the distribution of n, say by targeting
their consumption levels appropriately. What is the DU’s optimal strategy in that situation? Such an extension to the model would inform population policies. The models I am reporting here suffer from a general weakness. The distribution of wealth in those countries where data are not unreliable would appear to be ‘‘Pareto;’’ it is not lognormal, at least not in the upper tail. Being a power function, the Pareto distribution has a thick upper tail, unlike the log-normal distribution, which is
thin-tailed. To be sure, economists have constructed models in which the asymptotic distribution of wealth is Pareto, but each cohort’s size in those models has been taken to be deterministic and several features of the models are somewhat ad hoc. Despite years of work we still have no deep understanding of the dynamics that shape the distribution of wealth. Arrow and Levin (6) offer an interesting new step toward a better understanding of one of the most perplexing problems in the social sciences.
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6. Arrow KJ, Levin SA (2009) Intergenerational resource transfers with random offspring numbers. Proc Natl Acad Sci USA 106:13702–13706. 7. Ramsey FP (1928) A mathematical theory of saving. Econ J 38:543–549. 8. Hotelling H (1931) The economics of exhaustible resources. J Polit Econ 39:137–175. 9. Mirrlees JA (1967) Optimum growth when the technology is changing. Rev Econ Stud 34:95–124. 10. Arrow KJ, Kurz M (1970) Public Investment, the Rate of Return and Optimal Fiscal Policy (Johns Hopkins Univ Press, Baltimore).
11. Levhari D, Srinivasan TN (1969) Optimal savings under uncertainty. Rev Econ Stud 36:153–163. 12. Brock WA (1982) Asset prices in a production model. The Economics of Information and Uncertainty, ed McCall J (Univ Chicago Press, Chicago), pp 1– 46. 13. Gollier C (2002) Discounting an uncertain future. J Public Econ 85:149 –166. 14. Dasgupta P (2008) Discounting climate change. J Risk Uncertainty 37:141–169. 15. Arrow KJ (2009) A note on uncertainty and discounting in models of economic growth. J Risk Uncertainty, in press.
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Dasgupta
