Reassessing Edgeworth’s conjecture when population dynamics is stochastic
Introduction
Different ways of defining the social welfare function have been proposed in the economic literature. Probably the most diffused approaches (at least in macroeconomic theory) are average and total (or classical) utilitarianism, respectively based on the so-called Millian and Benthamite criterion. The former says that social welfare coincides with per-capita utility while the latter that social welfare is the sum of individual utility across the population (per-capita utility multiplied by the population size, if agents are homogeneous). Several studies try to figure out how the way we define social welfare is going to affect the final outcome of the model, and in particular which criterion leads to faster economic and demographic growth. The issue is particularly popular in the optimal population size1 literature, which wonders what is the optimal, or the most advantageous, number of lives in a population under given circumstances; the typical view on such a question is that the optimal population has to ensure the largest social welfare, and thus how social welfare is defined has clear implications on the possible answer.
Edgeworth (1925) is the first to conjecture that the Benthamite criterion leads to a larger population size and a lower economic performance. Following studies (as Nerlove et al., 1982, Nerlove et al., 1985) support his intuition: in a purely static context, the Benthamite principle implies larger population size and lower standards of living. The argument is simple: total utilitarianism allows for a substitutability between number of people and their individual wellbeing; since marginal utility is decreasing, to a large extent it is convenient to grow a larger population even if this may subtract resources from individual consumption. More recent analysis shows that in a dynamic framework of endogenous growth, the result is not so obvious: some works (Palivos and Yip, 1993, Boucekkine and Fabbri, 2013) show that the Benthamite criterion leads to smaller population size and higher economic growth, while others (Razin and Yuen, 1995) demonstrate exactly the opposite.2 Palivos and Yip (1993) analyze a canonical AK-type growth model where agents’ utility depends both on consumption and fertility, and capital dilutes linearly with population growth. Boucekkine and Fabbri (2013) consider a similar model where fertility does not affect utility but is related to capital accumulation by a nonlinear dilution function. Razin and Yuen (1995) deal with a different framework where fertility affects (linearly) human capital accumulation and (nonlinearly) population growth, in order to stress the role of the quality-quantity trade-off of children. In Palivos and Yip (1993), the direct effect of population growth on welfare is twofold since individuals gain utility from the number of children they have (i.e., fertility is an argument of the utility function) and this determines the size of the household (i.e., population size multiplies utility); in both Razin and Yuen (1995) and Boucekkine and Fabbri (2013), population growth has a direct effect on welfare only through the latter channel.
In this paper we try to shed some light on the utilitarian trade-off in dynamic models of endogenous growth, and in particular we aim at reassessing Edgeworth’s conjecture by extending the analysis to a stochastic context, since uncertainty about future population is an important aspect that has received only little attention so far. Demographic shocks, which may represent population booms, wars, catastrophic events, changes in immigration, family or health policies, are thought to have serious effects of macroeconomic variables, especially on growth rates (Robertson, 2002). Even the United Nations (UN), which regularly publish the “World population prospects” (WPP), probably the projections most widely used by international organizations, governments and researchers for planning and monitoring activities, recognize the importance of demographic uncertainty for future world development. Indeed, since the first publication of the WPP three different (deterministic) scenario projections (“low”, “medium” and “high”) have been presented in order to emphasize the potential implications of uncertainty on world population; as a recognition of the dramatically increasing role of uncertainty in the current world of fixed resources with rising and aging population, in its latest versions the UN have recently introduced a formal stochastic component3 in their predictions (UN, 2013). Despite this compelling need to take into account demographic uncertainty, apart from some sporadic attempt,4 the economic literature has remained silent on the topic, and no study has tried to relate the issue to the optimal population size problem.
To the best of our knowledge, all the existing works, both in a static and dynamic framework, focus on the deterministic version of the problem, and uncertainty about population change is not taken into account. In this paper we try to fill this gap by adopting a simple approach allowing for analytical results. Specifically, we assume, as standard in growth theory, that the instantaneous utility function depends only on consumption (as in Razin and Yuen, 1995; and Boucekkine and Fabbri, 2013) and we consider two different specifications of the model. In Section 2, we first analyze a simple framework where population dynamics is completely exogenous (as in Strulik, 2005, Bucci, 2008; and Marsiglio and La Torre, 2012a) in order to obtain an obvious benchmark for the following analysis; this assumption thus allows us to evaluate only the implications of different utilitarian approaches on economic performance. We show that which utilitarian criterion leads to faster growth depends on two factors: the magnitude of the inverse of the intertemporal elasticity of substitution and the features of the (completely exogenous) random process driving population dynamics. The simplification is removed in Section 3, where we endogenize the trend of population growth by assuming that capital accumulation is nonlinearly affected by demographic growth (as in Marsiglio, 2011, Boikos et al., 2013; and Boucekkine and Fabbri, 2013). Thus, since population change is endogenously determined, we can evaluate also the effects of different utilitarian approaches on population size. We show that, also in this case, the magnitude of the intertemporal elasticity of substitution and the characteristics of the random process driving population dynamics affect the utilitarian trade-off; however, a third factor, that is the magnitude of the (linear) dilution coefficient, determines the overall result. We show that Edgeworth’s conjecture may hold if certain parameter conditions are met, and through numerical simulations we assess how large is the range of parameter values allowing for such an outcome to occur. Section 4 as usual contains concluding remarks and proposes directions for future research.
Section snippets
Exogenous population growth
The model is a standard Ramsey-type Ramsey (1928) model of optimal growth where the social planner seeks to maximize the welfare of the society subject to the economic and demographic constraints, choosing the consumption level, , of the representative agent. The welfare is the infinite discounted sum of the product of the instantaneous utility function (assumed to be iso-elastic, , where is the inverse of the intertemporal elasticity of substitution) and the
Endogenous population growth
Edgeworth’s (1925) hypothesis concerns the effect of alternative utilitarian criteria on both the economic and population growth rates. Since we considered population dynamics to be completely exogenous in the previous section, we could just assess the former portion of his conjecture. In order to assess also the latter part, we need to endegenize the trend of population growth. In doing so and in order to maintain the same utility function traditionally considered in growth theory (depending
Conclusion
This paper reassesses Edgeworth’s (1925) conjecture, suggesting that the Benthamite criterion leads to a larger population size and a lower economic performance than the Millian criterion. In a static setup, it has been widely confirmed that Edgeworth’s hypothesis is verified (Nerlove et al., 1982, Nerlove et al., 1985). In a dynamic framework of endogenous growth, the result is unclear: some works (Palivos and Yip, 1993, Boucekkine and Fabbri, 2013) show this is no longer the case, while
Acknowledgements
I am grateful to Pavel Brunovsky, Alberto Bucci, Davide La Torre and Fabio Privileggi for insightful discussions. I also thank participants at workshops and conferences held in Marseille, Strasbourg, Alessandria and Bratislava in 2012 for their helpful comments and suggestions. I wish also to thank the Editor, Ping Wang, and two anonymous referees for their constructive comments on earlier drafts of the paper. All remaining errors and omissions are my own sole responsibility.
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