WELFARE COMPARISONS WHEN
POPULATIONS DIFFER IN SIZE
Thèse présentée
à la Faculté des études supérieures de l’Université Laval dans le cadre du programme de doctorat en Économique pour l’obtention du grade de Philosophiae Doctor (Ph.D.)
FACULTÉ DES SCIENCES SOCIALES UNIVERSITÉ LAVAL
QUÉBEC
2011
c
L’objectif principal de cette thèse est de faire des comparaisons sociales impli-quant des populations de taille différente. Ceci est pertinent pour deux raisons. En premier lieu, l’évaluation des politiques publiques implique souvent des comparaisons de situations où le nombre d’individus diffère d’une situation à une autre. En second lieu, les fondements théoriques de l’évaluation sociale dans le cadre des populations de taille variable fournissent peu d’indications sur la façon dont les changements dans la taille et dans la distribution des populations peuvent être socialement évalués.
Après avoir fait une revue de la littérature sur les problèmes de populations et particulièrement sur les questions liées à la taille et au bien-être des populations, et après avoir examiné comment évaluer socialement des populations de taille dif-férente, nous utilisons l’utilitarisme généralisé de niveau critique comme fonction d’évaluation sociale. Cette fonction a des fondements éthiques satisfaisants et est appropriée pour l’évaluation sociale des populations de taille variable. Mais elle re-quiert l’usage d’une valeur duniveau critique, un paramètre clé dans cette approche. Nous proposons un cadre de dominance basé sur l’utilitarisme généralisé de niveau critique. Nous montrons comment cette dominance est reliée à la dominance stochastique enpauvreté. Ceci est présenté dans le premier essai.
Dans le deuxième essai, nous développons un cadre théorique, normatif et statis-tique pour estimer des bornes inférieures et supérieures robustes des niveaux cri-tiques sur lesquelles les distributions des populations peuvent être ordonnées. Nous illustrons les résultats théoriques en utilisant des données réelles du Canada tirées d’enquêtes auprès des ménages.
Nous étendons les applications à une échelle nationale, régionale et mondiale. Les résultats indiquent de manière convaincante que la valeur de l’humanité peut être considérée comme ayant globalement augmenté entre 1990 et 2005, mais pas pour beaucoup de régions du monde. Ceci fait l’objet du troisième essai.
The main objective of this thesis is to make welfare comparisons involving dif-ferent population sizes. This is relevant for two reasons. First, the evaluation of public policies often implies comparisons of situations where the number of individ-uals differs from one situation to another. Second, the theoretical foundations of social evaluation provide little measurement guidance on how changes in population size and population distribution can be socially evaluated.
After a literature review on population problems and particularly questions re-lated to population sizes and social well-being, and after discussing how variable populations are socially evaluated, we use critical-level generalized utilitarianism as a social evaluation function. This function exhibits ethically desirable founda-tions and is shown to be more convenient for comparing well-being between variable populations. But it requires a value of the critical level, a key parameter in this approach.
We propose a dominance framework based on critical-level generalized utilitar-ianism. We show how this dominance is related to stochastic poverty dominance. This is presented in the first essay. In the second essay, we develop a theoretical, nor-mative and statistical framework to estimate some robust lower and upper bounds of critical levels within which population distributions can be ordered. We illustrate our theoretical results by using real data from Canada’s household surveys. We extend the applications to national, regional and world scales. The results indicate that the value of humanity can be persuasively shown to have increased globally between 1990 and 2005, but not so for many of the world’s regions. This is done in the third essay.
Je dois l’accomplissement de ce travail au professeur Jean-Yves Duclos et à plusieurs personnes que je voudrais remercier sans avoir la prétention de réussir à être exhaustive.
Tout d’abord, je remercie sincèrement mon directeur de thèse le professeur Jean-Yves Duclos pour sa disponibilité et tout ce qu’il m’a patiemment et généreusement enseigné. Je ne saurais oublier son soutien sans faille et l’appui considérable qu’il m’a accordés dans la rédaction de cette thèse, notamment pendant les moments difficiles. Sans son précieux soutien, cette thèse ne serait pas arrivée à ce stade. Je suis également reconnaissante envers le professeur Jean-Yves Duclos pour m’avoir préparée et soutenue pour la recherche de l’emploi.
Je remercie également mon co-directeur le professeur John Cockburn pour sa précieuse contribution à la réalisation de la thèse. Les discussions que j’ai eues avec lui m’ont apporté une aide précieuse dans mes travaux de recherche. Ses conseils et ses suggestions m’ont été d’une grande utilité pour la rédaction de ma thèse.
Je tiens à remercier la professeure Mme Lucie Samson, la directrice du départe-ment, le professeur Sylvain Dessy, le directeur de programme de doctorat, le pro-fesseur Patrick Gonzalez et le propro-fesseur Kevin Moran pour leurs conseils.
Je remercie le professeur Guy Lacroix, le directeur du Centre Interuniversitaire sur le Risque, les Politiques Économiques et l’Emploi (CIRPÉE), pour l’appui dans l’obtention de mon stage à la Banque Mondiale.
Je remercie tous les professeurs du département pour la qualité de l’enseignement que j’ai reçu. Je remercie Abdelkrim Araar et Sami Bibi pour m’avoir soutenue dans l’écriture des codes en Stata. Mes remerciements vont également à Mme Sonia
Moreau, Mme Gaétane Marcoux et à tout le personnel administratif avec qui j’ai échangé durant ces années.
Pendant ma visite à la direction de la recherche en développement (DECRG) de la Banque Mondiale, j’ai eu le privilège de connaître et de sympathiser avec des personnes très accueillantes que je tiens à remercier: Peter Lanjouw, Branko Milanovic, Dominique Van De Walle, Quy-Toan Do et Roy Van Der Weide.
Je voudrais aussi remercier les différents organismes qui m’ont apporté un sup-port financier tout le long de ma formation: le réseau de recherche sur les Politiques Économiques et la Pauvreté (PEP), le Centre Interuniversitaire sur le Risque, les Politiques Économiques et l’Emploi (CIRPÉE) et le Fonds Québécois de Recherche sur la Société et la Culture (FQRSC).
Un Merci tout particulier :
À mon frère Henri pour son soutien inconditionnel dont il a fait preuve pour moi durant mes études doctorales.
À ma mère Blandine Zoungrana, mon père Mathias Zabsonré, mes frères Clé-ment, feu Éric, Frédéric, Jean De Dieu et Patrice, mes sœurs Claire, Clarisse, Denise et Jeannine, pour leurs conseils et aussi leur patience durant ces années passées loin d’eux.
À mon amie Nadège pour son soutien quotidien, moral et inconditionnel et les discussions fructueuses que j’ai eues avec elle.
À mes amis, Firmin Doko, Prosper, Raphaël, Clarence, Fulbert, Rachid, Etienne et Alice pour leurs encouragements qui m’ont été bénéfiques.
À la chorale des étudiants catholiques de l’Université Laval, au père Jean Abud et à Papa Jean Bouda pour leur soutien spirituel.
Et enfin à tous les collègues étudiants du département et particulièrement à Legrand, Firmin Vlavonou, Bouba, Yélé, Komi, Habib et Aboudrahyme pour les rapports entretenus et les discussions enrichissantes pour la réalisation de cette thèse.
A ma mère Blandine, à mon père Mathias, à mes frères Clément, feu Éric, Frédéric, Henri, Jean De Dieu et Patrice et à mes sœurs Claire, Clarisse, Denise et Jeannine.
Résumé ii
Abstract iv
Avant-propos v
Contents viii
List of Figures xi
List of Tables xiii
1 Introduction 1
2 Population problems and social evaluations 4
2.1 Is a bigger society a better one? . . . 4
2.1.1 More people leads to greater human ingenuity and higherper capita income . . . 5
2.1.2 More people, less welfare . . . 9
2.1.3 When more people and more poverty go together . . . 11
2.1.4 More people is likely to bring more social happiness . . . 15
2.2 Social evaluations with variable population sizes . . . 18
2.2.1 Critical-level utilitarianism . . . 19
2.2.2 Optimal population . . . 21
3 Welfare comparisons with different population sizes: a theoretical analysis 25 3.1 Introduction . . . 26
3.2.1 Definition of CLGU . . . 28
3.2.2 Definition of dominance orderings . . . 29
3.3 CLGU and FGT dominance equivalence . . . 32
3.3.1 Poverty dominance . . . 32
3.3.2 CLGU dominance . . . 33
3.4 Critical level and dominance relations . . . 37
3.5 Generalization to a larger set of welfare indices . . . 38
3.6 Conclusion . . . 41 3.7 Appendix . . . 42 3.7.1 Proof of Proposition1 . . . 42 3.7.2 Proof of Proposition2 . . . 44 3.7.3 Proof of Corollary1 . . . 46 3.7.4 Proof of footnote2 . . . 47
4 Testing for social orderings when populations differ in size 49 4.1 Introduction . . . 50
4.2 Definitions of dominance relations . . . 53
4.3 Statistical inference . . . 55
4.3.1 Testing dominance . . . 56
4.3.2 Estimating robust ranges of critical levels . . . 61
4.4 A few simulations . . . 69
4.5 Illustration using Canadian data . . . 75
4.6 Conclusion . . . 83
4.7 Appendix . . . 85
4.7.1 Graphical illustrations of higher orders of dominance . . . 85
4.7.2 Proof of Theorems 2and 3 . . . 87
4.7.3 Asymptotic equivalence of statistics . . . 93
5 Has global welfare improved between 1990 and 2005? A critical-level utilitarian approach 102 5.1 Introduction . . . 103
5.2 Related literature . . . 106
5.3 Definitions and dominance criteria . . . 109
5.4 Robust ranges of critical levels . . . 111
5.5.1 Data description . . . 114
5.5.2 Some estimated values of the critical level . . . 119
5.5.3 Comparison between CLGU and per capita approaches . . . . 123
5.6 Conclusion . . . 126
5.7 Appendix . . . 127
5.7.1 Critical level bounds for developing countries . . . 127
5.7.2 Developing countries not included in PovcalNet data . . . 131
5.7.3 High-income countries included in the analysis . . . 132
5.7.4 Comparison of PovcalNet and real data . . . 133
6 Conclusion 134
3.1 Smaller dominates larger when z+ < α . . . . 36
3.2 Larger dominates smaller . . . 36
4.1 Poverty incidence curves adjusted for differences in population sizes . 64 4.2 Poverty incidence curves withα =α1 adjusted for differences in pop-ulation sizes . . . 64
4.3 Poverty incidence curves withα =α1 adjusted for differences in pop-ulation sizes . . . 65
4.4 Population poverty incidence curves and dominance of the larger pop-ulation . . . 71
4.5 Population P2 curves and dominance of the larger population . . . . 71
4.6 Population poverty incidence curves and dominance of the smaller population . . . 72
4.7 Population P2 curves and dominance of the smaller population . . . . 73
4.8 Canadian cumulative distributions . . . 76
4.9 Cumulative distributions of 1976 and 1996 and the critical level . . . 80
4.10 Relation between z+ and α 1 . . . 82
4.11 Ps curves and dominance of the larger population . . . . 85
4.12 Ps curves and dominance of the smaller population (case 1) . . . . . 86
4.13 Ps curves and dominance of the smaller population (case 2) . . . . . 86
5.1 Poverty incidence curves with α=α1 adjusted for differences in pop-ulation sizes . . . 113
5.2 Poverty incidence curves withα =α1 adjusted for differences in pop-ulation sizes . . . 113
5.3 Non-dominance of 2005 over 1990 . . . 117
5.5 World 2005 dominates world 1990 . . . 120 5.6 1990 dominates 2005 for ECA and SSA . . . 123 5.7 The increase in the absolute number of poor leads to more poverty in
4.1 Population sizes and upper bounds of the critical level — large
dom-inates small . . . 73
4.2 Population sizes and lower bounds of the critical level — small dom-inates large . . . 74
4.3 Asymptotic standard errors of the bounds of the critical levels . . . . 75
4.4 First-order dominance tests . . . 77
4.5 Estimates of the upper bound of ranges of critical levels over which the larger population dominates the smaller one . . . 78
4.6 Estimates of the upper bound of the range of critical levels over which the larger population dominates the smaller one . . . 79
4.7 Estimates of lower bound of the critical level . . . 81
5.1 Population size and average income . . . 115
5.2 Estimation of upper bounds: 2005 dominates 1990 . . . 119
5.3 Values of the utilitarian social evaluation index (in billion $) . . . 121
5.4 Estimation of lower bounds: 1990 dominates 2005 . . . 121
Introduction
Does the “value” of a society increase with its population size? How can we answer this in a normatively robust framework? What sort of statistical procedures can be performed to test this? What does the empirical evidence suggest? To address these questions is the main contribution of this thesis.
This thesis consists of three essays, but they are quite close. All these essays deal with social evaluations when populations differ in size. The motivation for such comparisons lies in the fact that the evaluation of public policies often implies situations where the number of individuals differ from one situation to another. This is also expected when comparisons are targeted toward different societies. Finally, comparisons involving different populations sizes are certainly the most generally encountered case in empirical analysis.
We therefore consider populations of different sizes and we are interested in the question of whether a society’s well-being increases with its population size. In Chapter2, we review the literature on this subject. Historically, there have been two opposite views about theideal population size. The first one follows the Malthusian view that small is better because of limited available resources. The second one is based on Bentham’s view and advocates that large is better. Although the recent literature develops some economic models to deal with this question, the results of
studies are sometimes ambivalent. Four different conclusions can emerge from the studies:
(i) more people lead to greater human ingenuity and higher per capita income; (ii) more people are likely to bring more social happiness;
(iii) more people, less welfare;
(iv) more people and more poverty go together.
The first conclusion derives essentially from positive arguments. The three last conclusions are based more on normative arguments. Given that most of these studies use some social evaluation objectives to deal with the question of size and value, we may be interested in knowing if such social evaluations are to be fa-vored, for instance, in case of comparing alternative public policies involving dif-ferent population sizes. Note that social evaluation rankings are easily made when populations have the same size. However, this is not the case when sizes differ. In that situation, the tradition has been to overcome the problem of variable sizes by calling on the replication invariance principle. In this case, social evaluation is made in per capita terms and then sizes do not substantially matter. However, as Blackorby, Bossert, and Donaldson (2005) have argued, population sizes should sometimes matter when comparing aggregate welfare. The thesis develops this idea and follows the critical-level principle.
In the literature,Blackorby and Donaldson (1984) were the first who introduced the notion of the critical-level principle. This principle suggests that further indi-viduals with welfare above a value of the critical level to an existing population can be considered as improving the social welfare of the population. In other words, the critical level reflects how much a life must be minimally worth to contribute positively to society’s welfare.
The critical-level principle can be used to socially evaluate populations of dif-ferent sizes. But this principle requires the choice of the value of the critical level, which remains the main problem of this approach. Until now, few studies have addressed this issue in the literature.
The first essay is presented in Chapter 3. It develops and presents some the-oretical results for stochastic dominance with varying population sizes. The pur-pose is to extend traditional critical-level generalized utilitarianism (CLGU) anal-ysis by considering arbitrary orders of social welfare dominance and ranges of poverty lines and values for the critical level. This essay draws from the proce-dure developed by Duclos and Makdissi (2004). Links between critical levels and orders of dominance are also investigated. We also introduce how to generalize the Blackorby and Donaldson (1984)’scritical-level principle.
The second essay is contained in Chapter 4. This essay performs a statistical inference analysis taking into account changes in population sizes and population distributions. A few simulations are made to show the effect of population size on social evaluation. The essay also derives the asymptotic distributions of some lower and upper bounds of robust ranges of critical levels. An empirical application is made using Canadian Surveys of Consumer Finances (SCF) for 1976 and 1986, and Canadian Surveys of Labour and Income Dynamics (SLID) for 1996 and 2006. Using asymptotic and bootstrap tests, we find that Canada’s welfare has globally improved in the last 35 years despite the substantial increase in population size.
The third essay, presented in Chapter 5, is primarily empirical. It aims at ex-tending the application of the CLGU approach to regional and world scales using data from most countries in the world. The objective is to assess whether the “value of humanity” has increased between 1990 and 2005 despite a substantial increase in world population size. We estimate the bounds of the critical levels for all develop-ing countries, for regions and for the entire world. In addition to results obtained in Chapter 4, such estimations show how the process of demographic transition, in which a large part of humanity has recently engaged, may be assessed through a CLGU approach. We also compare the CLGU approach to the traditionalper capita one.
Population problems and social
evaluations
This chapter reviews the linkages between population size and population well-being. It explores whether societies are better off with smaller populations or bigger populations. This requires using social evaluation functions that can take account differences in population size to assess society’s value as a whole. The choice of these functions is guided by economic objectives and also by some ethical and normative foundations. In such a setting, social evaluations also lead to important implications in terms of public policies.
The first section summarizes the principal conclusions that emerge from the links between a population’s size and its well-being. The second section discusses how changes in population size and population distribution are socially evaluated.
2.1
Is a bigger society a better one?
This question sparks off many reactions according to our understanding of what we call a“better society”. Does it implyhigher per capita income? Ahappier society?
We can focus on two views: those who believe that a larger population is usually better (the positive view) and those who do not (the negative view).
The first view’s argument is generally based on the idea that a large population favors development. This stems from the neoclassical theory of the impact of pop-ulation growth on economic growth. Poppop-ulation growth induces infrastructure and market development, and influences positively technological change and innovation. This in turn raises per capita income and growth, and therefore leads to a higher level of welfare.
The second view emphasizes that population growth leads to more deprivation in common dimensions (standards of living, education, health, etc.). The gist of the second view seems to be guided by an “observable fact”: poor countries are associated with rapid population growth and lower per capita income, while rich countries have lower rates of population growth and higher per capita income. The two views are motivated by both positive and normative arguments.
2.1.1
More people leads to greater human ingenuity and
higher
per capita
income
The literature that supports this positive view relies on the arguments of Boserup (1965), Boserup (1981) and Simon (1981) that a greater population is an important driver of technological change and innovation and encourages or-ganizational and institutional change. A larger population also allows for scale economies in production and consumption. The historical features of Boserup and Simon’s argument are the empirical investigations of Kremer (1993) and Acemoglu, Johnson, and Robinson (2005).
Using a model of population and endogenous technological change, Kremer sup-poses that technology is a pure public good. He also assumes that each individual’s research productivity is independent of population size and that the population growth rate is limited by the state of food production. He suggests a linear relation-ship between population growth and population size. He shows that larger initial
populations without technological contact enjoy faster technological progress and population growth.
Acemoglu, Johnson and Robinson show that the growth of urbanization between 1300 and 1850, with the access to Atlantic Trade and a favourable institutional envi-ronment, are the factors that explain the modern economic development of Western Europe. Using Kremer’s model, Klasen and Nestmann (2006) incorporate popula-tion density as an addipopula-tional positive determinant of technological change. Even if an increase in population leads to a greater number of potential suppliers of new technology, high density generates the linkages. It facilitates communication and exchange and raises the demand and the diffusion of technological innovations, which could in turn increase the quantity of available resources. This suggests why some people think that “human inventiveness and ingenuity are capable of enhanc-ing Earth’s capacity to support the species indefinitely and to support it at a high standard of living” (Dasgupta 2005, p. 416).
Easterly and Levine (1997) also emphasize the importance of population density. They argue that low density in Africa has some negative effects such as greater ethnic divisions. In addition, they conclude that ethnic divisions have something to do with Africa’s poor economic development.
All these models are consistent with endogenous growth models that find that the size of populations and also population density have a positive impact on the growth of per capita income (see, for instance, Grossman 1991, Aghion and Howitt 1992, Aghion and Howitt 1998 and Jones 1999). Some implications of the endogenous growth theory are the external benefits of having more people. More people means a higher level of knowledge, skills and human resources avalaible for future labor shortages. This in turn increases production. This idea may be found in the French philosophersBodin (1576) andQuesnay (1758), who consider the individual as play-ing a central role in the production of wealth. Accordplay-ing to them, there exists a bi-directional relation between population size and standards of living. Individuals represent the labour force and the strength of the army. According to Bodin, a large population can help a country to become rich and powerful:
citizens, for the strength of the commonwealth consists in men. Moreover the greater the multitude of citizens, the greater check there is on factions seditions. For there will be many in an intermediate position between the rich and the poor, the good and the bad, the wise and the foolish. There is nothing more dangerous to the commonwealth than that its subjects should be divided into two factions, with none to mediate between them. This is the normal situation in a small commonwealth of few citizens”. (Six Books of the Commonwealth, Book V, chapter II, translated by Tooley 1955)
In Quesnay’s point of view, it is more the availability of resources that leads to population growth:
“That the sovereign and the nation should never lose sight of the fact that the land is the unique source of wealth, and that it is agriculture which causes wealth to increase. For the growth of wealth ensures the growth of the population”. (General Maxims for the Economic Government of an Agricultural Kingdom, translated by Meek 1963, p. 232)
One advantage of a larger population that Bodin underlines is the sake of defense against foreign attacks.1
This is consistent with the view of those who see in a “better society” a peaceful one. A larger population itself contributes to society’s welfare, in the sense that it defends a nation against its enemies. There is some literature on this issue.
For instance, McNicoll (1984) reviews the positive effects of having more people on military capabilities. First, populous nations, other things equal, tend to have more influence in international decisions and world affairs. Second, as nations be-come alike in technology and in institutional organization, an increase in population is an important source of national power, both military and industrial. Because there is a great difference in technological levels among states in the world today,
1
Although the strength of the commonwealth is not expressed in concrete terms in the quo-tation, it is apparent in Bodin’s definition of the sovereignty of a nation. See for example
the achievement of technological equality is a distant prospect. However, for some developing countries and moreover for developed countries that are characterized by low fertility rates, population can account for a significant power factor. When the population size is very low, the survival of human race can be threatened. Us-ing the example of a nuclear holocaust which causes the problem of endangered species, Hurka (1983) argues that it is desirable in that situation to encourage pop-ulation to increase, even if average well-being decreases as poppop-ulation increases. De La Croix and Dottori (2008) also notice that the benefit of an increase in pop-ulation size was enhanced by the need for a large army. Investigating the Easter Island’s collapse and the quest for greater bargaining power between conflicting groups, the biggest group had the highest probability to win the war.
Nerlove, Razin, and Sadka (1986) (p. 601) go on to quote Edgeworth’s benefit of being larger: “being must be secured before well-being”. Furthermore, they extend Edgeworth’s argument to public goods and claim that a larger population has an advantage in providingpure public goods. The most interesting example is national defense, because theper capitacost of providing that public good falls as the popula-tion becomes larger. More generally, public goods such as infrastructure, electricity network, telecommunications and research are recognized as having cost-reduction benefits when there is more people.
Alesina and Spolaore (2003) enumerate five types of benefits of having a large population: (i) lower per capita costs of public goods and more efficient taxation. The public goods are monetary and financial institutions, the judicial system, in-frastructures for communication, police and crime prevention, public health, and so on. The second type of benefits is: (ii) greater military power and lower per capita defense and military costs; (iii) increase in productivity thanks to substantial skills and large markets (however, country openness to international market can limit this factor); (iv) providing insurance; (v) greater opportunities for income redistribution (pp. 3-4).
A slow population growth also has negative consequences mainly for countries with AN ageing population. Because of large proportions of elderly, pay-as-you-go transfer systems become difficult to sustain. There are also fewer and fewer workers
to finance investment in public goods.2
Kravdal (2010) notes the same problems and remarks that less populated regions may experience difficulties in surviving. In 2005, the European Commission publishes an official document beginning by a rather alarming content:
“Europe is facing today unprecedented demographic change. In 2003, the natural population increase in Europe was just 0.04 per cent per annum (...). The fertility rate everywhere is below the threshold needed to renew the population (around 2.1 children per woman), and has even fallen below 1.5 children per woman in many Member States.”
(Commission of the European Communities 2005, p. 2)3
Michel Rocard, a former French prime minister set the tone two decades ago in the following terms:
“La plupart des états d’Europe Occidentale sont en train de se suicider, de se suicider par la démographie, sans même en avoir conscience” (Michel Rocard, January 20, 1989, “Conférence des Familles”).4
2.1.2
More people, less welfare
Although there can be external benefits and advantages in the provision of public goods as populations become larger, there can also be pernicious effects of being larger. Larger populations are likely to be more heterogeneous than homogeneous. This gives rise to more diverse preferences, cultures, religions and languages within the same population. As individuals differ in many respects, they do not share the same objectives. This can result for example in conflicts and in an inefficient use of public goods. In fact, larger populations can intensify the use of non renewable resources and non-pure public goods, namelycommon-pool resources andclub goods.
2
Unless the country adopts migration policies to adjust labor supply and to correct imbalances, the problem will worsen. In fact, some governments in developed countries now focus on maximizing the benefits of economic migration.
3
See Bloom, Canning, Fink, and Finlay (2010) for the reasons justifying the rapid decline in fertility in Europe. From society’s point of view, parents do not have enough children when the social benefits of having children are higher than the private benefits.
4
In that situation, additional people may increase per capita costs, because resources are subject to the problem of congestion. This problem generates additional costs for society that can have a negative impact on society’s welfare. All these ideas seem to have sustained the philosophical and political debate in the past. Indeed, Aristotle (in Politics, Book IV, chap. IV) defended that a country should be small enough for the citizens to know and listen to each other. The entire territory should be small enough to be surveyed from a hill (Russell 2004). Rousseau (1772) goes beyond Aristotle thought by stating that it is difficult to control great nations:
“Large populations, vast territories! There you have the first and fore-most reason for the misfortunes of mankind, above all the countless calamities that weaken and destroy polite peoples. Almost all small states, republics and monarchies alike, prosper, simply because they are small, because all their citizens know each other and keep an eye on each other”. (p. 25)
Baron de Montesquieu (1748) had the same view and believes that large countries are necessarily diverse and thus require strong governments, resulting in monarchy or even despotism. He also points out the problem related to public goods congestion occuring in a large society:
“In an extensive republic the public good is sacrificed to a thousand pri-vate views; it is subordinate to exceptions, and depends on accidents. In a small one, the interest of the public is more obvious, better understood, and more within the reach of every citizen; abuses have less extent, and of course are less protected”. (The Spirit of Laws, Book VIII, Chap.16, translated by Thomas Nugent 1752)
According to these philosophers, small nations can easily supervise individuals and can control their activities. Besides, they think that the less populated states would seem to maintain social cohesion between members and thus can consolidate peace. Rousseau and Montesquieu would then associate the well-being of a society with social cohesion. Montesquieu’s view is often used by anti-federalists to forcefully contest the federalism system. As they often hammer home, a republic of diverse interests could not survive because it is affected by factions and fragile stability.
2.1.3
When more people and more poverty go together
A large society can also involve a risk of excessive use of environmental resources with possible deleterious effects on individual well-being. Much literature supporting this idea exists. The main argument is based on the effects of demographic behaviour onper capita income. The most basic description of such effects was first proposed byMalthus (1798) in two scenarios. The first reveals a positive effect of the standard of living on population growth, given that population size is small. As the standard of living is high, population will grow as a result of reproduction and population becomes larger. There is here a consistent view with that of Quesnay. The second describes the negative feedback due to the fact that the rise in population size exerts pressure on resources and therefore leads to a low standard of living. This forces population to be reduced as a consequence of low fertility and high mortality.
Starting from Malthus, fertility behaviour has been analyzed in different eco-nomic models, for instance Becker (1960) and Becker and Lewis (1973). They as-sume the fertility behavior of families to be determined by economic variables (num-ber and quality of children, consumption and other commodities) which are, in turn, influenced by fertility behavior. Parents face a quality and quantity trade-off in their decision on children. Economic models using these foundations have tried to explain the negative relation between income and population growth (see Barro and Becker 1989). Moreover, Galor and Weil (1999) extend these models to characterize the economic transition experienced by developed countries: the tran-sition from high population growth and low per capita income to low population growth and high per capita income.
Birdsall (1988) reviews the effects of the high fertility underlying rapid popula-tion growth in developing countries on their economic growth. Although she finds little direct connection between fertility rates and incomesper capita, she highlights the importance of the determinants of individual fertility decisions on the process of economic development. The determinants are essentially education, availability and distribution of health care services, work, access to family planning and prevailing religious views.5
Empirical work on the subject concludes that the social costs of
5
high fertility exceed private costs, as experienced by societal and parental difficul-ties in educating children and investing in their health (e.g. in some parts of Asia and Sub-Saharan Africa, where population growth rates remain high andper capita income is low).
In the fertility model of Barro and Becker, De La Croix and Doepke (2003) in-troduce a differential fertility effect between the rich and the poor for the long run relationship between inequality and growth. Using data on 68 countries, they find that higher inequality between rich and poor slows down economic growth and de-velopment because poor people have a higher fertility rate than the rich. They invest less in education than the rich and their population share increases with fertility and this lowers the average level of human capital in the society. Finally, stronger in-equality delays the fertility transition and reduces income per capita growth. These results suggest why rapid population growth rates may exacerbate the development problems of poor countries.
This has been reinforced in the last decade by the growing awareness of the problems of environmental degradation, particularly in poor countries. Along this line, Ehrlich and Ehrlich (1990) generalize Malthus’ “pessimist” idea on population growth and present population growth as a serious jeopardy that humans might face. They argue that overpopulation creates environmental damages and threatens world’s survival. In their book,Ehrlich and Ehrlich (1990) enumerate a set of envi-ronmental problems. The main problems are the rapid depletion of natural resources such as water and land, soil erosion and desertification, ecological destruction, cli-mate change and global warming. Environmental problems were widely discussed by The Club of Rome, an informal and international group of decision makers who commissioned the publication of Limits to Growth in 1972. This book addresses questions about the long term consequences of the world’s population growth, in-dustrialization, pollution, food production and resource depletion. Since then, there has been a growing interest in environmental issues, which has given rise to many concepts such as sustainable development. Al Gore (1992), a former American Vice
Caldwell and Caldwell (1990) emphasize this point to explain why fertility does not decline in Sub-Saharan Africa.
President and Nobel Peace Price winner, voices similar concerns. He draws attention to environmental and ecological issues as well as global warming:
“No goal is more crucial to healing the global environment than stabilizing human population. The rapid explosion in the number of people since the beginning of the scientic revolution and especially during the latter half of this century is the clearest single example of the dramatic change in the overall relationship between the human species and the earth’s ecological system.” (p. 380)
Stabilizing human population growth has become more important today because of concerns with the earth’s capacity. Moreover, population growth, especially among the poorer inevitably raises poverty. Most people think that this leads to a deterio-ration in overall welfare as it causes undernourishment, starvation and diseases. For example, let us consider the following point of view of John Seager (2009):
“Population growth in the poorest places on earth undermines quality of life. It destroys resources necessary to sustain healthy families. It creates conditions in which strife and conflict can flourish. It dooms billions to abject poverty.”
(http://goliveinternet.economist.com/debate/days/view/364).
Some empirical studies have attempted to provide evidence of a positive link between population increase and environmental degradation. For instance, in the context of rural Sub-Saharan Africa, Cleaver and Schreiber (1994) claim that a rapid growth of the population causes environmental damages (impoverishment of the soil, de-struction of the forests) and deteriorates well-being. Filmer and Pritchett (1996) report a positive link between fertility and deterioration of the local environmental resources in a set of villages in Pakistan.
However, these investigations fail somewhat in ignoring the poverty phenomenon. The effect of poverty on fertility is shown in most developing countries by the large differentials in household sizes across variables like income, education, health and other variables that reflect poverty. At the same time, poverty is also often seen
as an outcome of high fertility. Large families are likely to devote smaller amounts of their budget to invest in their children’s health and education. In their cross-country studyEastwood and Lipton (1999) estimate regressions to assess the impact of fertility on poverty. They find a positive correlation between population growth and the magnitude of absolute poverty. Wagstaff (2002) reviews the evidence on poverty and health inequalities between poor and non poor people. The two-way relationship between poverty and health is emphasized, whether comparisons are made between countries or within countries: poverty breeds ill-health, and ill-health keeps poor people poor.
People now regard determinants of fertility, poverty, and environmental degrada-tion as interconnected. According toDasgupta (2000),“poverty, household size, and environmental degradation would reinforce one another in an escalating spiral” (p. 635). See also Birdsall (1994). The positive feedback mechanism that links these elements enables us to associate many characteristics to poor households: they con-tribute to high fertility and infant mortality, to less investment in children education, to less health care for them and to dependence on natural resources.
The interdependence between determinants of fertility, poverty and environmen-tal quality requires treating all these variables as endogenous. Economists have widely recognized this in the case of fertility: parents determine the number of chil-dren they want to have in response to the economic constraints they face in such a way as to maximize their own utility and possibly the welfare of their children.
Although empirical studies at the micro and country level suggest that high fertility worsens poverty, the effects of population growth on poverty are more elusive and difficult to explore at the world scale. Empirical studies on such effects generally lead to inconclusive results. We cannot therefore establish a clear-cut relationship between world poverty and overpopulation, because there is no evidence to support that high population growth causes or exacerbates poverty:
“The problem of global poverty, in and of itself, cannot in an empiri-cal sense be defined as a world population crisis – unless one means it is a crisis that so many people today should be suffering from poverty. But it is a fundamental lapse in logic to assume that poverty is a
pop-ulation problem simply because it is manifest today in large numbers of human beings. The proper name for that logical error is the fallacy of composition.” (Eberstadt 2007, p.7)
Clearly, the links between overpopulation and poverty are not easy to exhibit. Some researchers who have attempted to describe the relationships between population growth and global poverty ended up presenting nuanced results. For instance, Ahlburg (1994) concluded that
“Although it is not clear whether population growth causes poverty in the long run or not, it is clear that high fertility leading to rapidly growing population will increase the number of people in poverty in the short run.” (Ahlburg 1994, p.143)
The complexity of population growth and poverty interactions can be understood by referring toCassen (1994)’s view about the relevant studies encompassing popu-lation growth and economic changes:
“The issue of whether per capita economic growth is reduced by population growth remains unsettled. Attempts to demonstrate such an effect em-pirically have produced no significant and reliable results.” (Cassen 1994, p.15)
One central normative difficulty with the endogenous fertility framework isthe defi-nition of the society whose welfare should be of concern. What type of social welfare function should be chosen to evaluate the outcome of fertility? This leads us to the discussion of the normative aspects of the relations considered above.
2.1.4
More people is likely to bring more social happiness
The question of how to value human well-being and society’s welfare is longstand-ing in several different fields (economics, psychology, philosophy, etc.). Answerlongstand-ing
this question is important for policy makers who seek to improve the welfare of the populations they serve.
Traditional welfare economics usually relies on utility (whether under-stood as the satisfaction of desires or as happiness) as a measure of hu-man well-being. Recent work seems to emphasize happiness rather than standard utility to evaluate well-being in order to guide public policy - see for example Kahneman, Diener, and Schwarz (1999), Kahneman (2000) and Kahneman and Krueger (2006). It is also recognized that having more people can create some external benefits that can raise society’s happiness. This in-cludes parents’ desire for a child (Easterlin 2005), relational goods (companionship, friendship, partners, marriages,etc.) and religious practice.
The happiness view takes root in Bentham’s notion of “pleasure”, even if Nussbaum (2008) claims that Bentham’s conception does not adequately capture what may be understood by happiness.6
To value a society’s welfare, Bentham chooses to maximize the sum of pleasures and to minimize the sum of pains. This aggregation gives rise to utilitarianism, whose principal criterion is the maximiza-tion of the “greatest happiness of the greatest number number”. Bentham is the father of utilitarianism and some further refinements are provided by Sidgwick. This is commonly known as total or classical utilitarianism. The point of view of Sidgwick (1966) is useful:
“So that, strictly conceived, the point up to which, on utilitarian princi-ples, population ought to be encouraged to increase, is not that at which average happiness is the greatest possible (...) but that at which the prod-uct formed by multiplying the number of persons living into the amount of average happiness reaches its maximum”. (pp. 415-416)
The implications of total utilitarianism are clear. If a social planner chose total
6
According toNussbaum (2008),“some pleasures are bad, namely, those that are closely associ-ated with bad activities. Rich people have pleasure in being ever rich and lording it over others ... Racists have pleasure in their racism, sexists in their sexism. In general, bad people have pleasure in their bad behavior” (p. 96). Nussbaum’s criticism is also based on the saying that “one man’s joy is another man’s sorrow”.
utilitarianism as the social objective, it is more likely that he would prefer a larger population to a smaller one. This is because total utilitarianism can encourage indefinitively an increase in the size of the population, even if such an increase leads to a very low average well-being. Parfit (1984) characterizes this situation as a “repugnant conclusion”. The latter is formulated as follows:
“For any possible population of at least ten billion people, all with a very high quality of life, there must be some much larger imaginable population whose existence, if other things are equal, would be better, even though its members have lives that are barely worth living.” (Parfit 1984, p. 388).
Parfit considers such a conclusion as being repugnant. It may indeed appear repug-nant that a better population be constituted only of very poor or miserable people. But such a judgment much relies on a valuation of individual well-being in terms of conventional material achievements, for instance, real income. As Narveson (2003) points out, “obviously, people can be unhappy though wealthy, and happy though poor”.7
Ng (1986a) and Arturo Barrios (2009) also believe that poor people are on the whole happy and are happy to have been born. Barrios expresses his view by referring directly to the third world, the part of the world where mass poverty is concentrated:
“So unlike the Economist reader elites who, having solved most of their existential problems, are constantly seeking problems to temper their well-being, most people in the third world are very happy to exist indeed, thank you very much. Being poor does not make one as unhappy as the Western elites imagine”.
(http://goliveinternet.economist.com/debate/days/view/364)
7
As the saying goes: money does not buy happiness. An analogous view is emphasized by
Easterlin (1973) on page 4: “In all societies, more money for the individual typically means more individual happiness. However, raising the incomes of all does not increase the happiness of all. The happiness-income relation provides a classic example of the logical fallacy of compo-sition - what is true for the individual is not true for society as a whole”. Thinking that the society would be happier if it were richer is also not consistent with findings in subjective data (Kahneman, Krueger, Schkade, Schwarz, and Stone 2006).
The happiness that poor people may enjoy can also be seen through religious lenses. In general, poor people do try to derive happiness out of life. Those who practice religion can feel happier, because religion makes them depend less on themselves and more on God. A religion that promises a better afterlife can also help the poor to live better in spite of their poverty.8
This religious dimension is important since it can make religion having a greater social impact in societies where members are believers and practicers.
The religious view is also often based on moral obligations that are to some extent consistent with the total utilitarianism principle. Whether or not a society experiences low material well-being, religious beliefs cannot discourage an increase in society’s population. Religion can encourage people’s procreative capacity to replenish the world and subdue nature, as Heyd (1992) underlines it:
“The value of the replication of God’s image is the reason given for Man’s creation. (...) Their number should be as large as possible so as to permeate the world with God’s image. (...) It is a unique command-ment, because it is the existential basis for the very possibility of all other commandments. It is conscious procreation rather than simple biological propagation which is the object of the first (moral) duty”. (Heyd 1992, p. 2)
2.2
Social evaluations with variable population sizes
Total utilitarianism is one of the most popular social evaluation functions for variable population sizes. However, it leads to the repugnant conclusion. A social planner who wants to avoid the repugnant conclusion will tend to be in favour of controlling population size from a social welfare perspective. A revised version of total utilitarianism that does this is average utilitarianism. Edgeworth (1925)at-8
Becchetti and Pelloni (2010) review the effects of religion on happiness. They conclude that the effects are positive and significant in most econometric studies. See also Baudin (2008) who uses French data to investigate the role of religion in fertility behavior and finds a similar result for religious practicers.
tributes it to John Stuart Mill, who chose a social objective on aper capita basis to justify limits to population size. The main idea of average utilitarianism is to make the average person as well as possible no matter how small the population, even if this results in a single person. A direct consequence of this is that a population with only one individual should be preferred to an arbitrarily larger one with almost the same average well-being (Blackorby, Bossert, and Donaldson 2005). Then size does not matter even in the case of ranking distributions with different population size. In such a case, the replication invariance principle which claims that an in-come distribution and its exact replication give the same level of social evaluation is implicitly used in the analysis.
Thus, it is well recognized that average utilitarianism is more conservative about population size (Sumner 1978, Broome 1992b). Sumner (1978, p. 99) supports this: “it is no accident that the average theory was devised strictly to handle ques-tions of population”. This can have important implicaques-tions for big populaques-tions (for instance, China and India). With the implementation in China of the one-child policy in response to population concerns, the consequences are such that women are forced to have abortions and more than 10 million of them are per-formed per year.9
Abortions are common especially when it comes to female fe-tuses, due to the discrimination against daughters in East and South Asia. For instance, Klasen and Wink (2003) estimate the number of “missing women” in the 1990s at nearly 41 million for China and 31 million for India. See also Sen (1990) who estimates that more than 100 million women are missing in the world. This outcome associated to average utilitarianism has been emphasized byCowen (1989), Broome (1992a) and Kanbur and Mukherjee (2007) in the case of the death of poor people, which can lead to an increase in society’s welfare. Such things seem to be morally difficult to accept.
2.2.1
Critical-level utilitarianism
When discussing the caveats of both total utilitarianism and average utilitarian-ism,Blackorby and Donaldson (1984) andBlackorby, Bossert, and Donaldson (2005)
9
investigate another version of utilitarianism called the“critical-level generalized util-itarianism” (CLGU). CLGU is a social evaluation function defined as the sum of the differences between a transformation of individual incomes or individual utilities and a transformation of a constant called critical level. From an ethical view, the critical level can be seen as being the minimum individual welfare needed for someone to add to the value of humanity. The critical level has been termed the value of living by Broome (1992b).
The principle of CLGU is to assess if new individuals that are added to an existing population result in a higher social welfare for the overall population. It is clear from its definition that CLGU is the same as the product of population size and the difference between therepresentative or average utility and the critical level. A social planner whose social objective is CLGU is committed to a possible trade-off between population size and average well-being in excess of the critical level. As long as the critical level is relatively low, the greater the population size, the higher the level of society’s welfare. But if the critical level is higher than average utility, society’s welfare will tend to be higher when population falls. As a consequence of this, a low value of the critical level leads to a preference for a large population whereas a high value leads to a small one. This version of utilitarianism proposed by Blackorby and Donalson avoids some of average utilitarianism’s problems, since the addition of a person is socially profitable only if his well-being is higher than the critical level although not necessarily higher than the average one. Hence, poor people with well-being above the critical level will be valued by the CLGU principle. CLGU also avoids the repugnant conclusion since it is socially undesirable to add individuals with well-being lower than the critical level.
The CLGU approach can serve to address important policy debates on popula-tions that actually occur in many countries in the world. For instance, one important policy use of the CLGU approach is to inform the issue of migration. Some devel-oped countries and particularly European countries are pursuing selective migration with the main reported objectives to increase population growth rates, which are judged to be too low, and to maintain a relatively high level of national welfare. This is consistent with CLGU. CLGU also justifies the policy of many developed
countries that are engaged in selective migration, and admit only those that enjoy a level of welfare at least equal to an implicit critical level.
The critical-level approach has been used in some previous works. For instance, Blackorby, Bossert, and Donaldson (2002) extend the approach to allow the critical level to depend on population size. Recent work tries to use the critical-level ap-proach in a context of dominance or social ranking. Trannoy and Weymark (2009) develop a second-order dominance criterion based on the CLGU. They show that this dominance criterion is equivalent to the dominance based on gener-alized Lorenz curves for a given value of the critical level. They also ex-plore the case where the critical level lies in an interval as is investigated by Blackorby, Bossert, and Donaldson (1996). Ethical foundations related to critical-level utilitarianism have been discussed by Ponthiere (2003). The main argument is that critical-level utilitarianism seems to be a coherent and intuitive approach for dealing with social evaluation of variable populations.
There exists in the literature other approaches for social evaluation in-volving variable population size. Indeed, a few studies on social ranking use neither the replication invariance principle or the critical-level approach. Aboudi, Thon, and Wallace (2010) make inequality comparisons between popula-tions of different sizes and show that a distribution is more equal than another one if the first distribution can be obtained from the second distribution by means of linear income transformations. Pogge (2007) follows a Pareto improvement criterion to socially rank distributions with different numbers of individuals.
2.2.2
Optimal population
As already mentioned above, the choice of the social evaluation objective has some implications for population concerns. The Benthamite function always leads to a larger population than the Millian one. The Blackorby and Donaldson one is an intermediate between the two. It may lead either to a larger or to a smaller population size. It is therefore important for the social planner to make a reasoned choice of social valuation in order to formulate appropriate policies for society’s
well-being.10
If both rapid population growth as well as low population growth can be detrimental to society’s well-being (as is discussed in Section 2.1), then it seems that there may exist an optimal population growth (or optimal size) for the society. The notion of an optimal size dates back to Plato. He quantified the optimal size of a state at 5,040 individuals. He said:
“The number of our citizens shall be 5040. This will be a convenient number; and these shall be owners of the land and protectors of the al-lotment. (...) Every legislator ought to know so much arithmetic as to be able to tell what number is most likely to be useful to all cities”.
Beyond its arithmetic virtues,11
Plato seemed to see in this number the size that would balance size and subsistence:
“The territory must be sufficient to maintain a certain number of inhab-itants in a moderate way of life more than this is not required; and the number of citizens should be sufficient to defend themselves against the injustice of their neighbours”.12
Plato’s view that the number of citizens must remain constant implies that there should be population control to ensure the stationarity. This number ensures and can thus avoid some disasters related to a large population (Stangeland 1904, pp. 24-26). Plato’s view is also consistent with arguments in favor of population stabilization and to a certain extent matches with Aristotle’s argument. According to Aristotle, the optimal size should not be too small or too high, but should rather lie between boundaries within which population ought to remain:
“The right number of citizens is not one fixed number, but any number within certain limits”. (The Nicomachean Ethics of Aristotle, translated by Peters 1886, pp. 312-313).
10
Ng (1986b) also investigates a new version of utilitarianism very much like total utilitarianism, called number-damped total utilitarianism. It is defined as the product between a function of population size and average well-being.
11
Plato adds that this number has 59 divisors with 10 divisors which are followed. This makes it possible to organize society in many different equal-sized groups.
12
The later Malthusian view on population and standards of living gave prominence to the concept ofoptimum population. Having discovered the risk of overpopulation, Malthus advocates policies that stabilize population size and that can lead to an “optimal population size”. Due to the concern of limited resources, this can help avoid some disasters related to a large population. The economists Cannan and Wicksell introduced the literature on optimum population as “what density of pop-ulation under given circumstances is most advantageous” (Gottlieb 1945, p. 290). John Stuart Mill was the first to give precise details on the optimum population concept: the one providing a largestper capita income.13
Optimum population has been successively investigated by Meade (1955), Mirrlees (1967), Dasgupta (1969), Lane (1975), Samuelson (1975), and Gigliotti (1983). Meade, Mirrlees, Dasgupta and Gigliotti maximize a discounted total utilitarianism whereas Lane and Samuel-son maximize discounted per capita utilitarianism. Meade shows how to choose the optimum population size at any given time. The rate of savings and the stock of capita are given. Dasgupta extends Meade’s framework by endogenizing the rate of savings. One limit to Meade’s and Dasgupta’s work is their implicit hypothesis that there is no cost for realizing optimum population. Lane imposes further con-straints by assuming that the population growth is entirely endogenous. Gigliotti uses a framework with overlapping generations and without technological change. Making a comparaison between the two criteria (total andper capitautilitarianism), Gigliotti finds significantly different results. For a discussion related to the choice of an optimal population size using the two utilitarian criteria under a context of sustainable development, see Asheim (2004).
Finally, if the existing population size is higher that what is given by optimum population analysis, population control may be justified (but not systematically). For this purpose, the general idea of resorting to contraception, child murder, abor-tion and any migratory movement, is to ensure populaabor-tion staabor-tionarity. However, these population control policies do not all have the same effects on living individ-uals, as some of them can be ethically unacceptable. In the case where population size is lower than the optimal one, as may be the case in developed countries, policies that encourage families to produce more children are to be implemented to favor a more rapid population growth.
13
The more recent literature uses an alternative approach to optimal popula-tion based on social choice theory. Blackorby, Bossert, and Donaldson (1995) de-rive an axiomatic representation for social orderings to determine an optimal pop-ulation size. This representation is critical-level utilitarianism already discussed above. Renström and Spataro (forthcoming) adopt critical-level utilitarianism as a dynamic welfare criterion to choose an optimal population growth. Their results reveal that total and average utilitarianism lead to unsatisfactory results for popu-lation growth rate in the sense that both cannot avoid corner solutions for the steady state. Total utilitarianism implies that is optimal to increase population as fast as possible. Hence, this leads to the repugnant conclusion. Average utilitarianism im-plies a population growth at a very low speed. However, using a positive value for the critical level, they find that critical-level utilitarianism yields an interior solution for the population growth rate. Other relevant contributions includeBroome (2003), Broome (2004) and Golosov, Jones, and Terthilt (2007). In the following chapter, we implement critical-level generalized utilitarianism in a dominance context. We perform a theoretical analysis by comparing socially two populations of different sizes.
Welfare comparisons with different
population sizes: a theoretical
analysis
This chapter focusses on welfare comparisons when populations differ in size. It considers welfare dominance based on “critical-level general-ized utilitarianism” in addition to “poverty” ranking and establishes an equivalence between a critical-level generalized utilitarianism dominance criterion and a poverty dominance criterion. This leads to dominance tests of arbitrary orders of dominance that involve possible choices of poverty lines (or censoring points) and possible values for critical levels. Links between critical levels and orders of dominance are also investi-gated.
Keywords: Critical-level generalized utilitarianism; Welfare domi-nance; Poverty domidomi-nance; Dominance equivalence.
3.1
Introduction
Traditionnally, welfare ranking has been established by using Lorenz and gen-eralized Lorenz curves. In this literature, Atkinson (1970) is commonly regarded as having laid the foundations for welfare analysis. His result applies only to com-parisons of income distribution over populations with equal sizes. However, many comparisons typically involve different population sizes. Some further results have been given byDasgupta, Sen, and Starret (1973),Sen (1973),Shorrocks (1983) and Kakwani (1984), which make it possible to rank welfare when populations differ in size. This is done by using the replication invariance principle. However, em-ploying this principle implicitly supposes that welfare should be assessed in per capita terms. In this context, social evaluations are based on average utilitari-anism. Using average utilitarianism as a social evaluation criterion implicitly as-sumes that population sizes should not matter. One consequence of this is that a population with only one individual will dominate any other population of ar-bitrarily larger size as long as those larger populations’ average utility is (per-haps only slightly) smaller than the single person’s utility level — see for in-stance Cowen (1989), Broome (1992a), Blackorby, Bossert, and Donaldson (2005), and Kanbur and Mukherjee (2007). This social evaluation framework would seem to be too biased against population size: it would say for instance that a society made of a single very rich person (Bill Gates for example) would be preferable to any other society of greater size but lower average utility.
An alternatively popular social evaluation criterion is total utilitarianism. Adopting total utilitarianism leads, however, to Parfit (1984)’s “repugnant con-clusion”. Parfit (1984)’s “repugnant conclusion” bemoans the implication that, with total utilitarianism, a sufficiently large population will necessarily be con-sidered better than any other smaller population, even if the larger popula-tion has a very low average utility. Blackorby and Donaldson (1984) introduce what they call the “Critical-level generalized utilitarianism” (CLGU). According to Blackorby, Bossert, and Donaldson (2000), CLGU satisfies some ethical require-ments:
ethically acceptable family, and we argue that there are good reasons for choosing one of them ”. (Blackorby, Bossert, and Donaldson 2000, p. 2)
By introducing a new criterion named “generalized concentration curves”, Trannoy and Weymark (2009) developed social welfare dominance criteria to com-pare distributions of utility using a CLGU function. They showed that the dominance criterion based on “generalized concentration curves” gives the same ordering as does the CLGU criterion. Their results are analogue to those obtained using generalized Lorenz dominance.
Our main objective is to extend the above work by using poverty curves in-stead of Lorenz curves. We believe that this approach is preferable because it makes it possible to test whether welfare has increased or decreased across time in a given society. It also makes possible to take into account changes in pop-ulation sizes and distributions. Given the fact that poverty continues to hold international organizations’ attention, as for example in the Millennium Devel-opment Goals, this approach has useful policy implications. Furthermore, it makes possible to have CLGU results based on various orders of dominance as in Foster and Shorrocks (1988b) and Duclos and Makdissi (2004). Section 3.2 defines social welfare dominance relations. It also discusses how this relates to well-known poverty dominance criteria which we call FGT dominance. This dominance context extends Blackorby and Donaldson (1984)’s focus on CLGU indices. It also builds on the theoretical contribution of Trannoy and Weymark (2009), who propose a CLGU dominance criterion that is an extension of generalized Lorenz dominance and second-order welfare dominance.
Section3.3establishes the equivalence between CLGU dominance and FGT dom-inance criteria. In Section3.4, we discuss how the critical level can be related to the order of dominance. Section3.5 explores how dominance relations can be extended to a larger class of social evaluation indices.
3.2
CLGU: an alternative social evaluation
3.2.1
Definition of CLGU
Blackorby and Donaldson (1984) have proposed CLGU as an alternative to (and in order to address the flaws of) average and total utilitarianism. To see how CLGU is defined, consider two populations of different sizes. The smaller population of size M has a distribution of incomes (or some other indicator of individual welfare) given by the vector u, and the larger population of size N has a distribution of incomes given by the vector v, with M < N. Let u := (u1, u2,..., uM), with ui
being the income of individual i, and v := (v1, v2,..., vN) with vj being the income
of individual j. Let the level of social welfare inu and v be given by
W(u;α) = M X i=1 (g(ui)−g(α)) (3.1) and W(v;α) = N X j=1 (g(vj)−g(α)), (3.2)
whereg is some increasing transformation of incomes andαis a “critical level”. Note that social welfare in the two populations remains unchanged when a new individual with income equal to αis added to the population. The smaller population exhibits greater social welfare than the larger one if and only if W(u;α)≥W(v;α).
CLGU thus aggregates the differences between transformations of individual in-comes and of a critical level. It can therefore avoid some of average utilitarianism’s problems, since the addition of a new person will be socially profitable if that person’s income is higher than the critical level, although that income may not necessarily be higher than average income. CLGU can also avoid the “repugnant conclusion” since it is socially undesirable to add individuals with incomes lower than the criti-cal level, regardless of how many there may be of them. Overall, CLGU provides a relatively appealing and transparent basis on which to make social evaluations and avoid the flaws associated with average and total utilitarianism.
3.2.2
Definition of dominance orderings
LetNbe the set of positive integers andR:= (−∞,∞). We denote by Ω, the set of possible income distributions: Ω = UN∈NRN. As we see, the distribution u ∈ Ω
and the distribution v∈Ω.
The welfare functions in 3.1 and 3.2 depend on g and α. One could choose a specific functional form forg and a specific value for α, but that would be inconve-nient in the sense that the welfare rankings of u and v could then be criticized as
depending on those choices. It is thus useful to consider making welfare rankings that are valid over classes of functions g and ranges of critical levels α. To do this, let s =1,2,..., stand for an order of “welfare dominance”. Consider Cs as the set of
functionsR−→R that are s times piecewise differentiable. Define the class Fs z−,z+ o
