Lorenz curves interpretations of the Bruss-Duerinckx theorem for resource dependent branching processes
aa r X i v : . [ q -f i n . E C ] A ug Lorenz curves interpretations of the Bruss-Duerinckxtheorem for resource dependent branching processes
Alexandre Jacquemain ∗ Université catholique de Louvain, 1348,Louvain-la-Neuve, Belgium
Thursday 6 th September, 2018
Abstract
The Bruss and Duerinckx theorem for resource dependent branching processes statesthat the survival of any society form is nested in an envelope formed by two extremepolicies. The objective of this paper is to give a novel interpretation of this theoremthrough the use of Lorenz curves. This representation helps us visualize how theparameters interplay. Besides, as we will show, it clarifies the impact of inequalityin consumption.
Keywords:
Consumption-inequality, perfect inequality, limited resources, Resourcedependent branching processes, Bruss-Duerinckx theorem, Lorenz curves ∗ Electronic address: [email protected]
Introduction
Bruss and Duerinckx (2015) [6] model the development of human populations and theinfluence of society forms through so-called resource dependent branching processes(RDBPs) . Throughout this paper, we focus solely on RDBPs and borrow the nota-tions of [6].In short, RDBPs are special models of branching processes in which individuals createand consume resources. A so-called policy (society rule) determines how resources aredistributed among the present individuals, and individuals have a means of interaction.RDBPs are models in discrete time with asexual reproduction. These assumptions aremade for simplicity. It should be pointed out, however, that the hypothesis of asexualreproduction, which may first seem inadequate, is justified by the notion of averagedreproduction rate of mating units defined in Bruss (1984) [4]. Indeed, Theorem 1 ofthe latter reference shows that for all relevant long-term questions concerning possiblesurvival and equilibria, the simplification is asymptotically perfectly in order.The remaining of this section is first dedicated to introduce formally RDBPs (section1.1) and then to present the concept of Lorenz curves (section 1.2). In section 2, weexamine the contribution of Lorenz curves for analyzing two specific policies, the weakest-first and strongest-first societies. Section 3 provides insightful interpretations of theenvelopment theorem. Further motivation to introduce Lorenz curves is provided insection 4 which introduces immigration. Finally, section 5 concludes. In the model of Bruss and Duerinckx, the two natural hypotheses and driving forces are • Hyp.1: Individuals want to survive and see a future for their descendants, and • Hyp.2: In general, individuals prefer a higher standard of living to a lower one,where Hyp.1. takes priority before Hyp.2, if these are incompatible. These hypothesesare modeled through a branching process revolving around four main aspects: reproduc-tion, resources, claims and policies.
Reproduction
Besides the simplification that reproduction is asexual, it is also assumed that all in-dividuals reproduce independently of each other and follow the same reproduction law( p j ) j , where p j denotes the probability that one individual has j descendants. In orderto avoid trivial cases, we suppose p > p j for at least some j > D kn ) n,k gathers i.i.d random variables D kn representing thenumber of descendants of the kth individual at the nth generation. Given what wepreviously stated, P ( D kn = j ) = p j . Besides, m ≡ E [ D kn ] < ∞ is the average number of For a more general reference on branching processes, the authors refer to Haccou, Jagers and Vatutin(2007) [9]. D ( k ) = D n ( k ) ≡ P kj =1 D jn is the total number of descendants of thenth generation given that this generation counts k individuals. Note that the n indexcan be omitted since we are dealing with i.i.d variables. Resource space
The resource space is viewed as a common pot that society distributes among its membersand is made of heritage, to which we add individual production and subtract individualconsumption.The resource creation matrix ( R kn ) n,k gathers all i.i.d individual resource creations R kn and r ≡ E [ R kn ] < ∞ is the average productivity of an individual. Finally, R ( k ) = R n ( k ) ≡ P kj =1 R jn is the total resource space. Claims
In RDBPs, individual have a means of interactions through claims. The basic idea isthat an individual stays in the society only if his claim is met, otherwise he leaves.The claim matrix ( X kn ) n,k gathers all i.i.d individual claims X kn . Besides F ( x ) = P ( X kn ≤ x ) is the distribution of the claims and µ ≡ E [ X kn ] is the average claim. Society
A policy π is defined as any function determining a priority order in the society. Indi-vidual claims are then met respecting this order until the resource space is exhausted. RDBP
A RDBP is any counting process (Γ t ) t with an initial ancestor Γ = 1 and for which thepopulation at the next period is determined byΓ n +1 = Q π ( D (Γ n ) , ( X , . . . , X D (Γ n ) ) , R (Γ n ))where Q π is the counting process based on the policy π . For further details we refer to[6]. For a more detailed motivation of [6] and implications of socio-economic interest werefer to Waijnberg (2014) [13] and Bruss (2016) [5].As we will elaborate later on, the following condition plays a central role in [6]: thepopulation cannot survive forever unless mF ( τ ) ≥ F is the continuous cumulative distribution function (CDF) of the random re-source claim (consumption) of an individual, and τ the unique solution of the implicitequation m Z τ xdF ( x ) = r (2)2quation (1), which will attract our interest in the present work, showed up in a moregeneral context already in Bruss and Robertson (1991) [7], and, in a further extension,in Steele (2016) [12] but we will have an innovative look at it. Our interest is to rephraseand re-interpret this condition in terms of the well-known notion of Lorenz curve . Lorenz curves have been extensively used in the context of income distributions to por-tray how the proportion of total income owned by the up-to- p poorest individuals evolvewith p . Though it was introduced as soon as 1905 by the American economist MaxLorenz [11] in order to picture social inequalities, it gained much more attention withthe work of Atkinson (1970) [2], which provided a normative rationale for the use ofLorenz curves to measure inequality.For the definition of the Lorenz curve, we follow Gastwirth (1971) [8]. The Lorenzcurve of F at ordinate p is defined as LC ( p ) ≡ µ Z p F − ( t ) dt Proceeding to the following change of variable t = F ( x ), we can rewrite the Lorenz curveat p as: LC ( p ) = 1 µ Z F − ( p )0 xf ( x ) dx = 1 µ Z F − ( p )0 xdF ( x )= 1 µ E [ XI [ F ( X ) ≤ p ]]In terms of properties, the LC passes through (0 ,
0) and (1 , andconvex.Turning to intuition, the Lorenz curve answers the following question: what share ofclaims do the up-to- p most modest individuals gather? If the curve is a straight line, wecall it the line of equality (LOE): the bottom 5% gather 5% of the mass of claims, the10% most modest gather 10%, and so on. Hence we have perfect equality. If the curveis right-angle shaped, all the claims are spoiled by the most demanding individual, thisis perfect inequality. Figure 1 displays a typical Lorenz curve (dotted) as well as thesituations of perfect equality (solid) and perfect inequality (dashed).We now have the necessary tools to define Lorenz dominance . We will say that F LC-dominates F if the following holds LC ( p ) ≥ LC ( p ) ∀ p ∈ [0 , p . Atkinson (1970) [2] provides normative reasons to useLC-dominance in order to rank societies in terms of inequality. The underlying idea is This is true provided that the variable of interest is nonnegative. This is not true in full generality. L ( p )Cumulative share of the population C u m u l a t i v e s h a r e o f t h e v a r i a b l e Figure 1: Representation of a Lorenz curve (solid) as well as perfect equality (dotted)and perfect inequality (dashed)that all sensible inequality indices will deem F to bear less inequality than F providedthat the Lorenz curve of F is above that of F . If instead the two curves cross, onecan find at least one pair of sensible inequality indices giving contradicting conclusions.The precise meaning of “sensible” is given in the latter reference. Obviously, the LOELC-dominates all other Lorenz curves.The author is well aware that the interpretation of Lorenz curves in the context ofRDBPs is in fact different from its usual sense. The latter is that the LC was originallyintended to measure the share of total wealth in a given state as a function of the shareof its effectives. In condition 1 for survival of a RDBP, it will intervene as the Lorenzcurve of claims, which is not the same as present wealth, but rather as consumption.However, the link bewteen these notions is sufficiently close. See also Waijnberg (2014)[13] for a free interpretation. Bruss and Duerinckx (2015) [6] present two extreme policies, the weakest-first and thestrongest-first societies. These policies present a particular interest as they will form anenvelope for the survival of any society in a sense we will make clear in section 3.
The weakest-first society (WFS) serves the less demanding first. Formally, it can bedefined through its counting process. N ( t, s ) = ( X < ,t> > ssup { ≤ k ≤ t : P kj =1 X
1] segment. Consequently, whatever thepolicy, we know society will finally die out or survive.4. Perfect inequality. Recall that the Lorenz curve in presence of perfect inequalityis triangle shaped. In this case, F ( τ ) = 1 and F ( θ ) = 0 and both areas vanish.Consequently, without knowing the underlying policy, there is nothing we can sayabout whether society will finally survive or die out. In a recent special course on RDBPs taught by Bruss at the Université catholique deLouvain it was seen that Lorenz curves also intervene whenever one brings immigra-tion into the picture and want to understand under which conditions the immigrant-population and the home-population can reach an equilibrium , both conditioned onsurvival. We confine here our interest to the simplest case where from some finite timeonward there are no new immigrants, and where the immigrants do not integrate intothe home-populatoin. In this case, the necessary condition for an equilibrium to existcan be expressed in terms of the condition m h F h ( τ ) = m i F i ( τ ) ≥ Note in passing that it also means that less inequality increases the chance of survival in the SFSand decreases it in the WFS. The equilibrium is meant as the asymptotic ratio α between the size of the immigrant-populationcompared to the home-population. τ is solution of m h Z τ xdF h ( x ) + αm i Z τ xdF i ( x ) = r h + αr i (4)and where F h (respectively F i ) denote the CDF of claims for individuals from the home-population (respectively immigrant-population). Besides, r h ( r i ) and m i ( m h ) are thecorresponding parameters of the two subpopulations. Note that the value τ is the same inboth integrals. Although we do not discuss equilibria of home-population and immigrant-population in the present short article, we wanted to mention the conditions (3) and (4)for a Bruss-equilibrium already in this present article. Indeed, this adds very much toour motivation to look at Lorenz curves under several angles of view, because (3) and (4)must be satisfied simultaneously, and the condition m F ( τ ) = m F ( τ ) is remarkablydemanding. How we could interpret these conditions in a single combined Lorenz curvegraphic remains a challenge. In the context of RDBPs, the Bruss and Duerinckx theorem states that the chance ofsurvival of any society is finally nested between two extreme cases, the weakest-first andthe strongest-first societies. As already pointed out in Bruss and Duerinckx (2015) [6],this envelope is impacted by the individual productivity, the multiplication rate of thepopulation and the distribution of claims. While the two first are simple parameters,the impact of the latter is more difficult to grasp.The contribution of the Lorenz curve is to disentangle the effect of the mean claimfrom its pure distributional aspects (i.e. inequality). By doing so, we observe thatinequality in consumption has a clear-cut impact on the stringency of the envelope. Lessinequality doesn’t necessarily translate into more or less chance of survival, it yields morecertainty. In practice, it means that less inequality increases the number of situationsfor which we are sure whether the society finally survives or dies out. We find this to bea remarkable property.Finally, note that the role of Hyp 2. listed in the introduction is only mentionedindirectly here. Hyp 1. has priority, but within this limitation the population is thenlikely to increase the standard of living so that the long-term multiplication factor mF ( τ )is likely to be chosen close to one, i.e. close to so-called criticality. RDBPs cannot bedirectly compared with Galton-Watson processes (as in Bruss (1978) [3]), or BPs in arandom environment, but there are some interesting links of results around criticality.In his course at UCL (2017), Bruss referred for further details to Jagers and Klebaner(2004) [10] and Afanasyev et al. (2005) [1]. References [1] Afanasyev V.I., Geiger J., Kersting G. and Vatutin V.A. (2005), Criticality in branch-ing processes in random environment,
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