Towards a Theory of Societal Co-Evolution: Individualism versus Collectivism
TTowards a Theory of Societal Co-Evolution: Individualism versus Collectivism
Kartik Ahuja ∗ , Simpson Zhang † and Mihaela van der Schaar ∗ Department of Electrical Engineering ∗ , Department of Economics † , UCLA Abstract
Substantial empirical research has shown that the level of individualism vs. collectivism is one of the mostcritical and important determinants of societal traits, such as economic growth, economic institutions and healthconditions. But the exact nature of this impact has thus far not been well understood in an analytical setting. In thiswork, we develop one of the first theoretical models that analytically studies the impact of individualism-collectivismon the society. We model the growth of an individual’s welfare (wealth, resources and health) as depending notonly on himself, but also on the level of collectivism, i.e. the level of dependence on the rest of the individualsin the society, which leads to a co-evolutionary setting. Based on our model, we are able to predict the impact ofindividualism-collectivism on various societal metrics, such as average welfare, average life-time, total population,cumulative welfare and average inequality. We analytically show that individualism has a positive impact on averagewelfare and cumulative welfare, but comes with the drawbacks of lower average life-time, lower total population andhigher average inequality.
I. I
NTRODUCTION
Why are some societies wealthier or healthier than others? Why do some societies have substantial inequalityamong their members while others have relatively little? And why do certain societies have a large population whileothers have a small population? Culture, specifically the level of individualism vs. collectivism in the society, playsan important and even central role in answering the above questions [1] [2].Landes [3] [4] and many others make the argument for the impact of culture on economic development.Furthermore, in [5] the authors argue that among the different dimensions of culture that affect long run growth, suchas individualism-collectivism, masculinity, power distance etc., the single most relevant dimension is individualism-collectivism. Thus understanding the impact of the level of individualism vs. collectivism on a society is of incredibleimportance in building a model of societal development. In individualistic societies, people tend to depend moreon themselves and less on society for growth in life, whereas in collectivistic societies, people tend to contributeto and depend on society to a greater extent. The level of collectivism in the society thus determines how muchthe growth of an individual is affected by the society, as well as how much the individual affects the developmentof the society, leading to a co-evolutionary setting. In this paper, collectivism represents a cultural element and notcommunism or a state (or religion) direction of activity.There has been substantial research [2] [6] [7] [8] towards analyzing the determinants of societal development.A significant thrust of this research has been on developing theories based on empirical tests [2] [7] [9]. Empiricalstudies have established the positive impact of individualism on economic parameters, namely GDP per capita andGDP of a country [8]. But there is also more inequality in the societies with higher levels of development both in a r X i v : . [ c s . S I] N ov conomic [10] [11] and health conditions [12]. Even though these empirical results exist, developing mathematicalmodels to understand such social systems is very important, because these mathematical models help us predictsocietal phenomenon and provide useful insights which can otherwise not be obtained just based on empirical tests.However, there are relatively few papers that analytically study the impact of individualism vs. collectivism. In [6]the author develops a mathematical model to show that individualism-collectivism is important in determining thestructure of economic institutions in the society. In [8] the authors come up with a mathematical model throughwhich they can predict that the individualistic societies promote more long run economic growth than collectivisticsocieties.In this work, we develop a mathematical model of the impact of individualism-collectivism on more generalparameters of a society, as opposed to only on economic institutions as in the above papers. Our mathematicalmodel helps us answer questions pertaining to the impact of individualism-collectivism on the socio-economicinequality in the society, the total population that can be sustained in the society and the average life-time ofindividuals, which cannot be answered with existing models. In our model, individuals are born into the societywith a fixed level of intrinsic quality, which determines the rate of change of their welfare. Welfare in our modelis an abstract quantity which represents an aggregate of the wealth, resources and health of an individual. Anindividual in our model will die either if its level of welfare drops too low or due to natural causes. Importantly,the level of collectivism determines the extent to which an individual’s welfare is affected by rest of the societyand vice-versa. Our objective is to compare societies with different levels of collectivism, levels of welfare requiredto survive while assuming the societies are identically impacted by other factors, such as economic institutions,government [13] [14] or geography, environment [15]. Our model is simplistic as we abstract away the impact ofeconomic institutions, government, geography and environment however, it still allows us to capture the impactof individualism-collectivism, as well as other forces, such as the level of welfare required to survive on societalmetrics, namely average welfare, average life-time, average inequality, and total population. From our model wecan make the following predictions:1. Although there is higher societal support given to individuals with low quality in a collectivistic society, thisdoes not increase the average welfare of individuals in collectivistic societies since the support from the rest ofindividuals in society comes at the expense of their own welfare levels. This implies lower average welfare levelsin a collectivistic society than in an individualistic society.2. Despite lower average welfare levels, average life-time may be higher in a collectivistic society because thesocial support given to lower quality individuals will allow them to survive for a longer amount of time. This alsomeans that collectivistic societies can sustain higher population levels.3. Cumulative welfare, defined as the total wealth, resources and health of a society, is lower in a collectivisticsociety. Although a collectivistic society supports a larger total population than an individualistic society, thisincrease is dominated by the decrease in the average welfare.4. The level of inequality in the society is higher in an individualistic society than in a collectivistic one, becauseindividualistic societies allow agents to reach higher personal welfare while giving less social support to individualsith low welfare levels.5. In addition we also study the impact of rate of birth, rate of natural deaths and the minimum welfare levelrequired to survive on the above societal metrics.Our analytical results are in general agreement with the existing empirical evidence, and we also provide somenew predictions that have so far not been tested empirically. We want to emphasize that the study here is verygeneral and potentially has a broader scope. Individualism-collectivism is a trait not particular to humans, and in abroad sense it can capture the collectivistic versus individualistic behavior of different biological species, such asbacteria [16]. Being able to mathematically understand individualism and collectivism is not only useful for societalevolution, but can also be of significant interest in biology.II. S YSTEM M ODEL
We consider an infinite horizon continuous-time model with a continuum of individuals living in a society. Eachindividual is characterized by his intrinsic quality, Q , which models his ability to develop in life, i.e. increase hiswealth, resources and health. The intrinsic quality is a random variable which can take either a good or a bad value,i.e. Q ∈ { , − } , where the probability that Q = 1 , P ( Q = 1) = . Due to space limitations, we only treat asimplistic model here, however our results can be extended for more general distributions of quality. We denote theindividual’s welfare, an abstract quantity representing aggregate wealth, resources and health of individual, at time t from birth as, X ( t ) and the welfare at birth is zero, X (0) = 0 . The rate at which the welfare of an individualincreases at any time t from birth is determined by the individual’s quality as well as the average quality of therest of society, and is given by R ( t ) (cid:44) dX ( t ) dt = (1 − w ) .Q + w. ¯ Q ( t ) , where ¯ Q ( t ) is the average quality of all theindividuals in the society at time t , and w ∈ [0 , is the level of dependence on society. This weight w is same forall individuals in the society and is a measure of collectivism in the society, i.e. w = 1 and w = 0 correspond to apurely collectivistic and purely individualistic society respectively. This mutual dependence amongst the individualsleads to a co-evolutionary setting.The individuals are born into the society at a rate of λ b mass per unit time, which means that the total mass ofindividuals entering the society in ∆ t time is λ b ∆ t . Individuals in the society can die either due to natural causesor due to poor welfare levels. The death due to natural causes is modelled as a Poisson arrival process with a rate λ d starting at the time of birth of the individual, and at the first arrival instance the individual dies, see Fig. 1. Thedeath due to poor welfare levels happens if the welfare levels fall below a threshold, − r which we call the deathboundary, see Fig. 1. Steady State of the Society:
As time increases the population increases up to a point where the rate of deathequals the rate of birth. Thus the total population will converge to a fixed mass, and the distribution of welfarelevels within the society will also converge to a constant. Thus in a steady state: a) the total population mass inthe society attains a fixed value, denoted by
P op ( λ b , λ d , r, w ) , at which the rate of birth will equal the rate ofdeath and b) the density of the population at a given welfare level x , p λ d ,λ d ,r,w ( x ) , see Fig. 2, and the mass of thepopulation with quality Q = q , M ( q ) , are also determined. We show below in Theorem 1 that there is always aunique steady state in our model given the exogenous parameters { λ b , λ d , r, w }, which characterize the society. = 1 Q = − w.Q Death
PoissonArrival
Birth w. ¯ Q ( λ b , λ d ,r,w )Society w. ¯ Q ( λ b , λ d ,r,w ) Society’scontributionto individualIndividual’scontributionto society
Death Boundary/Poisson Arrival w. ( Q = − w. ( Q = 1) Bad QualityIndividual Good QualityIndividual
Figure 1. Life-time of good and bad quality individuals.
Theorem 1.
Every society has a unique steady state.The detailed proofs can be found in the appendix (Section V) given at the end.
Lemma 1 . Good and bad quality individuals attain positive and negative welfare values respectively in the steadystate.Bad quality individuals can die either due to a Poisson arrival or due to poor welfare levels. As a result theproportion of the bad quality individuals is lower than that of good quality ones, which leads to a positive averagequality ¯ Q ( λ b , λ d , r, w ) . Hence, good quality individuals cannot take negative welfare values. Also, it can be shownthat the bad quality individuals cannot take positive welfare values, see the appendix (Section V) at the end fordetails.In the unique steady state the population density, p λ d ,λ d ,r,w ( x ) decays exponentially in both positive and negativedirections, see Fig. 2. We illustrate the life-time of an individual with good (bad) quality, i.e. Q = 1 ( Q = − in steady state in Fig. 1. The positive (negative) welfare levels are attained by good (bad) quality individuals inthe population. The rate at which the welfare of a bad quality individual decays in time is typically lesser thanthe rate of growth of good quality individuals, (due to the opposing effects of the negative quality and positivesocietal support for a bad quality individual), this leads to a higher decay in the population density of bad qualityindividuals as compared to good quality individuals, see Fig. 2. We focus on understanding the impact of theexogenous parameters on the properties of this steady state. To do so we denominate some important societalmetrics which help understand the properties of the steady state. Definition 1. Average quality:
The average quality of individuals represents the net impact the society has onrate of growth of welfare of each individual and is defined as ¯ Q ( λ b , λ d , r, w ) = 1 M ( Q =1) P op ( λ b ,λ d ,r,w ) − . M ( Q = − P op ( λ b ,λ d ,r,w ) . Definition 2. Average welfare:
The average value of welfare of the population, a measure of the average wealth,resources and health of an individual in the society, is defined as ¯ X ( λ b , λ d , r, w ) = ´ ∞− r x p λb,λd,r,w ( x ) P op ( λ b ,λ d ,r,w ) dx .Let T denote the random variable corresponding to the life-time of an individual in steady state. Let R be therate of growth of the individual in steady state where R = (1 − w ) .Q + w. ¯ Q ( λ b , λ d , r, w )) and Q is the qualityof the individual. If R ≥ , then the individual’s welfare will always be above zero, hence the individual will Individuals with Q = − Q = +1 c e − λ x - r , Death Boundary c . λ λ .e λ x (0 , p λ b , λ d ,r,w ( x )Populationdensity Welfare level, x Figure 2. Steady State Distribution of population density as a function of welfare levels. only die when there is a Poisson arrival. Therefore, T in this case will be an exponential random variable, T (cid:48) ,with mean λ d . If R < then the death will happen either at time T ( λ b , λ d , r, w ) = r − w (1+ ¯ Q ( λ b ,λ d ,r,w )) , where T ( λ b , λ d , r, w ) is the time taken to reach the death boundary, or if there is a Poisson arrival before T ( λ b , λ d , r, w ) .Hence, T = min { T (cid:48) , T ( λ b , λ d , r, w ) } , Definition 3. Average life-time:
The average life-time of an individual is defined as the expected value of thelife-time (unconditional on individual’s quality), ¯ T ( λ b , λ d , r, w ) = E λ b ,λ d ,r,w [ T ] .The next societal metric is a measure of average inequality in the welfare levels of individuals. Definition 4. Average inequality:
Average inequality, a measure of disparity in the society, is defined as thevariance of welfare ,
V ar X ( λ b , λ d , r, w ) = ´ ∞− r ( x − ¯ X ( λ b , λ d , r, w )) p λb,λd,r,w ( x ) P op ( λ b ,λ d ,r,w ) dx .Next, we come up with a notion of Cumulative welfare, which is the aggregate amount of welfare in the society,a measure of total wealth and resources. Definition 5. Cumulative welfare:
The cumulative welfare, a measure of total welfare of society accumulatedtogether, is defined as CF ( λ b , λ d , r, w ) = P op ( λ b , λ d , r, w ) ¯ X ( λ b , λ d , r, w ) .In the above societal metrics, average life-time, total population and average quality are more intuitive tounderstand, while average welfare is similar to GDP per capita [8], cumulative welfare is similar to the GDP[8] and average inequality is related to GINI coefficient [10] [11].III. R ESULTS
In this section, we will compare different societal metrics across societies differing either in the level ofcollectivism, w or the other exogenous parameters. Lemma 2. a) The average quality ¯ Q ( λ b , λ d , r, w ) and the average welfare ¯ X ( λ b , λ d , r, w ) of an individualdecrease as the level of collectivism w increases. b) ¯ Q ( λ b , λ d , r, w ) and ¯ X ( λ b , λ d , r, w ) decrease as the rate ofnatural deaths λ d increases. c) ¯ Q ( λ b , λ d , r, w ) and ¯ X ( λ b , λ d , r, w ) decrease as the death boundary − r decreases.In part a), as the level of collectivism is increased, the support from the society slows the rate at which thewelfare of a bad quality individual decays with time, causing a larger proportion of the population to be of lowquality. The good quality individuals contribute more to this support as well and as a result their own growth islowed. As a result, there is a negative impact both on the average quality and average welfare of the individuals.Parts b) and c) are straightforward, see the appendix (Section V) at the end for detail. This lemma is supportedby the empirical studies showing lower per capita income in collectivistic societies in comparison to individualisticsocieties [8] .Theorem 2. a) Total population P op ( λ b , λ d , r, w ) increases as the rate of birth λ b increases. b) P op ( λ b , λ d , r, w ) increases as the level of collectivism w increases. c) P op ( λ b , λ d , r, w ) increases as the death boundary − r decreases.d) If w < then P op ( λ b , λ d , r, w ) increases as the rate of natural deaths λ d decreases.Part a) and c) are easier to comprehend, see the appendix (Section V) at the end for details. For part b), as thelevel of collectivism increases the support from the society slows the rate at which the welfare of a bad qualityindividual decays with time. As a result, the proportion of individuals dying at the death boundary decreases, whichmeans that the population level at which the mass of population dying equals the mass of population being bornis higher. This agrees with the empirical studies which show collectivistic societies have less income per workerand have a larger population [17] [18]. In part d), as the rate at which natural deaths occur decreases, the rate ofdeaths due to achieving poor welfare levels through hitting the death boundary can increase. However, if the levelof dependence on the society is low then the decrease in the rate of natural deaths dominates, and as a result thetotal population increases such that the mass of deaths equals mass of birth. Theorem 3: a) Cumulative welfare CF ( λ b , λ d , r, w ) decreases as the rate of birth λ b decreases. b) CF ( λ b , λ d , r, w ) decreases as the rate of natural deaths λ d increases. c) If λ d r ≤ (cid:15) < and w < − (cid:15) with (cid:15) > , then CF ( λ b , λ d , r, w ) decreases as the death boundary − r decreases. d) CF ( λ b , λ d , r, w ) decreases as the level ofcollectivism w increases.Since CF ( λ b , λ d , r, w ) ∝ P op ( λ b , λ d , r, w ) , part a) follows from Theorem 2. For part b), as the rate of naturaldeath increases the average welfare of an individual decreases (Lemma 2) and the total population also decreases(Theorem 2), if the level of collectivism is not high. This shows the result for part b), when the collectivism is nothigh. However, it can be shown that even if the level of collectivism is high then as well there will be a decreasein cumulative welfare owing to a significant decrease in the average welfare (see the appendix (Section V)). Forpart c), as the death boundary decreases, the total population in the society increases whereas the average welfareof an individual decreases, leading to opposing effects. Therefore, if the λ d r is sufficiently low then the proportionof the population with bad quality is sufficiently low as well. Also, if the level of collectivism, w is low then thenthe rate at which the welfare of bad quality individuals decays with time is high, hence the effect of decreasing thedeath boundary on the average welfare is high. Under these conditions the decrease in average welfare dominatesthe increase in population. For part d), increasing the level of collectivism increases the total population (Theorem2), but it decreases the average welfare of an individual (Lemma 2). Interestingly, it can be shown that the decreasein the average welfare of an individual dominates the increase in population (see the appendix (Section V) fortechnical detail). This result is also aligned with the empirical tests showing higher GDPs for an individualisticsociety [8]. heorem 4. a) Average life time ¯ T ( λ b , λ d , r, w ) decreases with an increase in rate of natural deaths λ d . b) If λ d r > θ ∗ = ln(1 + √ ) then ¯ T ( λ b , λ d , r, w ) increases with an increase in level of collectivism w else, it firstdecreases and then increases with an increase in level of collectivism w . c), If λ d r > θ ∗ , then ¯ T ( λ b , λ d , r, w ) increases with a decrease in death boundary − r else, it first decreases and then increases with a decrease in deathboundary − r .For part a), an increase in the rate of deaths will affect the life-time of both good and bad quality individualsnegatively, thus leading to the result. For part b), increasing the level of collectivism slows the rate at which thewelfare of individuals with bad quality decays with time resulting in an increase in their life-time. It also leads to anincrease in the proportion of individuals with bad quality, but note that individuals with good quality have a higherlife-time than individuals with bad quality. This leads to an opposing effect. However, if λ d r is high i.e. λ d r > θ ∗ ,then the proportion of the individuals with bad quality is high enough, implying that the increase in the life-timeof individuals with bad quality has a dominating effect in comparison to the decrease resulting from a decreasingproportion of individuals with good quality. The proportion of the population of bad quality individuals increaseswith an increase in the level of collectivism. If λ d r ≤ θ ∗ and the level of collectivism is sufficiently high, therewill be a sufficiently high proportion of bad quality individuals, and so if level of collectivism is increased thenthere will be an increase in the average life-time. However, if the level of collectivism is not high then there willbe a decrease in the average life-time with an increase in the level of collectivism. A similar explanation appliesto part c). In Fig. 3, it is shown that if λ d r is sufficiently high the average life-time increases with the level ofcollectivism, otherwise, the average life-time decreases and then increases. It is important at this point to note thatin part b), we compare two societies with different levels of collectivism while other parameters remain the samewhich may include medical facilities, health awareness etc. that are also crucial determinants of life-time. Also, ourmodel does not yet consider the impact of cumulative welfare on the rate of natural deaths λ d and is an importantdirection for future research. Theorem 5.
The average inequality
V ar X ( λ b , λ d , r, w ) is always more in an individualistic society w = 0 as compared to a collectivistic society w = 1 . Also if the person only dies a natural death, i.e. r → ∞ , then a) lim r →∞ V ar X ( λ b , λ d , r, w ) decreases with an increase in level of collectivism w and b) lim r →∞ V ar X ( λ b , λ d , r, w ) decreases with an increase in rate of natural deaths λ d .For part a), the case when an individual only dies a natural death there is a symmetry in the proportion ofindividuals with good and bad quality. Hence, the average quality of an individual is zero. Therefore, the rate ofdecay (growth) for an individual with bad (good) quality is − w . Hence, increasing w slows the rate of decayand growth, thereby allowing individuals to neither take too low or too high welfare values, which leads to alower average inequality. Having higher levels of inequality in individualistic societies has also been observedin calculations of GINI coefficient for various countries [10], [11]. Also, having higher inequality has been animportant factor affecting the health of the society [12], this observation supports our result on the negative impactof individualism on average life-time in Theorem 4. For part b), it is clear that a higher rate of death, λ d implies A v e r ag e L i f e - T i m e , ¯ T ( λ b , λ d , r , w ) λ d r ≤ θ ∗ Level of collectivism, w Increasing λ d r .,i.e. proportionof bad qualityindividuals A v e r ag e L i f e - t i m e ¯ T ( λ b , λ d , r , w ) Level of collectivism, w Increasing λ d r ,i.e. proportionof bad qualityindividuals λ d r > θ ∗ Figure 3. Illustration of part b) of Theorem 4. that individuals with very high or low welfare levels are less likely to exist, thus leading to lesser inequality.IV. C
ONCLUSION
We propose a mathematical model to study societal co-evolution under the forces of individualism and collec-tivism. This work serves as an important step towards understanding the exact nature of the impact of individualism-collectivism on various societal facets. Through our model we can show that the average welfare of individuals ishigher in an individualistic society, however the average life-time is typically lower in comparison to a collectivisticsociety. A larger life-time in collectivistic society does allow for a larger population to be sustained, however thecumulative welfare is still lesser. Moreover, the average inequality is more in an individualistic society owing tothe lack of social support. Our results show concordance with existing empirical tests.V. A
PPENDIX
Theorem 1:
Every society has a unique steady state.
Proof:
We will start by deriving the population density at a given welfare level x , p λ b ,λ d ,r,w ( x ) and the totalpopulation mass P op ( λ d , λ b , r, w ) in the steady state and show that they are unique. To do so we first arrive at theexpression for the normalized population density f λ b ,λ d ,r,w ( x ) . The relation between f λ b ,λ d ,r,w ( x ) , p λ b ,λ d ,r,w ( x ) and P op ( λ d , λ b , r, w ) is given as, P op ( λ d , λ b , r, w ) = ´ ∞−∞ p λ b ,λ d ,r,w ( x ) dx, f λ b ,λ d ,r,w ( x ) = p λb,λd,r,w ( x ) P op ( λ d ,λ b ,r,w ) . Insteady state the average impact of the society, i.e. ¯ Q ( λ b , λ d , r, w ) is determined since the proportion of individualswith Q = q , i.e. M ( Q = q ) do not change. Hence, the rate at which the welfare of an individual grows can take onlytwo values depending on his quality, R = (1 − w ) . w. ¯ Q ( λ b , λ d , r, w ) , R − = (1 − w ) . − w. ¯ Q ( λ b , λ d , r, w ) , here R and R − are the rate of growth of good and bad quality individual respectively. To derive the densities insteady state, we will first show that in the steady state R and R − will be positive and negative respectively. Let’sassume that R and R − are both positive, i.e. all the individuals in the society experience a positive growth. Insuch a case the individuals can only die due to a Poisson arrival. Also, we know that an individual who is born isas likely to be good as he is to be bad. Hence, the population mass at which the rate of death will equal the rate ofbirth of good/bad quality individual is the same for both the types of individuals, i.e. M ( Q = +1) = M ( Q = − .s a result, the average quality ¯ Q ( λ b , λ d , r, w ) = 0 . Substituting this back in the expressions for the rate we get, R = (1 − w ) and R − = (1 − w ) . − . Therefore, R − is negative this contradicts the supposition that the both therates are positive. Next, let’s assume that both R and R − are negative. In this case the individuals can die eitherdue to a Poisson arrival or due to hitting the death boundary. In such a case the welfare values attained will onlybe negative. Let f λ b ,λ d ,r,w ( x ) correspond to the joint density that the individual of good quality attains a welfarelevel of x . Similarly, we can define f − λ b ,λ d ,r,w ( x ) to be the joint density for a bad quality individual at a givenwelfare level of x . In steady state although the density of population in a given welfare level is fixed, howeverthe individuals comprising the density at a given welfare level is not the same owing to change of welfare levels,births and deaths that happen continually. As a result, at any instant of time the mass of individuals that attain agiven welfare level will equal the mass of indiduals that leave that welfare level either due to change in welfareor due to dying. Consider an infinitesimal interval h , the mass of the population with quality Q = 1 between x − h and x at time t , where x ≤ , is given as, f λ b ,λ d ,r,w ( x ) .h . Consider a time interval t (cid:48) after which this massof individuals, f λ b ,λ d ,r,w ( x ) .h will either die or will attain a different welfare level between, y − h and y , here y = x + R .t (cid:48) . The probability that an individual does not die a natural death in time interval t (cid:48) is e − λ d t (cid:48) . Hence, theproportion of the mass of individuals who do not die a natural death and a result attain a welfare between y − h and y is e − λt (cid:48) f λ b ,λ d ,r,w ( x ) .h = f λ b ,λ d ,r,w ( y ) .h . This can be expressed as f λ b ,λ d ,r,w ( y ) = e − λ d y − xR f λ b ,λ d ,r,w ( x ) and f λ b ,λ d ,r,w ( y ) = C .e − λ d yR where f λ b ,λ d ,r,w (0) = C . Similarly, for y ≤ we can get f − λ b ,λ d ,r,w ( y ) = C − .e λ d yR − where f − λ b ,λ d ,r,w (0) = C − . Note that both f λ b ,λ d ,r,w ( x ) and f − λ b ,λ d ,r,w ( x ) are zero for positive welfare valuessince both good and bad quality individuals are assumed to have a negative rate of growth. Also, the rate at whichindividuals of good quality and bad quality are born is the same given as λ b . Hence, we can equate the mass ofgood (bad) quality individuals which enter the society in time δt , i.e. λ b δt to the mass of individuals betweenwelfare level of and δx ( and δx ) , i.e. C δx ( C − δx − ). This gives, C − R − = C R = C . Since the f λ b ,λ d ,r,w ( x ) and f − λ b ,λ d ,r,w ( x ) are joint density functions the integral of the sum of these joint densities should be . ´ ∞−∞ f λ b ,λ d ,r,w ( x ) dx + ´ ∞−∞ f − λ b ,λ d ,r,w ( x ) dx = 1 C R λ d (1 − e λdR r ) + C − R − λ d (1 − e λdR − r ) = 1 C = λ d − e λdR r − e λdR − r From this we can calculate the mass of the individuals with Q = 1 and Q = − , i.e. M ( Q = +1) = Cλ d (1 − e λdR r ) and M ( Q = −
1) = Cλ d (1 − e λdR − r ) . Since R > R − we can see that M ( Q = +1) > M ( Q = − . This yieldsthat the ¯ Q ( λ b , λ d , r, w ) > and thereby R > . This contradicts the supposition that both the rates are negative.Also, since R > R − the only case left is R is positive while R − is negative. In this case the good andbad quality individuals take positive and negative welfare values respectively. We can calculate the joint densitiesin the same manner as described above and thus the resulting density is f λ b ,λ d ,r,w ( x ) = C e − λdR x , x > and f − λ b ,λ d ,r,w ( x ) = C − e λdR − x , x < , with C R = C − R − . To solve for the constants we need to proceed in aimilar manner as above: ´ f λ b ,λ d ,r,w ( x ) dx + ´ f − λ b ,λ d ,r,w ( x ) dx = 1 C R λ d + C − R − λ d (1 − e − λd (1 − w ) . − w ¯ Q ( λb,λd,r,w ) . r ) = 1 C = λ d − e − λ (1 − w ) . − w ¯ Q ( λb,λd,r,w ) . r ) For simplification of notation, we introduce auxiliary notation, λ = λ d (1 − w )+ w. ¯ Q ( λ b ,λ d ,r,w ) and λ = λ d (1 − w ) − w. ¯ Q ( λ b ,λ d ,r,w ) .Hence, the density functions are denoted as follows, f λ b ,λ d ,r,w ( x ) = λ − e − λ r e − λ x , x > and f − λ b ,λ d ,r,w ( x ) = λ − e − λ r e λ x , x < . Also, we can deduce that the marginal density f λ b ,λ d ,r,w ( x ) = f λ b ,λ d ,r,w ( x ) , x > and f λ b ,λ d ,r,w ( x ) = f − λ b ,λ d ,r,w ( x ) , x < . Using the density computed above we can calculate M ( Q = 1) = − e − λ r and M ( Q = −
1) = − e − λ r − e − λ r . Also, the average quality needs to be consistent with the average quality computedusing the distributions derived above. This is formally stated as ¯ Q ( λ b , λ d , r, w ) = e − λdr − w − w. ¯ Q ( λb,λd,r,w ) − e − λdr − w − w. ¯ Q ( λb,λd,r,w ) = e − λ r − e λ r (1)Next, we compute the total population mass by equating rate of births to the rate of deaths. The rate of deathsis comprised of two terms, the first term is the rate of natural deaths occurring due to Poisson shocks and thenext term is the rate of deaths due to hitting the death boundary, f λ b ,λ d ,r,w ( − r ) . (1 − w − w. ¯ Q ( λ b , λ d , r, w )) corresponds to the density of the individuals hitting the death boundary per unit time. Hence, the rate of deathsis λ d .P op ( λ b , λ d , r, w ) + f λ b ,λ d ,r,w ( − r ) . (1 − w − w. ¯ Q ( λ b , λ d , r, w )) .P op ( λ b , λ d , r, w ) , . Equating rate of births torate of deaths we get the following. λ b = ( λ d + f λ b ,λ d ,r,w ( − r ) . (1 − w − w. ¯ Q ( λ b , λ d , r, w )) P op ( λ b , λ d , r, w ) λ b = ( λ d + λ d . e − λ r − e − λ r ) .P op ( λ b , λ d , r, w ) P op ( λ b , λ d , r, w ) = λ b λ d (1 + ¯ Q ( λ b , λ d , r, w )) (2)Now that we have both the normalized density and the total population’s expressions, we can arrive at theexpression of the population density p λ b ,λ d ,r,w ( x ) which is just a product of the two, formally given as follows. p λ b ,λ d ,r,w ( x ) = P op ( λ b , λ d , r, w ) . λ − e − λ r e − λ x , if x > P op ( λ b , λ d , r, w ) . λ − e − λ r e λ x , if x <0If we can show that there is a unique average quality, ¯ Q ( λ b , λ d , r, w ) satisfying (1) then both the total populationmass (2) and the population density (3) are uniquely determined. We know that Q ∈ {− , } hence, ¯ Q ( λ b , λ d , r, w ) ∈ [ − , . To solve for ¯ Q ( λ b , λ d , r, w ) , we need to solve z = g ( z ) , where g ( z ) = e − λdr (1 − w ) − wz. r − e − λdr − w − w.z r and z ∈ [ − , . Wewill first show that there exists a solution in the set, [ − , . Let z = − and z = min { , − ww }. If w < then, = 1 else z = − ww . g ( z ) = e − λdr − w − e − λdr − w and g ( z ) > z . If w < then g ( z ) = e − λdr − w − e − λdr − w which is less thanor equal to z = 1 , i.e. g ( z ) ≤ z . Based on this and since the function z and g ( z ) are continuous in the range [ − , − ww ) , there has to be a point in the interval [ − , ⊂ [ − , − ww ) where g ( z ) = z . Also, g ( z ) is decreasing inthe range [ − , − ww ) , this can be seen from the expression for g (cid:48) ( z ) = − λ d rw (1 − w − wz ) (2 e λdr − w − wz − e λdr − w − wz and z is strictly increasing function. Therefore, g ( z ) − z is a strictly decreasing function in [ − , − ww ) , which implies thatthe root is unique. When w = , z = 1 we can see that g ( z ) > z holds, but g ( z ) is not continuous at z . This isnot a problem as we know that the function is continuous everywhere from [ − , z ) and lim (cid:48) z → z g ( z ) = 0 , where lim (cid:48) z → z g ( z ) corresponds to the left hand limit, hence lim (cid:48) z → z g ( z ) < z . Hence, the same argument as above canbe applied. In the case when w > then we will show that there exists a unique solution for g ( z ) = z in the range [ − , . We know that g ( z ) = e − λdr − w − e − λdr − w , but since w > we need to be careful about the case when w = 1 .For now we can assume that < w < . Hence, we know that g ( z ) > z . Here z = − ww and g ( z ) will not becontinuous at z . But we can show that lim (cid:48) z → z g ( z ) = 0 , where lim (cid:48) z → z g ( z ) corresponds to the left hand limit,and lim (cid:48) z → z g ( z ) < z . Hence, from the decreasing nature of g ( z ) − z we know that there is a unique solution inthe range [ − , − ww ) . Since > w > then [ − , − ww ) ⊂ [ − , we need to show that there is no solution in therange ( − ww , . In the range ( − ww , the function g ( z ) is not necessarily continuous. There exists a discontinuityif e λdr − w − wz − and z ∈ ( − ww , . Let’s assume that there is a discontinuity. In that case, the function g ( z ) will decrease values from − to −∞ , then to the right of the discontinuity at e λdr − w − wz − the functiondecreases from ∞ to e λdr − w − . Since w > and e λdr − w − wz − for some z ∈ ( − ww ,
1] 1 > e λdr − w − > we can say that e λdr − w − > . Hence, there is no point in the range in [ − , which intersects with this function.In the case, when there is no discontinuity it is straightforward to show that there is no solution of g ( z ) = z asthe function g ( z ) will only take negative values less than − . Also, when w = 1 the individuals welfare is fixed tozero all the time, hence there is a symmetry in the proportion of good and bad quality individuals, which leads toa unique solution ¯ Q ( λ b , λ d , r, w ) = 0 . Lemma 1.
Good and bad quality individuals attain positive and negative welfare values respectively.
Proof:
The proof of theorem 1, already contains the proof for this lemma as we show that R and R − attainpositive and negative welfare values respectively. Lemma 2.
The average quality ¯ Q ( λ b , λ d , r, w ) and the average welfare ¯ X ( λ b , λ d , r, w ) of an individual a).Decrease as the level of collectivism, w is increased., b). Decrease as the rate of natural deaths, λ d increases., c).Decrease as the the death boundary, − r decreases. Proof:
We already know that the solution for ¯ Q ( λ b , λ d , r, w ) requires solving a transcendental equation (1),which means that we do not have a closed form analytical expression for it. It can be shown that the expression for ¯ X ( λ b , λ d , r, w ) expressed in terms of the ¯ Q ( λ b , λ d , r, w ) is ( r + λ d ) . ¯ Q ( λ b , λ d , r, w ) . From Theorem 1, we knowthat for every set of parameters there does exist a solution ¯ Q ( λ b , λ d , r, w ) . For part a), as the level of collectivism isincreased let us assume that the average quality ¯ Q ( λ b , λ d , r, w ) increases. However, if there is an increase in both thecollectivism and the average quality, the expression g ( ¯ Q ( λ b , λ d , r, w )) = e − λdr − w − w. ¯ Q ( λb,λd,r,w ) − e − λdr − w − w. ¯ Q ( λb,λd,r,w ) decreases whichontradicts the increase in ¯ Q ( λ b , λ d , r, w ) . Hence, ¯ Q ( λ b , λ d , r, w ) has to decrease with an increase in collectivism.And from the expression of ¯ X ( λ b , λ d , r, w ) expressed in terms of ¯ Q ( λ b , λ d , r, w ) it is straightforward that theaverage welfare also decreases with an increase in the level of collectivism. For part b), again as the rate ofnatural deaths increases assume that ¯ Q ( λ b , λ d , r, w ) increases. However the decrease in e − λdr − w − w. ¯ Q ( λb,λd,r,w ) − e − λdr − w − w. ¯ Q ( λb,λd,r,w ) will contradict the assumption. With an increase in λ d the first term in the expression of ¯ X ( λ b , λ d , r, w ) whichinversely related to λ d has to decrease, this combined with the decrease in ¯ Q ( λ b , λ d , r, w ) leads to a decrease inthe average welfare. For part c), we arrive at the expression of the derivative of average quality ¯ Q ( λ b , λ d , r, w ) w.r.t. r , − λ d ( d )( d +1)(1 − w − wd )(1 − w − wd ) + λ d rw ( d +1) which is negative. Hence, we know that the average quality indeed decreases withan increase in r . For average welfare we give an intuitive explanation first, increasing r decreases the averagequality as a result of which the growth of a good quality individual slows down and the decay of a bad qualityindividual becomes faster. As a result the average welfare levels attained by a good and bad quality individual arelower. Moreover, increase in r increases the proportion of the bad quality individuals which further has a negativeeffect on the average welfare. To prove this formally we will show that the average welfare of both good andbad quality individuals decreases and the proportion of the bad quality individuals increases. Since the averagewelfare value of a bad quality individual is always lower than that of a good quality individual this is sufficientto show the result. The average welfare of good quality individuals is given as λ = − w + w ¯ Q ( λ b ,λ d ,r,w ) λ d . This canbe derived as follows, the distribution of the welfare conditional on the fact that individuals are of good quality f λ b ,λ d ,r,w ( x | Q = +1) can be shown to be an exponential distribution with parameter λ exactly on the same linesas we derived the joint densities f λ b ,λ d ,r,w ( x ) in Theorem 1. Since ¯ Q ( λ b , λ d , r, w ) decreases as a function of r ,the average welfare of a good quality individual also decreases as a function of r . Similarly we need to arrive atthe distribution f λ b ,λ d ,r,w ( x | Q = − , which turns out to be f λ b ,λ d ,r,w ( x | Q = −
1) = λ − e − λ r e λ x , x < . Theaverage welfare value of bad quality individual can be arrived at using this distribution and it turns out to be, − λ + re − λ r − e − λ r . As r is increased, ¯ Q ( λ b , λ d , r, w ) decreases and thus λ decreases as well. The partial derivativeof average welfare of bad quality individual − λ + re − λ r − e − λ r w.r.t. r is given as e λ r − λ re λ r − e λ r − and this expressionturns out to be negative for ( λ , r ) ∈ R . Also, it can be shown that the partial derivative of − λ + re − λ r − e − λ r w.r.t λ is given as ( λ ) − r e λ r ( e λ r − and this expression turns out to be positive. Hence, from the sign of these partialderivatives we can easily see the result. Theorem 2. a) Total population
P op ( λ b , λ d , r, w ) increases as the rate of birth λ b increases. b) P op ( λ b , λ d , r, w ) increases as the level of collectivism w increases. c) P op ( λ b , λ d , r, w ) increases as the death boundary − r decreases.d) If w < then P op ( λ b , λ d , r, w ) increases as the rate of natural deaths λ d decreases. Proof:
In order to compute the total population in the steady state, we need to have the rate of birth equals therate of death which is formally stated as follows, ( λ d + f λ b ,λ d ,r,w ( − r ) . (1 − w − w. ¯ Q ( λ b , λ d , r, w ))) .P op ( λ b , λ d , r, w ) = λ b op ( λ b , λ d , r, w ) = λ b λ d . (1 + ¯ Q ( λ b , λ d , r, w )) For part a), ¯ Q ( λ b , λ d , r, w ) does not depend on the rate of births and it is clear that the result holds since thepopulation is directly proportional to λ b . For part b) as well it can be seen that the only term in the expressionwhich depends on w is ¯ Q ( λ b , λ d , r, w ) which will decrease as w is increased (Lemma 2). Therefore, it is clearthat the population has to increase with level of collectivism. For part c), again we can see that the only term inthe expression which depends on the death boundary − r is ¯ Q ( λ b , λ d , r, w ) . We know that as the death boundarydecreases ¯ Q ( λ b , λ d , r, w ) decreases as well (Lemma 2), thereby leading to an increase in the population. In part d),as the rate at which natural deaths occur decreases, the rate of deaths due to achieving poor welfare levels or hittingthe death boundary can increase. However, if the level of dependence on the society is low then the decrease inthe rate of natural deaths dominates, as a result the total population increases such that the mass of deaths equalsmass of birth. We now show this formally. Let us take the derivative of the term in the denominator w.r.t λ d , (1 + ¯ Q ( λ b , λ d , r, w )) + d ¯ Q ( λ b , λ d , r, w ) dλ d (1+ ¯ Q ( λ b , λ d , r, w ))( (1 − w − w. ¯ Q ( λ b , λ d , r, w )) + λ d rw ¯ Q ( λ b , λ d , r, w ) − λ d r ¯ Q ( λ b , λ d , r, w )(1 − w − w. ¯ Q ( λ b , λ d , r, w ))(1 − w − w. ¯ Q ( λ b , λ d , r, w )) + λ d rw ¯ Q ( λ b , λ d , r, w )(1+ ¯ Q ( λ b , λ d , r, w ))( (1 − w − w. ¯ Q ( λ b , λ d , r, w ))(1 − w − w. ¯ Q ( λ b , λ d , r, w ) − λ d r ¯ Q ( λ b , λ d , r, w )) + λ d rw ¯ Q ( λ b , λ d , r, w )(1 − w − w. ¯ Q ( λ b , λ d , r, w )) + λ d rw ¯ Q ( λ b , λ d , r, w ) ) If we can show that (1 − w − w. ¯ Q ( λ b , λ d , r, w ) − λ d r ¯ Q ( λ b , λ d , r, w )) > then the above expression will be positive.We know from lemma 2 that ¯ Q ( λ b , λ d , r, ≥ ¯ Q ( λ b , λ d , r, w ) , ∀ w ∈ [0 , . This leads to ¯ Q ( λ b , λ d , r, < − ww + λ d r which is a sufficient for the above derivative to be positive. It can be checked that this condition is satisfied if w < . Theorem 3: a) Cumulative welfare CF ( λ b , λ d , r, w ) decreases as the rate of birth λ b decreases. b) CF ( λ b , λ d , r, w ) decreases as the rate of natural deaths λ d increases. c) If λ d r ≤ (cid:15) < & w < − (cid:15) with (cid:15) > , then CF ( λ b , λ d , r, w ) decreases as the death boundary − r decreases. d) CF ( λ b , λ d , r, w ) decreases as the level of collectivism w increases. Proof:
For part a), we know that CF ( λ b , λ d , r, w ) = ¯ X ( λ b , λ d , r, w ) P op ( λ b , λ d , r, w ) . Also, since the averagewelfare of an individual is independent of λ b we only need to consider the effect on total population which wealready know from Theorem 2. For part b), let us simplify the expression of cumulative welfare, CF ( λ b , λ d , r, w ) =( r + λ d ) . λ b λ d . ¯ Q ( λ b ,λ d ,r,w )(1+ ¯ Q ( λ b ,λ d ,r,w )) . From this expression we can see that as λ d increases the term ( r + λ d ) . λ b λ d willdefinitely decrease. In fact the other term will also decrease, as can be seen from the derivative of the secondterm w.r.t. λ d , Q ( λ b ,λ d ,r,w )) d ¯ Q ( λ b ,λ d ,r,w ) dλ d and this combined with Lemma 2. For part d), we can see thatonly ¯ Q ( λ b ,λ d ,r,w )(1+ ¯ Q ( λ b ,λ d ,r,w )) depends on the weight w and its derivative w.r.t. w is Q ( λ b ,λ d ,r,w )) d ¯ Q ( λ b ,λ d ,r,w ) dw . Thisexpression of the derivative and Lemma 2, lead us to the result. For part c), as the death boundary decreases, theotal population in the society increases whereas the average welfare of an individual decreases, leading to opposingeffects. Therefore, if the λ d r is sufficiently low then the proportion of the population with bad quality is sufficientlylow as well. Also, if the level of collectivism, w is low then then the rate at which the welfare of bad qualityindividuals decays with time is high, hence the effect of decreasing the death boundary on the average welfare ishigh. Under these conditions the decrease in average welfare dominates the increase in population. We next showthis formally. The derivative of cumulative welfare w.r.t. r is given as, (cid:16) ¯ Q ( λ b , λ d , r, w )¯ Q ( λ b , λ d , r, w ) + 1 (cid:17) . ( (1 − w − w. ¯ Q ( λ b , λ d , r, w ) − ( λ d r + 1)(1 − w − w. ¯ Q ( λ b , λ d , r, w )))(1 − w − w. ¯ Q ( λ b , λ d , r, w ) + λ d rwd ( d + 1) ) (cid:16) ¯ Q ( λ b , λ d , r, w )¯ Q ( λ b , λ d , r, w ) + 1 (cid:17) . ( ( w. ( ¯ Q ( λ b , λ d , r, w )) . ( − (1 − w. (1 + ¯ Q ( λ b , λ d , r, w )) + λ d r + λ d r ¯ Q ( λ b , λ d , r, w )) − λ d r (1 − w − w. ¯ Q ( λ b , λ d , r, w ) + λ d rwd ( d + 1) ) If − (1 − w. (1 + ¯ Q ( λ b , λ d , r, w )) + λ d r + λ d r ¯ Q ( λ b , λ d , r, w ) < then the above derivative is negative. Note ¯ Q ( λ b , λ d , r, < − w − λ d rw + λ d r is sufficient for this condition to hold and it leads to the following condition, w < − (cid:15) and λ d r ≤ (cid:15) < where (cid:15) > . This proves part c. Theorem 4. a) Average life time ¯ T ( λ b , λ d , r, w ) decreases with an increase in rate of natural deaths λ d . b) If λ d r > θ ∗ = ln(1 + √ ) , then ¯ T ( λ b , λ d , r, w ) increases with an increase in level of collectivism w else, it firstdecreases and then increases with an increase in level of collectivism w . c), If λ d r > θ ∗ , then ¯ T ( λ b , λ d , r, w ) increases with a decrease in death boundary − r else, it first decreases and then increases with a decrease in deathboundary − r . Proof:
The expression for the average life-time of an individual ¯ T ( λ b , λ d , r, w ) involves the computation of theaverage life-time of good quality individuals and bad quality individuals separately and then combining the twousing the conditional probabilities. Hence, ¯ T ( λ b , λ d , r, w ) = λ d + ( λ d ) ¯ Q ( λ b ,λ d ,r,w ) − ¯ Q ( λ b ,λ d ,r,w ) ¯ Q ( λ b ,λ d ,r,w )+1 . The derivative of ¯ T ( λ b , λ d , r, w ) w.r.t w can be expressed as λ d ¯ Q ( λ b ,λ d ,r,w ) +2 ¯ Q ( λ b ,λ d ,r,w ) − Q ( λ b ,λ d ,r,w )+1) d ¯ Q ( λ b ,λ d ,r,w ) dw . If ¯ Q ( λ b , λ d , r, < √ − then the above derivative is positive. This leads to the condition λ d r > ln(1 + √ ) . However, if λ d r < ln(1 + √ ) then ¯ Q ( λ b , λ d , r, > √ − and as a result the derivative is negative. However, ¯ Q ( λ b , λ d , r, w ) will decrease withincrease in w and it can be observed that at w = 1 , ¯ Q ( λ b , λ d , r, w ) will be zero, this is due to the fact that theindividuals completely depend on the society and the rate of growth is zero for all individuals. Hence, for some w = w ∗ the ¯ Q ( λ b , λ d , r, w ∗ ) = √ − where the life-time will take the minimum value. Therefore, we know that in theregion w > w ∗ , the life-time will increase. This explains part b). For part c), a similar explanation can be given. Theexpression for the derivative changes to λ d ¯ Q ( λ b ,λ d ,r,w ) +2 ¯ Q ( λ b ,λ d ,r,w ) − Q ( λ b ,λ d ,r,w )+1) d ¯ Q ( λ b ,λ d ,r,w ) dr and the rest of the explanationfollows from above and Lemma 2. For part a), we will first show that the average life-time of both a good and badquality individual decrease. Then, we will show that the proportion of the bad quality individuals increase. Since theaverage life-time of a bad quality individual is always lesser than that of a good quality individual, this will lead to a θ ∗ is a fixed constant which in general will depend on P ( Q = 1) , and when P ( Q = 1) = it is ln(1 + √ ) . ecrease in the average life-time unconditional on the quality of the individual. First of all the average life-time of agood quality individual is λ d and it decreases with λ d . Next, the average life-time of an individual with bad qualityis arrived at by computing the expectation of min { T (cid:48) , T ( λ b , λ d , r, w ) = r − w (1+ ¯ Q ( λ b ,λ d ,r,w )) , } where T (cid:48) is anexponential random variable with mean λ d . The life-time of a bad quality individual is λ d . (1 − e − λdr − w − w. ¯ Q ( λb,λd,r,w ) ) ,the derivative of this expression is − λ d . (1 − e − λdr − w − w. ¯ Q ( λb,λd,r,w ) − λ d r − w − w ¯ Q ( λ b ,λ d ,r,w ) e − λdr − w − w. ¯ Q ( λb,λd,r,w ) ) + r.w − w − w. ¯ Q ( λ b ,λ d ,r,w ) d ¯ Q ( λ b ,λ d ,r,w ) dλ d . The term (1 − e − λdr − w − w. ¯ Q ( λb,λd,r,w ) − λ d r − w − w ¯ Q ( λ b ,λ d ,r,w ) e − λdr − w − w. ¯ Q ( λb,λd,r,w ) ) hasto be positive since ( x + 1) e − x < . Hence, we can see that the derivative is negative which implies the result. Theorem 5.
The average inequality
V ar X ( λ b , λ d , r, w ) is always more in an individualistic society w = 0 as compared to a collectivistic society w = 1 . Also if the person only dies a natural death, i.e. r → ∞ , then a) lim r →∞ V ar X ( λ b , λ d , r, w ) decreases with an increase in level of collectivism w and b) lim r →∞ V ar X ( λ b , λ d , r, w ) decreases with an increase in rate of natural deaths λ d . Proof:
V ar X ( λ b , λ d , r, w = 1) = 0 since all the individuals have the same welfare value of zero. So, we needto show that V ar X ( λ b , λ d , r, w = 0) > . The expression for variance is, V ar X ( λ b , λ d , r, w = 0) = ( 1 λ d ) (8 e λ d r + e λ d r ( − λ d r ) + 4 λ d r − − λ d r + ( λ d r ) + 1)(2 e λ d r − It can be shown that the expression in the numerator of the above expression is indeed positive. To do sowe show that at any point ( λ d , r ) ∈ R the partial derivative w.r.t to either λ d or r is positive and also that V ar X ( λ b , λ d = 0 , r = 0 , w = 0) > which helps us establish the result.For part a), the case when an individual only dies a natural death there is a symmetry in the proportion ofindividuals with good and bad quality. Hence, the average quality of an individual is zero. Therefore, the rate ofdecay for an individual with bad quality is − w and the same is the rate of growth for an individual with goodquality. Hence, increasing w slows the rate of decay and growth, thereby allowing individuals to neither take toolow or too high welfare values, which leads to a lower average disparity. Formally if r → ∞ , ¯ Q ( λ b , λ d , r, w ) = 0 ,this leads to the density distribution given as, f λ b ,λ d ,r,w ( x ) = λ d − w e − λd − w x and f − λ b ,λ d ,r,w ( x ) = λ d − w e λd − w x . Thisleads to the expression of the variance given as, ( − wλ d ) and therefore, part a) and b) follow directly from this.R EFERENCES[1] G. Hofstede, “Culture and organizations,”
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