Towards a Theory of Societal Co-Evolution: Individualism versus Collectivism

Toards 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 Theorem 1: Every society has a unique steady state. I. APPENDIX Proof: We ill start by deriving the population density at a given elfare level x, p λb,,r,(x) and the total population mass P op(, λ b, r, ) in the steady state and sho that they are unique. To do so e first arrive at the expression for the normalized population density f λb,,r,(x). The relation beteen f λb,,r,(x), p λb,,r,(x) and P op(, λ b, r, ) is given as, P op(, λ b, r, ) = p λ b,,r,(x)dx, f λb,,r,(x) = p λ b,,r,(x) P op(,λ b,r,). In steady state the average impact of the society, i.e. Q(λb,, r, ) is determined since the proportion of individuals ith Q = q, i.e. M(Q = q) do not change. Hence, the rate at hich the elfare of an individual gros can take only to values depending on his quality, R 1 = (1 ).1 +. Q(λ b,, r, ), R 1 = (1 ). 1 +. Q(λ b,, r, ), here R 1 and R 1 are the rate of groth of good and bad quality individual respectively. To derive the densities in steady state, e ill first sho that in the steady state R 1 and R 1 ill be positive and negative respectively. Let s assume that R 1 and R 1 are both positive, i.e. all the individuals in the society experience a positive groth. In such a case the individuals can only die due to a Poisson arrival. Also, e kno that an individual ho is born is as likely to be good as he is to be bad. Hence, the population mass at hich the rate of death ill equal the rate of birth of good/bad quality individual is the same for both the types of individuals, i.e. M(Q = +1) = M(Q = 1). As a result, the average quality Q(λ b,, r, ) = 0. Substituting this back in the expressions for the rate e get, R 1 = (1 ) and R 1 = (1 ). 1. Therefore, R 1 is negative this contradicts the supposition that the both the rates are positive. Next, let s assume that both R 1 and R 1 are negative. In this case the individuals can die either due to a Poisson arrival or due to hitting the death boundary. In such a case the elfare values attained ill only be negative. Let fλ 1 b,,r,(x) correspond to the joint density that the individual of good quality attains a elfare level of x. Similarly, e can define f 1 λ b,,r,(x) to be the joint density for a bad quality individual at a given elfare level of x. In steady state although the density of population in a given elfare level is fixed, hoever the individuals comprising the density at a given elfare level is not the same oing to change of elfare levels, births and deaths that happen continually. As a result, at any instant of time the mass of individuals that attain a given elfare level ill equal the mass of indiduals that leave that elfare level either due to change in elfare or due to dying. Consider an infinitesimal interval h, the mass of the population ith quality Q = 1 beteen

x h and x at time t, here x 0, is given as, fλ 1 b,,r,(x).h. Consider a time interval t after hich this mass of individuals, fλ 1 b,,r,(x).h ill either die or ill attain a different elfare level beteen, y h and y, here y = x+r 1.t. The probability that an individual does not die a natural death in time interval t is e t. Hence, the proportion of the mass of individuals ho do not die a natural death and a result attain a elfare beteen y h and y is e λt fλ 1 b,,r, (x).h = f λ 1 b,,r, (y).h. This can be expressed as f λ 1 b,,r, (y) = e y x R 1 fλ 1 b,,r,(x) and f 1 λ b,,r, (y) = C 1.e y R 1 here fλ 1 b,,r, (0) = C 1. Similarly, for y 0 e can get f 1 λ b,,r, (y) = C 1.e here f 1 λ b,,r, (0) = C 1. Note that both fλ 1 1 b,,r,(x) and fλ b,,r,(x) are zero for positive elfare values since both good and bad quality individuals are assumed to have a negative rate of groth. Also, the rate at hich individuals of good quality and bad quality are born is the same given as λ b 2. Hence, e can equate the mass of good (bad) quality individuals hich enter the society in time δt, i.e. y R 1 λ b 2 δt to the mass of individuals beteen elfare level of 0 and δx 1 (0 and δx 2 ), i.e. C 1 δx 1 (C 1 δx 1 ). This gives, C 1 R 1 = C 1 R 1 = C. Since the fλ 1 1 b,,r,(x) and fλ b,,r,(x) are joint density functions the integral of the sum of these joint densities should be 1. f 1 λ b,,r, (x)dx + f 1 λ b,,r, (x)dx = 1 C 1R 1 (1 e λd C = R 1 r ) + C 1R 1 λ d r r 2 e R 1 e R 1 (1 e λd R 1 r ) = 1 From this e can calculate the mass of the individuals ith Q = 1 and Q = 1, i.e. M(Q = +1) = C (1 e R 1 r ) and M(Q = 1) = C (1 e R 1 r ). Since R1 > R 1 e can see that M(Q = +1) > M(Q = 1). This yields that the Q(λ b,, r, ) > 0 and thereby R 1 > 0. This contradicts the supposition that both the rates are negative. Also, since R 1 > R 1 the only case left is R 1 is positive hile R 1 is negative. In this case the good and bad quality individuals take positive and negative elfare values respectively. We can calculate the joint densities in the same manner as described above and thus the resulting density is f 1 λ b,,r, (x) = C 1e R 1 x, x > 0 and f 1 λ b,,r, (x) = C 1e λd R 1 x, x < 0, ith C 1 R 1 = C 1 R 1. To solve for the constants e need to proceed in a similar manner as above: C 1R 1 f 1 λb,,r, (x)dx + f 1 λ b,,r, (x)dx = 1 + C 1R 1 (1 e λd (1 ).1 Q(λ b,,r,). r ) = 1 C = 2 e λ (1 ).1 Q(λ b,,r,). r ) For simplification of notation, e introduce auxiliary notation, λ 1 = (1 )+. Q(λ b,,r,) and λ 2 = (1 ). Q(λ b,,r,). Hence, the density functions are denoted as follos, fλ 1 b,,r, (x) = λ 1 e λ1x, x > 0 and f 1 2 e λ 2 r λ b,,r, (x) = λ 2 e λ2x, x < 0. Also, e can deduce that the marginal density f 2 e λ 2 r λb,,r,(x) = fλ 1 b,,r,(x), x > 0 and f λb,,r,(x) = f 1 λ b,,r, (x), x < 0. Using the density computed above e can calculate M(Q = 1) = 1 2 e λ 2 r and M(Q = 1) = 1 e λ 2 r 2 e λ 2 r. Also, the average quality needs to be consistent ith the average quality computed

using the distributions derived above. This is formally stated as Q(λ b,, r, ) = r e 1. Q(λ b,,r,) 2 e r 1. Q(λ b,,r,) = e λ2r 2 e λ2r (1) Next, e compute the total population mass by equating rate of births to the rate of deaths. The rate of deaths is comprised of to terms, the first term is the rate of natural deaths occurring due to Poisson shocks and the next term is the rate of deaths due to hitting the death boundary, f λb,,r,( r).(1. Q(λ b,, r, )) corresponds to the density of the individuals hitting the death boundary per unit time. Hence, the rate of deaths is.p op(λ b,, r, ) + f λb,,r,( r).(1. Q(λ b,, r, )).P op(λ b,, r, ),. Equating rate of births to rate of deaths e get the folloing. λ b = ( + f λb,,r,( r).(1. Q(λ b,, r, ))P op(λ b,, r, ) e λ2r λ b = ( +. 2 e λ2r ).P op(λ b,, r, ) P op(λ b,, r, ) = λ b (1 + Q(λ b,, r, )) No that e have both the normalized density and the total population s expressions, e can arrive at the expression of the population density p λb,,r,(x) hich is just a product of the to, formally given as follos. λ P op(λ b,, r, ). 1 e λ1x, if x > 0 2 e p λb,,r,(x) = λ 2 r λ P op(λ b,, r, ). 2 e λ2x, if x<0 2 e λ 2 r If e can sho that there is a unique average quality, Q(λb,, r, ) satisfying (1) then both the total population mass (2) and the population density (3) are uniquely determined. We kno that Q { 1, 1} hence, Q(λ b,, r, ) [ 1, 1]. To solve for Q(λ b,, r, ), e need to solve z = g(z), here g(z) = e r (1 ) z. r 2 e r 1.z r (2) and z [ 1, 1]. We ill first sho that there exists a solution in the set, [ 1, 1]. Let z 1 = 1 and z 2 = min{1, 1 }. If < 1 2 then, z 2 = 1 else z 2 = 1. g(z 1) = e r 1 2 e r 1 and g(z 1 ) > z 1. If < 1 2 then g(z 2) = r e 1 2 2 e r 1 2 hich is less than or equal to z 2 = 1, i.e. g(z 2 ) z 2. Based on this and since the function z and g(z) are continuous in the range [ 1, 1 ), there has to be a point in the interval [ 1, 1] [ 1, 1 ) here g(z) = z. Also, g(z) is decreasing in the range [ 1, 1 ), this can be seen from the expression for g (z) = r r 2e (1 z) 2 (2e 1 z 1) 2 is strictly increasing function. Therefore, g(z) z is a strictly decreasing function in [ 1, 1 r 1 z and z ), hich implies that the root is unique. When = 1 2, z 2 = 1 e can see that g(z 1 ) > z 1 holds, but g(z) is not continuous at z 2. This is not a problem as e kno that the function is continuous everyhere from [ 1, z 2 ) and lim z z 2 g(z) = 0, here lim z z 2 g(z) corresponds to the left hand limit, hence lim z z 2 g(z) < z 2. Hence, the same argument as above can be applied. In the case hen > 1 2 then e ill sho that there exists a unique solution for g(z) = z in the range [ 1, 1]. We kno that g(z 1 ) = r e 1 2 e r 1, but since > 1 2 e need to be careful about the case hen = 1. For no e can assume that 1 2 < < 1. Hence, e kno that g(z 1) > z 1. Here z 2 = 1 and g(z) ill not be

continuous at z 2. But e can sho that lim z z 2 g(z) = 0, here lim z z 2 g(z) corresponds to the left hand limit, and lim z z 2 g(z) < z 2. Hence, from the decreasing nature of g(z) z e kno that there is a unique solution in the range [ 1, 1 ). Since 1 > > 1 2 1 then [ 1, ) [ 1, 1] e need to sho that there is no solution in the range ( 1 1, 1]. In the range (, 1] the function g(z) is not necessarily continuous. There exists a discontinuity if 2e r 1 z 1 = 0 and z ( 1, 1]. Let s assume that there is a discontinuity. In that case, the function g(z) ill decrease values from 1 to, then to the right of the discontinuity at 2e decreases from to 1 e can say that 1 r 2e 1 2 1 r 2e 1 2 1. Since > 1 2 and 2e r 1 z r 1 z 1 = 0 for some z ( 1 1 = 0 the function λdr, 1] 1 > 2e 1 2 1 > 0 > 1. Hence, there is no point in the range in [ 1, 1] hich intersects ith this function. In the case, hen there is no discontinuity it is straightforard to sho that there is no solution of g(z) = z as the function g(z) ill only take negative values less than 1. Also, hen = 1 the individuals elfare is fixed to zero all the time, hence there is a symmetry in the proportion of good and bad quality individuals, hich leads to a unique solution Q(λ b,, r, ) = 0. Lemma 1. Good and bad quality individuals attain positive and negative elfare values respectively. Proof: The proof of theorem 1, already contains the proof for this lemma as e sho that R 1 and R 1 attain positive and negative elfare values respectively. Lemma 2. The average quality Q(λ b,, r, ) and the average elfare X(λ b,, r, ) of an individual a). Decrease as the level of collectivism, is increased., b). Decrease as the rate of natural deaths, increases., c). Decrease as the the death boundary, r decreases. Proof: We already kno that the solution for Q(λb,, r, ) requires solving a transcendental equation (1), hich means that e do not have a closed form analytical expression for it. It can be shon that the expression for X(λ b,, r, ) expressed in terms of the Q(λ b,, r, ) is (r + 1 ). Q(λ b,, r, ). From Theorem 1, e kno that for every set of parameters there does exist a solution Q(λ b,, r, ). For part a), as the level of collectivism is increased let us assume that the average quality Q(λ b,, r, ) increases. Hoever, if there is an increase in both the collectivism and the average quality, the expression g( Q(λ b,, r, )) = r e 1. Q(λ b,,r,) 2 e r 1. Q(λ b,,r,) decreases hich contradicts the increase in Q(λ b,, r, ). Hence, Q(λb,, r, ) has to decrease ith an increase in collectivism. And from the expression of X(λb,, r, ) expressed in terms of Q(λb,, r, ) it is straightforard that the average elfare also decreases ith an increase in the level of collectivism. For part b), again as the rate of natural deaths increases assume that Q(λb,, r, ) increases. Hoever the decrease in e r 1. Q(λ b,,r,) 2 e r 1. Q(λ b,,r,) ill contradict the assumption. With an increase in the first term in the expression of X(λb,, r, ) hich inversely related to has to decrease, this combined ith the decrease in Q(λ b,, r, ) leads to a decrease in the average elfare. For part c), e arrive at the expression of the derivative of average quality Q(λ b,, r, ).r.t. r, (d)(d+1)(1 d) (1 d) 2 + r(d+1) hich is negative. Hence, e kno that the average quality indeed decreases ith an increase in r. For average elfare e give an intuitive explanation first, increasing r decreases the average quality as a result of hich the groth of a good quality individual slos don and the decay of a bad quality individual becomes faster. As a result the average elfare levels attained by a good and bad quality individual are

loer. Moreover, increase in r increases the proportion of the bad quality individuals hich further has a negative effect on the average elfare. To prove this formally e ill sho that the average elfare of both good and bad quality individuals decreases and the proportion of the bad quality individuals increases. Since the average elfare value of a bad quality individual is alays loer than that of a good quality individual this is sufficient to sho the result. The average elfare of good quality individuals is given as 1 λ 1 = 1 + Q(λ b,,r,). This can be derived as follos, the distribution of the elfare conditional on the fact that individuals are of good quality f λb,,r,(x Q = +1) can be shon to be an exponential distribution ith parameter λ 1 exactly on the same lines as e derived the joint densities f 1 λ b,,r, (x) in Theorem 1. Since Q(λ b,, r, ) decreases as a function of r, the average elfare of a good quality individual also decreases as a function of r. Similarly e need to arrive at the distribution f λb,,r,(x Q = 1),hich turns out to be f λb,,r,(x Q = 1) = λ 2 1 e λ 2 r e λ2x, x < 0. The average elfare value of bad quality individual can be arrived at using this distribution and it turns out to be, 1 λ 2 + re λ 2 r 1 e λ 2 r. As r is increased, Q(λb,, r, ) decreases and thus λ 2 decreases as ell. The partial derivative of average elfare of bad quality individual 1 λ 2 + re λ 2 r.r.t. r is given as eλ2r λ 2re λ2r 1 and this expression 1 e λ 2 r (e λ 2 r 1) 2 turns out to be negative for (λ 2, r) R 2 +. Also, it can be shon that the partial derivative of 1 λ 2 + re λ 2 r.r.t λ 2 is given as ( 1 λ 2 ) 2 1 e λ 2 r r2 e λ 2 r (e λ 2 r 1) 2 and this expression turns out to be positive. Hence, from the sign of these partial derivatives e can easily see the result. Theorem 2. a) Total population P op(λ b,, r, ) increases as the rate of birth λ b increases. b) P op(λ b,, r, ) increases as the level of collectivism increases. c) P op(λ b,, r, ) increases as the death boundary r decreases. d) If < 1 2 then P op(λ b,, r, ) increases as the rate of natural deaths decreases. Proof: In order to compute the total population in the steady state, e need to have the rate of birth equals the rate of death hich is formally stated as follos, ( + f λb,,r,( r).(1. Q(λ b,, r, ))).P op(λ b,, r, ) = λ b P op(λ b,, r, ) = λ b.(1 + Q(λ b,, r, )) For part a), Q(λb,, r, ) does not depend on the rate of births and it is clear that the result holds since the population is directly proportional to λ b. For part b) as ell it can be seen that the only term in the expression hich depends on is Q(λ b,, r, ) hich ill decrease as is increased (Lemma 2). Therefore, it is clear that the population has to increase ith level of collectivism. For part c), again e can see that the only term in the expression hich depends on the death boundary r is Q(λ b,, r, ). We kno that as the death boundary decreases Q(λ b,, r, ) decreases as ell (Lemma 2), thereby leading to an increase in the population. In part d), as the rate at hich natural deaths occur decreases, the rate of deaths due to achieving poor elfare levels or hitting the death boundary can increase. Hoever, if the level of dependence on the society is lo then the decrease in the rate of natural deaths dominates, as a result the total population increases such that the mass of deaths equals mass of birth. We no sho this formally. Let us take the derivative of the term in the denominator.r.t,

(1 + Q(λ b,, r, )) + d Q(λ b,, r, ) d (1+ Q(λ b,, r, ))( (1. Q(λ b,, r, )) 2 + r Q(λ b,, r, ) r Q(λ b,, r, )(1. Q(λ b,, r, )) (1. Q(λ b,, r, )) 2 + r Q(λ b,, r, ) (1+ Q(λ b,, r, ))( (1. Q(λ b,, r, ))(1. Q(λ b,, r, ) r Q(λ b,, r, )) + r Q(λ b,, r, ) (1. Q(λ b,, r, )) 2 + r Q(λ ) b,, r, ) If e can sho that (1. Q(λ b,, r, ) r Q(λ b,, r, )) > 0 then the above expression ill be positive. We kno from lemma 2 that Q(λ b,, r, 0) Q(λ b,, r, ), [0, 1]. This leads to Q(λ b,, r, 0) < 1 + r hich is a sufficient for the above derivative to be positive. It can be checked that this condition is satisfied if < 1 2. Theorem 3: a) Cumulative elfare CF (λ b,, r, ) decreases as the rate of birth λ b decreases. b) CF (λ b,, r, ) decreases as the rate of natural deaths increases. c) If r ɛ < 1 2 & < 1 2 ɛ ith ɛ > 0, then CF (λ b,, r, ) decreases as the death boundary r decreases. d) CF (λ b,, r, ) decreases as the level of collectivism increases. Proof: For part a), e kno that CF (λ b,, r, ) = X(λ b,, r, )P op(λ b,, r, ). Also, since the average elfare of an individual is independent of λ b e only need to consider the effect on total population hich e already kno from Theorem 2. For part b), let us simplify the expression of cumulative elfare, CF (λ b,, r, ) = (r + 1 ). λ b. Q(λ b,,r,) (1+ Q(λ b,,r,)). From this expression e can see that as increases the term (r + 1 ). λ b definitely decrease. In fact the other term ill also decrease, as can be seen from the derivative of the second term.r.t., 1 only (1+ Q(λ d Q(λ b,,r,) b,,r,)) 2 d ill and this combined ith Lemma 2. For part d), e can see that Q(λ b,,r,) (1+ Q(λ b,,r,)) depends on the eight and its derivative.r.t. is 1 (1+ Q(λ d Q(λ b,,r,) b,,r,)) 2 d. This expression of the derivative and Lemma 2, lead us to the result. For part c), as the death boundary decreases, the total population in the society increases hereas the average elfare of an individual decreases, leading to opposing effects. Therefore, if the r is sufficiently lo then the proportion of the population ith bad quality is sufficiently lo as ell. Also, if the level of collectivism, is lo then then the rate at hich the elfare of bad quality individuals decays ith time is high, hence the effect of decreasing the death boundary on the average elfare is high. Under these conditions the decrease in average elfare dominates the increase in population. We next sho this formally. The derivative of cumulative elfare.r.t. r is given as, ( Q(λb,, r, ) ).( (1. Q(λ b,, r, ) 2 ( r + 1)(1. Q(λ b,, r, ))) Q(λ b,, r, ) + 1 (1. Q(λ ) b,, r, ) 2 + rd(d + 1) ( Q(λb,, r, ) ).( (.( Q(λ b,, r, )).( (1.(1 + Q(λ b,, r, )) + r + r Q(λ b,, r, )) r Q(λ b,, r, ) + 1 (1. Q(λ ) b,, r, ) 2 + rd(d + 1)

If (1.(1 + Q(λ b,, r, )) + r + r Q(λ b,, r, ) < 0 then the above derivative is negative. Note Q(λ b,, r, 0) < 1 r + r and r ɛ < 1 2 is sufficient for this condition to hold and it leads to the folloing condition, < 1 2 ɛ here ɛ > 0. This proves part c. Theorem 4. a) Average life time T (λ b,, r, ) decreases ith an increase in rate of natural deaths. b) If r > θ = ln(1 + 2 2 )1, then T (λ b,, r, ) increases ith an increase in level of collectivism else, it first decreases and then increases ith an increase in level of collectivism. c), If r > θ, then T (λ b,, r, ) increases ith a decrease in death boundary r else, it first decreases and then increases ith a decrease in death boundary r. Proof: The expression for the average life-time of an individual T (λ b,, r, ) involves the computation of the average life-time of good quality individuals and bad quality individuals separately and then combining the to using the conditional probabilities. Hence, T (λb,, r, ) = 1 + ( 1 ) Q(λ b,,r,) Q(λ b,,r,) 2 Q(λ b,,r,)+1. The derivative of T (λ b,, r, ).r.t can be expressed as 1 Q(λb,,r,) 2 +2 Q(λ b,,r,) 1 d Q(λ b,,r,) (Q(λ b,,r,)+1) 2 d. If Q(λ b,, r, 0) < 2 1 then the above derivative is positive. This leads to the condition r > ln(1 + 2 2 ). Hoever, if r < ln(1 + 2 2 ) then Q(λ b,, r, 0) > 2 1 and as a result the derivative is negative. Hoever, Q(λb,, r, ) ill decrease ith increase in and it can be observed that at = 1, Q(λb,, r, ) ill be zero, this is due to the fact that the individuals completely depend on the society and the rate of groth is zero for all individuals. Hence, for some = the Q(λ b,, r, ) = 2 1 here the life-time ill take the minimum value. Therefore, e kno that in the region >, the life-time ill increase. This explains part b). For part c), a similar explanation can be given. The expression for the derivative changes to 1 Q(λb,,r,) 2 +2 Q(λ b,,r,) 1 d Q(λ b,,r,) (Q(λ b,,r,)+1) 2 dr and the rest of the explanation follos from above and Lemma 2. For part a), e ill first sho that the average life-time of both a good and bad quality individual decrease. Then, e ill sho that the proportion of the bad quality individuals increase. Since the average life-time of a bad quality individual is alays lesser than that of a good quality individual, this ill lead to a decrease in the average life-time unconditional on the quality of the individual. First of all the average life-time of a good quality individual is 1 is arrived at by computing the expectation of min{t, T 2 (λ b,, r, ) = and it decreases ith. Next, the average life-time of an individual ith bad quality r 1 (1+ Q(λ b,,r,)), } here T exponential random variable ith mean 1. The life-time of a bad quality individual is 1.(1 e the derivative of this expression is 1 λ 2 d.(1 e r 1. Q(λ b,,r,) r. 1. Q(λ d Q(λ b,,r,) b,,r,) d. The term (1 e λdr r 1 Q(λ b,,r,) e 1. Q(λ b,,r,) λ d r 1 Q(λ b,,r,) e is an λdr 1. Q(λ b,,r,) ), λdr 1. Q(λ b,,r,) ) + λdr 1. Q(λ b,,r,) ) has to be positive since (x + 1)e x < 1. Hence, e can see that the derivative is negative hich implies the result. Theorem 5. The average inequality V ar X (λ b,, r, ) is alays more in an individualistic society = 0 as compared to a collectivistic society = 1. Also if the person only dies a natural death, i.e. r, then a) lim r V ar X (λ b,, r, ) decreases ith an increase in level of collectivism and b) lim r V ar X (λ b,, r, ) decreases ith an increase in rate of natural deaths. 1 θ is a fixed constant hich in general ill depend on P (Q = 1), and hen P (Q = 1) = 1 2 it is ln(1 + 2 2 ).

Proof: V ar X (λ b,, r, = 1) = 0 since all the individuals have the same elfare value of zero. So, e need to sho that V ar X (λ b,, r, = 0) > 0. The expression for variance is, V ar X (λ b,, r, = 0) = ( 1 ) 2 (8e2r + e r ( 2( r) 2 + 4 r 8) 3 r + ( r) 2 + 1) (2e r 1) 2 It can be shon that the expression in the numerator of the above expression is indeed positive. To do so e sho that at any point (, r) R 2 +the partial derivative.r.t to either or r is positive and also that V ar X (λ b, = 0, r = 0, = 0) > 0 hich helps us establish the result. For part a), the case hen an individual only dies a natural death there is a symmetry in the proportion of individuals ith good and bad quality. Hence, the average quality of an individual is zero. Therefore, the rate of decay for an individual ith bad quality is 1 and the same is the rate of groth for an individual ith good quality. Hence, increasing slos the rate of decay and groth, thereby alloing individuals to neither take too lo or too high elfare values, hich leads to a loer average disparity. Formally if r, Q(λb,, r, ) = 0, this leads to the density distribution given as, f 1 λ b,,r, (x) = λ λd d 1 e 1 x and f 1 λ b,,r, (x) = 1 e λd 1 x. This leads to the expression of the variance given as, ( 1 ) 2 and therefore, part a) and b) follo directly from this.