Evolution, Heritable Risk, and Skewness Loving
EEvolution, Heritable Risk, and Skewness Loving ∗ Yuval Heller † and Arthur Robson ‡ August 7, 2020
Abstract
Our understanding of risk preferences can be sharpened by considering their evo-lutionary basis. The existing literature has focused on two sources of risk: idiosyn-cratic risk and aggregate risk. We introduce a new source of risk, heritable risk, inwhich there is a positive correlation between the fitness of a newborn agent and thefitness of her parent. Heritable risk was plausibly common in our evolutionary pastand it leads to a strictly higher growth rate than the other sources of risk. We showthat the presence of heritable risk in the evolutionary past may explain the tendencyof people to exhibit skewness loving today.
JEL Classification : D81, D91.
Keywords : evolution of preferences, risk atti-tude, risk interdependence, long-run growth rate, fertility rate.Final pre-print of a manuscript accepted for publication in
Theoretical Economics . Our understanding of risk preferences can be sharpened by considering their evolutionarybasis (see Robson and Samuelson, 2011, for a survey). This claim was advanced in theeconomics literature by Robson (1996), for example, who presented a model in which ∗ We thank Erol Akcay, Gilad Bavly, Ben Golub, Aviad Heifetz, Laurent Lehmann, John McNamara,Jonathan Newton, Debraj Ray, Roberto Robatto, Larry Samuelson, Bal`azs Szentes, the editor, two anony-mous referees, and LEG2019 conference participants for helpful comments. We thank Ron Peretz for hiskind help in extending Theorem 1. We thank Renana Heller for writing the simulation used in Section 6.1. † Bar-Ilan University, Department of Economics. (email: [email protected], homepage:https://sites.google.com/site/yuval26/). Heller is grateful to the European Research Council for its fi-nancial support (ERC starting grant ‡ Simon Fraser University, Department of Economics (email: [email protected], homepage:https://sites.google.com/view/arthurrobson/home/). Robson thanks the Social Sciences and HumanitiesResearch Council of Canada for support. a r X i v : . [ ec on . T H ] A ug ach agent lives a single period and faces a choice between lotteries over the number ofoffspring. (See also related models in Lewontin and Cohen, 1969; McNamara, 1995.) Someof the feasible lotteries involve aggregate risk (when all agents obtain the same realization).Robson (1996) showed that idiosyncratic risk (independent across individuals) induces ahigher long-run growth rate (henceforth “growth rate”) than aggregate risk, and as a resultnatural selection should induce agents to be more risk averse with respect to aggregate risk. This result has been put into an intriguing new light by Robatto and Szentes (2017)who reconsider the model in continuous time. In such a framework it is appealing toformulate both consumption and the production of offspring as rates . Once this is doneaggregate risk becomes equivalent to idiosyncratic risk as long as fertility and mortalityare age-independent. (See Robson and Samuelson, 2019, and Section 7 of this paper.)The way in which idiosyncratic risk has been modeled in the previous literature captureswell coin flips concerning fertility that only affect a particular individual. However, itis compelling that, in the evolutionary past, there were plausibly many cases in whichthe “outcome of the flip” persisted from parents to offspring. In this paper we capturethis persistence by introducing a new source of risk, heritable risk, which is basicallyidiosyncratic risk, but allows a positive correlation between the fitness of a newborn agentand the fitness of her parent.Heritable risk in this sense must have been common in the evolutionary past of humanbeings. Such risk is induced if the agent’s fitness is heritable due to imitation of theparent’s behavior or genetic inheritance. For example, a foraging technique in prehistorichunter-gatherer societies would be inherited if an individual copied her parent’s technique.Alternatively, risk is heritable if the choice an individual makes is controlled genetically,and this gene is passed down from mother to daughter. The key properties are just: (1)there is a positive correlation between the fitness of an agent and that of her parent, and(2) by contrast, there is little correlation between the fitness of two randomly chosen agentsin the population.We show that this heritable risk yields a strictly higher growth rate than the othersources of risk. We derive this result in Robatto and Szentes’s (2017) setup, as it is morestriking to see the advantage of heritable risk in a setup in which all other sources of riskare equivalent. It is relatively simple to show that heritable risk is also advantageous inother setups considered in the literature. See Heller (2014) for a discussion of why this might explain people’s tendency to overestimate theaccuracy of their private information. ighlights of the model Consider a simple setup in which agents occasionally redrawa lottery over their consumption rate, and the realized consumption determines the fertilityrate through a concave increasing function ψ . Specifically, assume that the lottery can yielda high consumption rate ( c h , inducing a fertility rate r h = ψ ( c h )) with probability q h or alow consumption rate ( c l , inducing a fertility rate r l = ψ ( c l )) with probability q l = 1 − q h .Each agent redraws her realized level of fertility at an annual rate of λ . For simplicityassume that there is no mortality. Our crucial departure from the existing literature isto assume that a newborn agent inherits the realized fertility rate of her parent and thevalues remain the same until either the parent or the offspring redraws their fertility rate. Key result
Theorem 1 shows that the growth rate x ∗ induced by heritable risk is both(1) strictly higher than the lottery’s expected fertility rate µ ≡ q ‘ · r ‘ + q h · r h , but x ∗ → µ as λ → ∞ , and (2) strictly below the highest realization r h , but x ∗ → r h as λ →
0. Tosee the intuition behind (2), consider the case where λ > r h .Our result has two main implications: (1) heritable risk induces a higher growth ratethan either aggregate risk or idiosyncratic risk (both of which induce a growth rate that isequal to the lottery’s expectation µ ), and (2) this difference in the growth rates is especiallylarge when dealing with positively skewed lotteries (since the growth rate can be made closeto r h in a way that is independent of the probability q h ).One can interpret our result as follows. The long-run impact of risk interdependencedepends on the “direction” of the interdependence (vertical or horizontal). The formof risk we introduce induces correlation between an agent’s outcome and her offspring’soutcome. This “vertical correlation” is helpful to the growth rate, as it allows successfulfamilies to have fast exponential growth. By contrast, this risk does not involve “horizontalcorrelation” of risk between agents of the same cohort, which would be harmful to thegrowth rate. The insight that vertical correlation increases the growth rate, but horizontalcorrelation decreases it, may be applicable in other domains of economics and finance. Risk attitude
We assume that individuals in our evolutionary past had different types,and that the agent’s type determines her risk attitude—in particular, how the agent choosesbetween a risky consumption option and a safe one. An agent is likely to have the same3ype as her parent due to genetic inheritance. Occasionally, new types are introduced intothe population following a genetic mutation. Observe that the population share of agentsof the type that induces the highest long-run growth rate will grow, until, in the long run,almost all agents are of this type.In Section 5 we show that our key result implies that the type with the highest growthrate is (1) risk averse with respect to most lotteries over consumption (due to the concavityof the function ψ relating consumption and fertility), but (2) risk loving with respect tosufficiently positively skewed lotteries. Since biological types evolve slowly, it is likely thatthis risk attitude persists in modern times, even though the birth rate may no longerbe increasing in the consumption rate. This finding fits the stylized empirical fact thatpeople, although being in general risk averse, are skewness loving. That is, people likelotteries involving a small probability of winning a high prize. (See, for example, Golecand Tamarkin, 1998; Garrett and Sobel, 1999.) Structure
The rest of the paper is organized as follows. Section 2 informally presentsthe essence of our key result. The model is presented in Section 3. Section 4 formallypresents our key result. In Section 5 we discuss the implications of our result for attitudesto risk. Section 6 extends our baseline model by allowing dependency between redraws ofheritable risk within each of a number of dynasties, with independence across dynasties,which seems plausible in various applications. We show that this extension does not affectour results for infinite populations. By contrast, this structure can affect the growth rateof finite populations, which we investigate by numerical simulations. We discuss severaladditional related references in Section 7 and conclude in Section 8.
The following example conveys the gist of our key result. Consider three populations, eachhaving a random fertility rate (which is independent of the agent’s age) with the samemarginal distribution. Each population has a probability q ‘ of having a low fertility rateof r ‘ , and a probability q h = 1 − q ‘ of having a high fertility rate of r h . For notationalcompactness, we now take as implicit the dependence of fertility on consumption rates c i , i = ‘, h . For simplicity, we focus on fertility, so that there is no mortality. The sourceof risk is independent across populations.In Population 1 risk is idiosyncratic; that is, the fertility rate of each agent is inde-pendent of the fertility rate of all other agents in the populations and, in particular, of4er parent’s fertility rate. Applying the law of large numbers, the number of agents inPopulation 1 at time t is equal to N ( t ) = e ( q ‘ · r ‘ + q h r h ) · t , where N (0) = 1, and the annualgrowth rate is t · ln N ( t ) = q ‘ · r ‘ + q h r h ≡ µ .In Population 2 risk is aggregate. There are two states: ‘ and h . In state ‘ , all agentshave fertility rate r ‘ , and in state h , all agents have fertility rate r h . There is a continuousprobability rate λ that the state is redrawn. If it is, the fertility rate is r ‘ with probability q ‘ and r h with probability q h . What is the (long-run) growth rate of the population exposedto this aggregate risk? If N ( t ) is the population at time t , and N (0) = 1, it follows thatln N ( t ) t = r ‘ · (time in state ‘ ) + r h · (time in state h ) t −→ q ‘ · r ‘ + q h · r h = µ, as t → ∞ , given the evident ergodicity of the process. Thus, as shown in Robatto andSzentes (2017), both idiosyncratic risk and aggregate risk induce the same growth rate.We introduce a novel form of risk in Population 3, called heritable risk. Each agentredraws her heritable birth rate independently of all other agents at a rate λ , and at eachredraw the agent gets a fertility rate r ‘ or r h with probability q ‘ or q h = 1 − q ‘ , respectively(independently of all other events). The previous literature makes an implicit assumptionthat each offspring is given a fresh draw, and so all offspring are equivalent and evolutionarysuccess entails simply counting these undifferentiated offspring. By contrast, suppose thateach offspring inherits the realized fertility rate of the parent. Since offspring are nowdifferentiated, the value of these offspring varies with type and simply counting them isinadequate. Our key result shows that in this case the growth rate is strictly higher thanthe expectation µ , and indeed converges to r h as λ → τ periods, which is comparableto an arrival rate of λ = 1 /τ . As before, the redrawn values of different agents areindependent. On each draw, a share q ‘ of the agents get r ‘ and the remaining agentsget r h . If the initial population is of size 1, then, after a time k · τ , the population is N ( k · τ ) = ( q ‘ · e r ‘ · τ + q h · e r h · τ ) k , so that1 k · τ ln N ( k · τ ) = 1 τ · ln( q ‘ · e r ‘ · τ + q h · e r h · τ ) ≡ ¯ g ( λ ) . It follows that the growth rate of the population, ¯ g ( λ ), is decreasing in λ , ¯ g ( λ ) → r h if λ → τ → ∞ ), and ¯ g ( λ ) → µ ≡ q ‘ · r ‘ + q h · r h , if λ → ∞ ( τ → µ , which is the5igure 1: Long-Run Growth Rate for a Binary Lottery ( r h = 5%, r l = 0%, and q h = 10%).growth rate induced by either idiosyncratic risk or aggregate risk with the same marginaldistribution.Recall that x ∗ is the growth rate in the general model. What Theorem 1 shows, moreprecisely, is that x ∗ > µ and x ∗ > r h − λ . This latter result implies that x ∗ → r h as λ → x ∗ < r h . Figure 1 illustrates our result for the values r h = 5%, r l = 0%, and q h = 10%;i.e., for a binary lottery that yields a high annual birth rate of 5% with probability 10%and a zero birth rate with probability 90%. When risk is either idiosyncratic or aggregatethe (long-run) growth rate is equal to the expected birth rate µ = 0 . µ . The figure also draws the exact growth rate induced by heritablerisk according to the explicit formula presented in Claim 1 (in Appendix C) for binarylotteries. As can be seen from the figure, when the redraw rate λ is very small (resp.,large) with respect to r h , then the growth rate is slightly above r h − λ (resp., µ ). The simplifying assumption that the intervals between redraws are deterministic (rather than stochas-tic intervals induced by a Poisson process) decreases the growth rate, and thus the above example mightyield a lower growth rate than the lower bound r h − λ of Theorem 1. Model
Consider a continuum population of an initial mass one. Time is continuous, indexed by t ∈ R + . To simplify matters, we assume that reproduction is asexual. The growth processdepends on the parameters ( δ, ( X, q x , λ x ) , ( Y, q y , λ y ) , ( Z, q z , λ z )), as described below.In what follows, we first present an intuitive description of Poisson processes on theindividual level that incorporate the probability of each agent dying, giving birth, andchanging her birth rate (parts (i) below). We then specify the corresponding exact evolutionof the large population that is assumed in our model (parts (ii) below).
1. (i) We suppose intuitively that each agent experiences a constant Poisson death rate δ ≥ independent of all other random variables and, in particular, of allcomponents of the birth rates.(ii) We assume precisely that, in each infinitesimal period of time between t and t + dt , a fraction δ · dt of the population dies, where this fraction is uniform acrossall components of the birth rate.Each individual i at time t has a birth rate b i ( t ) = x i ( t ) + y i ( t ) + z ( t ) with threecomponents. These components are constructed as follows:2. (i) The random variable x i ( t ) ≥ x i has a finite support X = supp ( x ) = { x , ..., x n } , where x < ... < x n and n ≥
2. The function q x : X → (0 , , P x ∈ X q x = 1 , assigns aprobability to each x ∈ X. Intuitively, in each infinitesimal period of time dt eachagent has a probability of λ x · dt of redrawing her heritable birth rate (where λ x > w ( t ) is the total population at time t and w k ( t ) is the mass of agents who areendowed with heritable component x k . Then the rate of increase of w k ( t ) is dw k ( t ) dt = w k ( t ) x k − λ x w k ( t ) + λ x w ( t ) q x ( x k ) − δw k ( t ) . (1) The formalization of the intuitive claim that the idiosyncratic Poisson process for the birth rate of anindividual in a large population implies the mean is exactly attained raises various technical difficulties.See Duffie and Sun (2012) (and the citations therein) for details. w k ( t ) due to offspring who are endowedwith x k . This captures the key characteristic of heritable risk that all offspring areendowed with the same component x k as their parent. Since this term is independentof λ x , it will follow that w k ( t ) grows at rate x k − δ when λ x →
0. The second termexpresses the loss from w k ( t ) of those agents who redraw. The third term representsthe increase due to all agents from w ( t ) (including those from w k ( t )) who redraw andobtain x k . The final term represents the loss from w k ( t ) due to death.3. (i) The random variable y i ( t ) ≥ y i has a finitesupport Y = supp ( y ) = n y , ..., y n y o . The function q y : Y → (0 , , P y ∈ Y q y = 1,assigns a probability to each y ∈ Y. In each infinitesimal period of time dt eachagent has a probability of λ y · dt of redrawing her idiosyncratic birth rate, and theseredrawing events are independent of all other events.(ii) The precise assumption is that the idiosyncratic component within any groupof agents always reflects the distribution q y . That is, the share of agents with id-iosyncratic outcome y ‘ , for example, in the group of agents with heritable outcome x k is exactly equal to q y ( y ‘ ) for any time t ≥
0. This implies that the idiosyncraticcomponent in any group of agents with any heritable component x k is exactly equalto the expectation µ y .4. The aggregate component of the birth rate z ( t ) ≥ z i has a finite support Z = supp ( z ( t )) = { z , ..., z n z } .The function q z : Z → (0 , , P z ∈ Z q z = 1 , assigns a probability to each z ∈ Z. At time t = 0 the aggregate birth rate z (0) is randomly determined according tothe distribution q z . In each infinitesimal period of time between t and t + dt a newrandom value of the aggregate birth rate is drawn independently (according to q z )with a probability of λ z · dt , where λ z >
0. This aggregate birth rate applies to allindividuals in the entire population equally.8
Key Result
Let w ( t ) denote the mass of the population at time t . We normalize w (0) = 1. We say thatthe growth process of w ( t ) given by ( δ, ( X, q x , λ x ) , ( Y, q y , λ y ) , ( Z, q z , λ z )) has an equivalent(long-run) growth rate g ∈ R if and only if lim t →∞ ln w ( t ) t = g, almost surely.Let µ x = P k x k · q x ( x k ) (resp., µ y = P k y k · q y ( y k ), µ z = P k z k · q z ( z k )) be the expectationof the heritable (resp., idiosyncratic, aggregate) birth rate. We show that the equivalentgrowth rate is the sum of four components: g = f ( X, q x , λ x ) + µ y + µ z − δ. The resultson the idiosyncratic and aggregate components of the overall growth rate accord with theexisting literature (Robatto and Szentes, 2017), namely, these components are equal to µ y and µ z , respectively. The novel part of the result is that the heritable birth componentsatisfies f ( X, q x , λ x ) ∈ (max ( µ x , x n − λ x ) , x n ) . That is, the heritable birth component is always larger than µ x , and it cannot be morethan λ x away from the highest realization x n . The first property shows that the desirabilityof heritable risk is that it induces a higher growth rate than comparable aggregate oridiosyncratic risk. The second property shows that the highest realization of the heritablerisk has a substantial influence, regardless of how low is its probability. That is, a lotteryin which x n > λ x induces a growth rate of at least x n − λ x regardless how small q x ( x n ) and µ x might be.The intuition is that the distribution of the heritable birth rate in the populationconverges to a distribution p ∈ ∆ ( X ) that first-order stochastically dominates q x . This isbecause, at each point in time, agents with a high heritable birth rate tend to have moreoffspring and these offspring share the parent’s heritable birth rate. Hence, in a steadystate, the share of agents with a high heritable birth rate is strictly higher than q . Highervalues of λ reduce this effect, as the offspring redraw more rapidly a new value for theirheritable birth rate (according to q x ).The final claim is that f ( X, q x , λ x ) increases following a mean-preserving spread of theheritable birth rate. The intuition is that a mean preserving spread increases the high x k ’s while decreasing the low x k ’s, and there is a net gain from this due to the over-representation of high x k -agents in the steady-state distribution.9 heorem 1. Let ( δ, ( X, q x , λ x ) , ( Y, q y , λ y ) , ( Z, q z , λ z )) be a growth process. Then its equiv-alent growth rate is equal to g = f ( X, q x , λ x ) + µ y + µ z − δ, where, setting f ( X, q x , λ x ) = x ∗ for compactness, x ∗ is the unique positive solution of x ∗ = λ x n X k =1 q k · x k λ x + x ∗ − x k ∈ (max ( µ x , x n − λ x ) , x n ) , with q k ≡ q x ( x k ) for each k ∈ { , .., n } . It follows that x ∗ → P nk =1 q k x k = µ x as λ x → ∞ and x ∗ → x n as λ x → . Moreover, if ( X , q x ) is a mean-preserving spread of ( X, q x ) , then f ( X , q x , λ x ) >f ( X, q x , λ x ) .Sketch of proof; The full proof is in Appendix A. Since the novel result here concerns her-itable risk, let us suppose, for simplicity, that there is no aggregate risk, idiosyncratic risk,or mortality. Suppose further that the size of the population at time t is w ( t ) and thata steady-state fraction p k of this population has birth rate x k . The net increase in eachinfinitesimal period dt of those agents with birth rate x k is then p k · x k · w ( t ) · dt (offspringborn to parents with a birth rate x k who inherit this rate) minus ( p k − q k ) · λ x · w ( t ) · dt .(Note that λ x · w ( t ) · dt agents have redrawn a fresh value for the heritable birth rate, andthe share of x k -agents among them has changed from p k to q k .) The increase in the totalmass of agents is P k p k · x k · w ( t ) · dt (the sum of offspring born to parents with each birthrate). The equilibrium value of p should match the ratio of the net increase of agents witha high heritable birth rate to the net increase of the population, such that p k = ( p k · x k + ( q k − p k ) · λ x ) · w ( t ) · dt P k p k · x k · w ( t ) · dt = p k · x k + ( q k − p k ) · λ x P k p k · x k · . Solving for p k yields (where x ∗ ≡ P k p k · x k ): p k = λ x · q k λ x + x ∗ − x k . (2)This solution assumes that p k is positive for all k so that x ∗ > x n − λ x . Next we multiplyeach k -th equation by x k and sum to an equation in one unknown: x ∗ = X k x k · λ x · q k λ x + x ∗ − x k . (3) The formal proof deals with the general case, and shows global convergence to the steady state. x ∗ > x n − λ x the LHS (resp., RHS) is increasing (resp.,decreasing) in x ∗ , which implies that there exists a unique solution x ∗ > x n − λ x to Eq.(3). Substituting this solution in Eq. (2) yields the unique steady-state distribution p .From Eq. (3) it follows that x ∗ = X k x k · q k x ∗ − x k λ x → X k x k · q k , (4)as λ x → ∞ . Since x ∗ ∈ ( x n − λ x , x n ) , it is also immediate that x ∗ → x n as λ x → . The final claim is proved as follows. Eq. (3) can be written as E x " x · λλ + x ∗ − x = x ∗ , (5)where x is the random variable ( X, q x ). The fact that x · λλ + x ∗ − x is a convex function of x implies that it increases following a mean-preserving spread. This, together with thefact that it is decreasing in x ∗ , implies that in order to maintain Eq. (5) following amean-preserving spread, the growth rate x ∗ must increase. We suppose that individuals in a large population may have different types, where the typerepresents the agent’s risk attitude—in particular, how the agent chooses between a riskyconsumption option and a safe one. An agent has the same type as her parent. Occasion-ally, new types may be introduced into the population as genetic mutations. Observe thatthe population share of agents that are endowed with the type that induces the highestlong-run growth rate for its practitioners will grow, until, in the long run, almost all agentsare of this type. For example, suppose that there are two types θ, θ in the population,each with an initial frequency of 50% that induce growth rates g ( θ ) , g ( θ ), respectively.After time t the share of agents having type θ will be e g ( θ ) t e g ( θ ) t + e g ( θ ) t , which converges to oneas t → ∞ , if g ( θ ) > g ( θ ). See Robson and Samuelson (2011) and the citations therein,for a more detailed argument of why natural selection induces agents to have types thatmaximize the long-run growth rate.Now consider a setup in which agents face choices between various alternatives, whereeach alternative corresponds to a lottery over the consumption rate. We assume thatthe birth rate is a concave increasing function of consumption, given by ψ : R + → R + .11o simplify the presentation, assume that the birth rate is entirely heritable; the resultremains qualitatively the same if the birth rate induced by consumption has all three riskcomponents (heritable, idiosyncratic, and aggregate). We now argue that a growth-rate-maximizing type induces agents (1) to be risk averse with respect to most lotteries overconsumption, and, yet, (2) to strictly prefer some fair lotteries that are sufficiently skewed.Thus, natural selection should induce agents to have a risk attitude combining risk aversionand skewness loving.For simplicity, assume that an agent faces choices among lotteries over consumption( C, q ) with a finite support C , where c > c ∈ C. Suppose probabilities areassigned by q : C → [0 , , P c ∈ C q ( c ) = 1. Let m = max { c ∈ C } be the maximal possiblerealization and let ¯ c = P c ∈ C q ( c ) · c be the mean. For any fixed lottery, we show that,once ψ is sufficiently concave, the constant consumption rate of ¯ c will induce a higher long-run growth rate than the lottery ( C, q ). This explains why the growth-rate-maximizingtype should induce the agents to be risk averse with respect to most lotteries, when ψ issufficiently concave. Consider, for example, the function ψ ( c ) = c β for β ∈ (0 , C, q ) to the mean ¯ c when β = 1 so that ψ ( c ) = c . However, if β is small enough this preference is reversed. This is formalized in thefollowing proposition that shows that, given any lottery over consumption, the individualwill prefer the mean consumption to the lottery if β is small enough. Proposition 1.
Suppose that ψ ( c ) = c β for β ∈ (0 , . Then, given any gamble ( C, q ) , themean ¯ c = P c ∈ C q ( c ) · c induces a higher growth rate than the lottery ( C, q ) , if β > isclose enough to . Proof
See Appendix B.On the other hand, for a fixed function ψ ( c ), if the lottery ( C, q ) is sufficiently skewed—i.e., if m is high enough and q ( m ) is low enough so that ψ ( m ) − λ x > ψ (¯ c )—then thelottery induces a strictly higher growth rate than the constant consumption rate of ¯ c . Thisfollows from Theorem 1 since the lottery’s long-run growth rate is bounded from below by ψ ( m ) − λ x . This implies that growth-rate-maximizing agents would prefer a sufficientlypositively skewed lottery to its expectation.The above argument suggests that natural selection has induced people to be generallyrisk averse and sometimes skewness loving. As biological types evolve slowly, it seemslikely that this risk attitude persists in modern times, in which, arguably, the birth rate isno longer increasing in the consumption rate. Thus, our findings fit the stylized fact thatpeople, although being in general risk averse, are skewness loving, in the sense of being risk12oving with respect to lotteries involving a small probability of winning a high prize (e.g.,buying state lottery tickets; see Golec and Tamarkin, 1998; Garrett and Sobel, 1999). In our baseline model, the event of an agent redrawing her heritable birth rate is inde-pendent of her parent’s redrawing event. In various environments, it seems plausible thatmembers of a dynasty may change their heritable birth rate together, while remainingindependent of other dynasties. For example, if heritable risk is induced by a foragingtechnique or a geographical location, and environmental changes affect the effectiveness ofthe foraging technique, then an entire dynasty of agents (who use the same foraging tech-nique or live in the same geographical location) may simultaneously change their heritablebirth rate.In this section we extend our baseline model by introducing dynasties, and allowingdependency between redraws of heritable risk within each dynasty. We show that thisextension does not affect our results for infinite populations. By contrast, this structure canaffect the growth rate of finite populations, which we investigate by numerical simulations.
Extended model
In what follows we extend our baseline model to a continuum ofdynasties. We adopt the same notation as in the baseline model. The processes according towhich agents die, are born, and change their idiosyncratic and aggregate birth componentsremain the same as in the baseline model. Importantly, each offspring is born into thesame dynasty as her parent.Let [0 ,
1] be the set of dynasties, where each agent i in the initial population (of massone) lives in a different dynasty i ∈ [0 , q x . Formally, we assume that the mass of dynastieshaving heritable birth component x k ∈ X is equal to q x ( x k ) . The heritable birth componentof each agent is tied to the heritable birth component of all members of her dynasty.There are two processes that change the heritable birth component of agents. Webegin with an intuitive description of two Poisson processes that change the heritablebirth component: migration and a dynasty’s redraw. We then specify the correspondingexact evolution of the distribution of the heritable birth component as the product of thesetwo processes. 13.
Migration : Intuitively, in each infinitesimal time dt each agent has a probability of λ m · dt (where λ m ≥
0) to leave her dynasty and move to a new random dynasty(distributed uniformly in the set of all dynasties [0,1]). These migration events areindependent of all other events. Following the migration, the agent is endowed withthe heritable birth component of her new dynasty.2.
Dynasty’s redraw : Intuitively, in each infinitesimal time dt each dynasty j ∈ [0 , λ r · dt to redraw a fresh value for its heritable birth component.These redrawing events are independent of all other events. When a dynasty redrawsits heritable component it changes the heritable component of all agents living inthat dynasty.Next, we formulate the precise dynamics of the mass of agents w k ( t ) who are endowedwith the heritable component x k that is induced by the combined effect of migration anda dynasty’s redraws. The rate of increase of w k ( t ) is dw k ( t ) dt = w k ( t ) x k − λ m w k ( t ) + λ m w ( t ) q x ( x k ) − λ r w k ( t ) + λ r w ( t ) q x ( x k ) − δw k ( t ) . (6)The first and final terms are identical to Eq. (1) of the baseline model. The first termexpresses the increase in w k ( t ) due to offspring who are endowed with x k . The final termrepresents the loss from w k ( t ) due to death.The second and third terms express the impact of migration. The second term ( − λ m w k ( t ))is the loss from w k ( t ) of agents who migrate out of dynasties with heritable component x k . The third term ( λ m w ( t ) q x ( x k )) represents the increase due to all agents from w ( t )(including those from w k ( t )) who migrate into dynasties with heritable component x k .The fourth and fifth terms express the impact of redraws of dynasties. The fourth term( − λ r w k ( t )) represents the loss from w k ( t ) of agents who live in dynasties with heritablecomponent x k that redraw a fresh draw. Finally, the fifth term ( λ r w ( t ) q x ( x k )) representsthe increase due to all agents from dynasties (including dynasties that already had x k )that draw a fresh value of heritable component x k .Observe that Eq. (6) is equivalent to Eq. (1) of the basic model except that λ x isreplaced with λ m + λ r . That is, the dynamics of w k ( t ) in the extended model is exactlythe same as in the baseline model with λ x = λ m + λ r . As the impact of the heritablecomponent on the growth rate is exactly captured by w k ( t ), this implies that all of ourresults hold in this extended setup with dynasties.14 ver-growing population with dying dynasties Consider a simple case in which:(1) λ m = 0, i.e., agents never migrate, and each dynasty is an isolated subpopulation, (2)all risk is heritable, (3) the growth rate predicted by the continuum model is positive, and(4) an aggregate birth rate with the same marginal distribution induces a negative growthrate. For example, assume that the heritable birth rate of each dynasty is randomly chosento be either x l = 0% or x h = 2% with equal probability, that there is a constant death rateof δ = 1 . λ r = 2%. Theorem 1 and Claim 1 imply that this heritable birth rate induces a positivegrowth rate of 0.014%, while if the birth rate were induced by aggregate risk with the samedistribution, then the growth rate would be negative: − .
4% = (0 . ·
0% + 0 . · − . − . t period is 2 t , which is the product of a tinyprobability of 0 . t of the dynasty surviving and the very large size of the dynasty (4 t ),conditional on surviving. If the population includes a continuum of mass one of dynasties,then (by applying an exact law of large numbers) after t periods the size of the populationis 2 t (with probability one), and this population is concentrated on a continuum of a smallmass of 0 . t of surviving large dynasties. Thus, the population’s size converges to infinity,even though the share of surviving dynasties converge to zero. By contrast, if the numberof dynasties were finite (instead of a continuum), then after a sufficiently long finite time,the population’s size would eventually be zero with probability one. Finite populations
The result of an ever-growing population in which each dynastyis eventually doomed cannot happen when the number of dynasties is finite. Since eachdynasty is doomed to extinction, so too is the overall population. However, the fact thatthe mean size of each subpopulation is growing implies that the overall population maygrow significantly in the interim. As the finite model converges to the continuum model,15his initial growth phase becomes more and more prolonged, and the inevitable ultimatedemise of the population is postponed indefinitely.When there is no migration, a large finite population tends to ultimately put all its eggsin one basket. That is, the distribution of the finite population over its subpopulationstends to become very unequal, often concentrated in just one subpopulation. Such largesubpopulations hold up the mean, which is the growth rate found here. Once the populationis concentrated like this, however, doom is inevitable because the heritable risk of a largesubpopulation, essentially, becomes an aggregate risk since it affects a large share of theentire population.Migration introduces a new element to these observations. In the finite model migrationhas a distinct effect from that of the redraw rate. If some subpopulations grow large,and others shrink, migration acts to redistribute the population. This means that thepopulation can exploit the numbers in the large subpopulations, while diversifying therisk. These observations motivate the simulations described below.
In this section we present simulations that test whether our theoretical results for contin-uum populations hold for finite populations.
Description of the Simulation
The simulation is a discrete-time version of the ex-tended model (with dynasties) described above. Specifically, the basic time step of thesimulation is one year, and we replace each continuous Poisson rate with the respectiveindependent per-year probability (e.g., an annual birth rate of 2% is replaced with an in-dependent probability of 2% of each agent giving birth in each year). The Python code(contributed by Renana Heller) is included in the online supplementary material.We describe here the results of 150 simulation runs, which comes from 15 runs of 10different parameter combinations. In each simulation run, the initial population includes3,000 agents that are initially randomly allocated to 300 dynasties. The aggregate birthrate and the idiosyncratic birth rate are both equal to zero (i.e., µ y = µ z = 0). Theheritable birth rate in each dynasty is randomly chosen to be either x l = 0% or x h = 2%with equal probabilities (i.e., q = 0 . λ m + λ r = 2%. We set the annual death rate at1.4%, which implies that the theoretical prediction for a continuum population (see Claim1 in Appendix C) is that: (1) the share of agents with a high heritable birth rate converges16o about 71%, and (2) the annual long-run growth rate will be about 0 . − .
4% = (0 . ·
0% + 0 . · − . extinction ). The various simulation runs study 10 different ratios λ m λ r of the migration rate relative to the dynastic risk redrawing rate (while maintaining λ m + λ r = 2%): 0.01, 0.02, 0.05, 0.1, 0.25, 0.5, 1, 2, 4, 10. Numerical Results
Figure 2 presents four representative simulation runs with ratios:0.01 ( λ m = 0 . λ r = 1 . λ m = 0 . λ r = 1 . λ m = 0 . λ r = 1 . λ m = λ r = 1%).Figure 2: Representative Simulation Runs for four Ratios of λ m λ r t in each year (i.e.,it shows g ( t ) = ln( w ( t )) t ).The figure shows that when the ratio λ m λ r is small (0.01 or 0.05), dynastic risk has similarproperties to aggregate risk. The low rate of migration implies that a couple of “successful”dynasties (which happen to have had a high heritable birth rate for a long time) containmost of the population. This causes the heritable risk, essentially, to be aggregate. Thefrequency of agents with a high heritable birth rate has large fluctuations, since a singlechange of the heritable birth rate of the most populated dynasty has a large impact on thisfrequency. This is shown in the top-right panel. The cumulative growth rate (bottom-rightpanel) is initially positive, but after a couple of thousand years it becomes negative andstarts converging to the negative growth predicted by aggregate risk, until the populationbecomes extinct (top-left panel).By contrast, Figure 2 shows that when the ratio λ m λ r is 0.25 (resp., 1), then the theoreticalprediction for the continuum case becomes relatively (resp., very) accurate for the finitepopulation. When the migration rate is sufficiently high, a “successful” dynasty spreadsits offspring to many other dynasties, staving off extinction. The bottom-left panel showsthat the frequency of agents living in the most populated dynasty is at most 10% (resp.,2%). This implies that the share of agents with a high heritable birth rate has a relatively(resp., very) small fluctuations around Claim 1’s predicted value of about 71%, as can beseen in the top-right panel. The cumulative growth rate (bottom-right panel) converges tothe positive value of 0.01%, as predicted in Claim 1, as is shown in the top-left panel.Figure 3 presents the mean long-run growth rate obtained in the 15 simulation runsfor each of the ten ratios of λ m λ r . The results show that conclusions drawn from the fourrepresentative simulation runs presented in Figure 2 are indeed valid for the entire set of150 simulation runs. Asexual reproduction
Our model, like the related literature, makes the simplifyingassumption that reproduction is asexual, where offspring are identical to the parent. Sim-18igure 3: Mean Long-Run Growth Rate for each Ratio of λ m λ r The black points describe the mean growth rate of 15 simulation runs for each ratio of λ m λ r .The vertical bars show intervals of one standard deviation on each side of the mean. Thelabels describe how many simulation runs ended in an extinction of the population.ilar results should hold if reproduction were sexual and haploid, where a single geneticvariant—an allele—that determines choice is inherited with probability 1 / Horizontal and vertical correlation
An insight of our model is that vertical corre-lation increases the growth rate, but horizontal correlation decreases it. Horizontal cor-relation is called within-generation bet hedging by Lehmann and Balloux (2007). Verticalcorrelation is called the multiplayer effect by McNamara and Dall (2011) who study anon-overlapping generations model in which an asexual species breeds annually in one ofa large number of breeding sites. Each site can be either good (high expected number ofoffspring) or bad. In each generation each site changes its type with probability less than0.5. Each animal observes a noisy signal about the quality of the site in which it was born,and it has to choose whether to stay or to migrate to a new site. McNamara and Dallshow that when the signal is sufficiently noisy, it is best for nature to induce each animalto ignore the signal, and always stay in its birth site because the mere fact that the animal19as born in the site makes it more likely that the site is good.
Additive separability
Our model assumes that the various component of risk are ad-ditively separable. This assumption clearly facilitates the analysis. It permits a directcomparison of the implications of the three types of risk. Separability seems intuitivelyunlikely to be crucial to the results. At the least, there ought to be approximate results fora general non-separable criterion and small aggregate, heritable and idiosyncratic compo-nents. Further, it seems that it would be possible to allow for arbitrary aggregate shockswith heritable and idiosyncratic shocks conditional on the aggregate state, much as inRobson (1996).
Age structure
Recently, a different approach was applied by Robson and Samuelson(2019) to show that risk interdependence matters in a continuous-time setting (see alsorelated results in Robson and Samuelson, 2009). Specifically, they show that adding agestructure to Robatto and Szentes’s (2017) setting (i.e., allowing the fertility rate to dependon the agent’s age) implies that interdependence of risk influences the growth rate. Bycontrast, the present paper shows that interdependence of risk is important for the inducedgrowth rate in a hierarchical population, even when the age structure is trivial, but stillin a continuous-time setting. It would be interesting for future research to study theimplications of heritable risk in age-structured populations.
Migration between fragmented habitats
Our numerical analysis suggests an impor-tant advantage to connecting isolated small habitats of an endangered species. The relatedexisting literature (e.g., Burkey, 1999; Smith and Hellmann, 2002) shows that having sev-eral isolated small habitats for a species induces a larger extinction probability relative toa situation in which the species lives in a single large habitat. This result holds in a setupin which the birth rates are decreasing in the population’s density, and are deterministic.The present paper shows that connecting isolated small habitats with migration increasesthe long-run growth rate. We adopt a complementary setup of the birth rate that does notdepend on the population’s density, but does have a dynastic stochastic component (theheritable component of the birth rate). 20
Conclusion
In this paper, we demonstrate that a crucial aspect of the evolution of a population exposedto risk is inheritance. If the actual choice made by a parent is inherited by her offspring,this induces a correlation between the parent’s risk and the offspring’s risk. A type thatdoes this will outperform types that are exposed to either idiosyncratic or aggregate risk.This result is a force favoring risk-taking. Although most risk-taking may be reversedby a sufficiently concave relationship between resources and offspring, positively skewedlotteries that involve high enough prizes, but relatively low means, will be taken.
A Proof of Theorem 1
The following global convergence result of Goh (1978) will be helpful in the proof
Lemma 1 (Goh, 1978, Theorems 1 and 2) . Consider the system of n differential equations dp k ( t ) dt = p k ( t ) · F k ( p ) , k = 1 , .., n, where each F k ( p ) is a continuous function of p ∈ R n + ≡ { p | p k > ∀ k ∈ { , .., n }} . Supposethere is a fixed point p ∗ > satisfying p ∗ k · F k ( p ∗ ) for each k . Assume further thatthere exists a constant matrix E such that for all p ∈ R n + : (1) ∂F k ( p ) ∂p k ≤ E kk < for each k ∈ { , .., n } , and (2) (cid:12)(cid:12)(cid:12) ∂F k ( p ) ∂p j (cid:12)(cid:12)(cid:12) ≤ E jk for each j = k , and all the leading principal minors of − E are positive. Then every trajectory p ( t ) starting at any initial state p (0) > convergesto p ∗ > . For each time t , let w k ( t ) be the mass of agents with heritable birth rate x k at time t (henceforth, x k -agents). Let p k ( t ) = w k ( t ) w ( t ) be the share of x k -agents at time t . Let¯ b ( t ) = P k p k ( t ) · x k + µ y + z ( t ) be the average birth rate at time t . Let b k ( t ) be theaverage birth rate of x k -agents in time t : b k ( t ) = x k + µ y + z ( t ). The mass of x k -agentsat time t + dt is given by (neglecting terms of O (cid:16) ( dt ) (cid:17) ): w k ( t + dt ) = w k ( t ) + dt · (( b k ( t ) − δ − λ ) · w k ( t ) + w ( t ) · λ x · q k ) ,Hence dw k ( t ) dt = ( b k ( t ) − δ − λ x ) · w k ( t ) + w ( t ) · λ x · q k . t + dt is given by w ( t + dt ) = w ( t ) + dt · (cid:16) ¯ b ( t ) − δ (cid:17) · w ( t ) , so that dwdt = (cid:16) ¯ b ( t ) − δ (cid:17) · w ( t ) . Since 1 p k ( t ) dp k ( t ) dt = 1 w k ( t ) dw k ( t ) dt − w ( t ) dw ( t ) dt , it follows that dp k ( t ) dt = (cid:16) b k ( t ) − ¯ b ( t ) − λ x (cid:17) · p k ( t ) + λ x · q k . Substituting ¯ x ( t ) ≡ P k p k ( t ) · x k , we obtain dp k ( t ) dt = (( x k − ¯ x ( t )) − λ x ) · p k ( t ) + λ x · q k = p k ( t ) · F k ( p ) . (7)Let ∆ n + ⊆ R n + be the interior of the simplex, ∆ n + = n p ∈ R n + | P k p k = 1 o . Since ddt P n p k ( t ) = 0 , it follows that p (0) ∈ ∆ n + implies that p ( t ) ∈ ∆ n + for each t .We now show that there exists a fixed point p ∗ ∈ ∆ n + . If dp k ( t ) dt = 0 and p k ( t ) = p ∗ k inEq. (7), then setting x ∗ ≡ P k p ∗ k · x k yields the requirement: p ∗ k = λ x · q k λ x + x ∗ − x k , (8)where it will be shown that the denominator is positive, for all k = 1 , ..., n . Next multiplyeach p ∗ k in (8) by x k and sum to obtain an equation in one unknown: x ∗ = X k x k · λ x · q k λ x + x ∗ − x k . (9)In the range x ∗ > x n − λ x the LHS (resp., RHS) is increasing (resp., decreasing) in x ∗ .Further, LHS
The growth rate derived from the mean ¯ c = P c ∈ C q ( c ) c is ¯ c β . The growth rate derivedfrom the lottery ( C, q ) is the unique solution for x ∗ > m β − λ x of x ∗ = λ x X c ∈ C q ( c ) · c β λ x + x ∗ − c β . (11)If β > x ∗ < ¯ c β if R ≡ ¯ c β > λ x X c ∈ C q ( c ) · c β λ x + ¯ c β − c β ≡ S. This is because the LHS of Eq. (11) is increasing in x ∗ and the RHS of Eq. (11) isdecreasing in x ∗ for x ∗ > m β − λ x . In addition, ¯ c β > m β − λ x , for sufficiently small β > R = S = 1 at β = 0. In addition, dSdβ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β =0 = (cid:18) λ x (cid:19) X c ∈ C q ( c ) ln c − λ x ln ¯ c < ln(¯ c ) = dRdβ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β =0 (12)Hence R > S for all small enough β > x ). C Explicit Solution for Binary Lotteries
Theorem 1 has derived the key properties of the growth rate, without calculating an explicitformula for f ( X, q x , λ x ). In what follows we present such an explicit formula in the caseof binary lotteries over the heritable birth rate, which is used to yield the theoreticalpredictions in Section 6.1 and in Figure 1. Specifically, we now assume that the heritablebirth rate has two possible realizations, i.e., X = { x l , x h } . Let µ x denote the lottery’sexpectation, let ∆ x = x h − x l denotes the lottery’s spread, and let q ≡ q x ( x h ) denote theprobability of the higher realization. Claim . The equivalent growth rate of a growth process with a binary heritable birth rate24s equal to g = f (∆ x, µ x , q, λ x ) + µ y + µ z − δ, where f (∆ x, µ x , q, λ x ) = µ x + ∆ x · (1 − · q ) − λ x + q (∆ x − λ x ) + 4 · q · ∆ x · λ x . (13)Moreover, f (∆ x, µ x , q, λ x ) is decreasing in λ x . Proof.
Substituting p h = p , p l = 1 − p , q h = q and q l = 1 − q in Eq. (2) yields: p = λ x · qλ x − (1 − p ) · ∆ x ⇔ p · ∆ x + p · ( λ x − ∆ x ) − q · λ x = 0 . This quadratic equation has a unique solution in (0 , p (∆ x, q, λ x ) = ∆ x − λ x + q (∆ x − λ x ) + 4 · q · ∆ x · λ x · ∆ x , (14)which yields (13), when substituting this solution into f (∆ x, µ x , q, λ x ) = p (∆ x, q, λ x ) · x h + (1 − p (∆ x, q, λ x )) · x ‘ = µ x + ( p (∆ x, q, λ x ) − q ) · ∆ x. Next we prove that f (∆ x, µ x , q, λ x ) is decreasing in λ x . Take the derivative of p (∆ x, q, λ x ): ∂p (∆ x, q, λ x ) ∂λ x = 12 · ∆ x − · (∆ x − λ x ) + 4 · q · ∆ x · q (∆ x − λ x ) + 4 · q · ∆ x · λ x − . We have to show that ∂p (∆ x,q,λ x ) ∂λ x is negative for any λ x >
0, which is true iff q (∆ x − λ x ) + 4 · q · ∆ x · λ x > ∆ x · (2 · q −
1) + λ x After some algebra, this condition holds if and only if q (1 − q ) > q ∈ (0 , References
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