On the Biological Foundation of Risk Preferences Roberto Robattoyand Balázs Szentesz September 23, 2015
Abstract This paper considers a continuous-time biological model in which the growth rate of a population is determined by the risk attitude of its individuals. We consider choices over lotteries which determine the number of o¤spring and involve both idiosyncratic and aggregate risks. We distinguish between two types of aggregate risk: environmental shocks and natural disasters. Environmental shocks in‡uence the death and birth rates, while natural disasters result in instantaneous drops in population size. Our main result is a utility representation of the evolutionary optimal behavior. The utility is additively separable in the two types of aggregate risk. The term involving environmental shocks is a von Neumann-Morgenstern utility which induces the same attitude towards both idiosyncratic and aggregate risk. The term involving disasters cannot be interpreted as an expected utility maximization and induces less tolerance towards aggregate risk.
1
Introduction
Most models in economics take preferences as given and derive the choices induced by those preferences. This paper does just the opposite. We entertain the hypothesis that choice behaviours are genetically determined and shaped by natural selection. The underlying individual preferences are then merely the representations of those evolutionary optimal choice behaviours. We work from the basic premise that in the long-run context of evolution, only the fastest-growing genes survive. As this paper focuses on risk preferences, we consider choices over lotteries that a¤ect the reproductive value of individuals. Our main result is a utility representation of the optimal choice behaviour. The corner stone of our analysis is a characterization of the long run dynamics of a population that inhabits a risky environment. Two types of aggregate risk are present, which we refer to as environmental shocks and natural disasters. Environmental shocks in‡uence the birth and death We have bene…ted from discussions with Eddie Dekel, Je¤ Ely, Arthur Robson, Phil Reny, Larry Samuelson. of Finance, University of Wisconsin-Madison, WI, USA. E-mail:
[email protected]. z Department of Economics, London School of Economics, London, UK. E-mail:
[email protected]. y Department
1
rates of individuals, determining the rate of increase of the population size. Meanwhile, natural disasters cause discrete drops in population size, a¤ecting the level as opposed to the slope. The reproductive …tness of an individual is also subject to idiosyncratic risk, that is, conditional on an environmental shock or a natural disaster, the birth rate and the survival probability are still random variables. Because the population is a¤ected by aggregate shocks, it does not grow at a constant rate. Our key result consists of a characterization of the asymptotic growth rate as a function of the ergodic distributions of the various types of risk. The main theorem of this paper states that this function is the utility representation of the evolutionary optimal choice behaviour over lotteries in an environment where the risk is determined by individuals’choices. In order to better explain our contribution and to contrast our results with those in existence, we …rst describe Robson’s (1996a) seminal paper on the evolution of risk preferences. Time is discrete and individuals live for one period. The number of o¤spring an individual has is determined by the realization of a lottery. A lottery, L, is described by a triple,
; G; fF ( j!)g!2
denotes the set of possible states of the world, G is the ergodic distribution on
, where
, and F ( j!) is
the distribution on the number of o¤spring if the state of the world is !. Conditional on !, the realization of F ( j!) is independent across individuals, so this element represents the idiosyncratic
component of the risk. The distribution G represents the aggregate risk, since ! determines the distribution of reproductive values in the population. Robson (1996a) shows that the asymptotic growth rate of the population is u (L) =
Z
log
Z
dF ( j!) dG (!) :
(1)
Since the gene inducing the choice of the lottery L grows at rate u (L) in the long run, Robson (1996a) interprets u as the utility representation of the evolutionary optimal behaviour. That is, the surviving gene chooses L over L0 if and only if u (L)
u (L0 ). The main implication of (1)
is that the evolved attitude towards risk depends strongly on whether the risk is idiosyncratic or aggregate. In fact, individuals will be relatively less tolerant of aggregate risk, as compared to idiosyncratic risk. To illustrate this observation, consider two lotteries, L1 and L2 . Suppose that both lotteries generate the same distribution of o¤spring, H. However, the risk induced by L1 R is purely idiosyncratic while L2 involves only aggregate risk.1 By (1), u (L1 ) = log dH ( ) > R log dH ( ) = u (L2 ). That is, even though L1 and L2 induce the same risk from the viewpoint
of a single individual, L1 is strictly preferred to L2 .
In contrast to Robson (1996a), we consider a continuous time model, which allows us to distinguish between two types of aggregate risk: environmental shocks and natural disasters. Recall that environmental shocks determine the rate of increase of the population size, while natural disasters cause instantaneous changes in the level. In discrete time, any aggregate shock necessarily results in a discrete change in the population size. It then becomes natural to ask whether the results 1 Formally,
0 if
L1 =
; G; fF1 ( j!)g!2
< ! and one otheriwise.
where F1 ( j!)
H ( ) and L2 = R+ ; H; fF2 ( j!)g!2
2
where F2 ( j!) =
obtained by Robson (1996a) apply only to natural disasters, or to aggregate risk in general. Our answer is that his results do not hold for environmental shocks: the evolutionary optimal behaviour induces the same attitude towards both idiosyncratic risk and aggregate risk due to environmental shocks. In order to state our representation theorem, let us describe our setup in detail. Time is continuous and population dynamics are determined by a lottery L = ( ; G; ; Fn ( j!) ; Fe ( j!)). The set
denotes the set of states of the world and G is the ergodic distribution on
function
:
. The
! R is the arrival rate of a disaster. The c.d.f. Fn ( j!) denotes the distribution of
the survival probability at the moment a natural disaster occurs. Fe ( j!) denotes the distribution
of the net birth-rate (birth-rate minus death-rate) of an individual conditional on !. Our main result is that the asymptotic growth rate, U (L), is Z Z Z Z U (L) = "dFe ("j!) dG (!) + (!) log
dFn ( j!) dG (!) :
(2)
In order to show that the function U can indeed be interpreted as a utility function of the evolutionary optimal choice behaviour in this context, we introduce the stochastic arrival of choice problems over lotteries. A gene is de…ned to be a choice behaviour, that is, a mapping from choice sets into choices. At each moment in time, the dynamics of the population are determined by the lottery chosen in the most recent choice problem. We show that only those genes which behave as if they were maximizing the function U will survive. The two additive terms in the utility representation (2) have very di¤erent economic interpretations. The …rst term, which is associated with environmental shocks, is a standard von Neumann-Morgenstern representation with Bernoulli utility as the identity function.2 This implies that the choice induced by U depends only on the expected value of the net birth-rate; it doesn’t matter whether the source of the risk is aggregate or idiosyncratic. The second term, which is associated with natural disasters, is analogous to (1) of Robson (1996a). Indeed, if
(!)
1,
which corresponds to the frequency of discrete changes in Robson (1996a), the two expressions coincide. Let us again point out that this term is not a von Neumann-Morgenstern representation of risk attitude. Interestingly, this term is formally identical to a smooth ambiguity-averse preference representation where the ambiguity is determined by G and the ambiguity aversion is determined by the logarithm function. For a discussion and axiomatic characterization of smooth ambiguity averse preferences, see Klibano¤ et al. (2005). We now consider why there is an expected utility representation of the choice behaviour associated with environmental shocks (see the …rst term of (2)). If we suppose that there are no 2 Since
the lotteries directly determine net birth-rates, the Bernoulli utility is the identity function, and hence,
these preferences exhibit risk-neutrality. However, it is not hard to generate risk-aversion in our model. If the lotteries were to determine reproductive resources instead of net birth-rates and these resources could be consumed and transformed into o¤spring according to a reproduction function, the Bernoulli utility would become the reproduction function. The concavity of such a reproduction function would induce risk-averse preferences.
3
disasters, then the population growth rate at each point in time is simply the population’s average net birth-rate. The time-average of these growth rates is needed to compute the asymptotic growth rate. By Birkho¤’s Theorem, the time-average of the population growth rates is given by the expectation over the ergodic distribution G. As a result, the evolutionary optimal criterion can be expressed as an expectation of net birth-rates. There are environments in which risk cannot naturally be characterized in terms of birthand death-rates, even in the absence of disasters. This may be exempli…ed by plant and animal species that reproduce periodically at particular times of the year. Consider an annual plant which reproduces only once in its lifetime and whose life cycle lasts one year. Although the plant might be exposed to various risks each day, this risk can intuitively be summarized using annual quantities, such as the probability that a seed survives until the reproductive season and the number of new seeds produced by the plant. As such, reproduction can be characterized in terms of factors rather than rates. The model of Robson (1996a) can be viewed as a description of the risk faced by such a plant in the case where the plant always survives until the reproductive season and produces seeds. This generates a non-expected utility representation, (1), for the following reason. The aggregate growth factor of the population each period is the average of each individual’s growth factors, in this case the number of seeds each plant produces. The population growth rate is given by the logarithm of this aggregate growth factor and will depend on the state of the world. As a result, the asymptotic growth rate is the expectation over the ergodic distribution G of the logarithm of the population average of individual growth factors, and the optimal criterion cannot be written as an expectation over individual growth factors. Indeed, the appearance of the logarithmic function in the optimal choice criterion was …rst noticed in the context of annual organisms, see for example Cohen (1966). Robson (1996a) takes this approach and applies it in a discrete model of human population dynamics. However, human reproduction occurs throughout the year rather than being con…ned to distinct breeding seasons.3 Approximating human population dynamics by a discrete model requires to group together all risks a¤ecting the population in a given time period and to characterize reproductive values in terms of factors. We consider a continuous-time model which allows us to describe risks in more details and to express reproductive values in terms of birth- and deathrates. In the absence of natural disasters, this leads to an expected-utility representation (see the …rst term in (2)), in contrast to Robson’s non-expected utility representation in (1). However, we also wish to consider events which cause the instantaneous and contemporaneous death of many individuals4 , such as earthquakes, forest …res and ‡oods. It is hard to see how jumps in human population size could ever go upward, or why one would characterize a downward jump as anything other than a natural disaster. It is not clear that these events should be considered more 3 Of
course, risks faced by humans may be seasonal and this can be incorporated into the stochastic evolution of
the states in our continuous model. 4 We consider an individual to be dead at the moment when her reproductive value drops to zero, rather than when she is clinically dead.
4
important than environmental shocks in determining the evolution of risk preferences. Given our results regarding environmental shocks, we interpret our representation theorem as providing at least a partial evolutionary justi…cation for von Neumann-Morgenstern preferences. Let us emphasize that in our model, environmental shocks do not by any means have to be associated with small changes. In fact, these shocks might have profound instantaneous e¤ects on death rates. The spread of a viral disease, for example, could triple the death-rate and wipe out a signi…cant fraction of the population within a short period of time. Dramatic changes in weather conditions could also be modelled as environmental shocks. A long lasting drought reduces food sources, and a large proportion of the population might starve to death within a summer. What these examples have in common is that the increased number of deaths occur through a shift in the rate, and do not occur at a single point in time. Of course, the evolutionary optimal behaviour tries to eliminate the risk of increased death rates associated with such environmental shocks. However, our model predicts that this type of risk is not treated any di¤erently from idiosyncratic risk. As a …nal step, we enhance the realism of our model by generalizing our results to account for the age-structure of the population. In particular, we relax the assumption in our baseline model that individuals’ reproductive values do not depend on their age. In this extension, we abstract from disasters and focus exclusively on environmental shocks. In the extended model, lotteries describe the risk faced by individuals in each cohort, i.e. they specify stochastic streams of reproductive values over individuals’life cycles. When the e¤ects of lotteries are age-dependent, choices over lotteries re‡ect both time preferences and risk preferences. Risk preferences, however, are de…ned over static lotteries rather than stochastic sequences of prizes. But if choices involve intertemporal trade-o¤s, how can one determine whether evolutionary optimal behaviour has an expected or a non-expected utility representation?5 In order to deal with this problem, we prove two results. First, in order to eliminate time preferences from our analysis, we restrict attention to those lotteries which do not involve intertemporal trade-o¤s, e.g. those that specify risk over age-independent death rates. We show that the optimal choices over this domain have an expected utility representation. Second, we consider arbitrary lotteries with intertemporal e¤ects. Since the population growth rate at a given point in time depends on both the state of the world and the population age-distribution, we de…ne an augmented state space which is the product of the original state space and the set of possible age distributions.6 We show that, in the continuous time model, the evolutionary optimal criterion can again be expressed as the expectation of an individual’s net birth-rate according to the ergodic distribution over the augmented state space.7 We will explain how this optimal criterion can be interpreted as the expected net birth-rate of a 5 The
di¢ culty of de…ning expected-utility representations over temporal risk were explored, for example, in
Markowitz (1959), Mossin (1969), Spence and Zeckhauser (1972), Dreze and Modigliani (1970), Kreps and Porteus (1979), and Machina (1984) in the context of decision theory. 6 For example, a positive shock to the birth-rate of old individuals results in a higher growth rate if the fraction of old individuals is higher in the population. 7 We also generalize the discrete model of Robson (1996a) and show that the logarithmic function appears in the optimal criterion as in (1).
5
randomly selected individual at a random moment in time. Since the age of a randomly selected individual depends on the population’s age distribution, augmenting the state space allows us to collapse intertemporal trade-o¤s to risk about age. Literature Review Numerous papers use evolution to at least partially explain preferences. The …rst is probably Becker (1976), who adopts an evolutionary argument to explain altruism. Overviews of the theories on the relationship between biology and economic behaviour can be found in Robson (2001 and 2002). Here we review only selected papers concerned with the evolution of risk preferences. Our continuous time assumption is not the only di¤erence between our paper and Robson (1996a). Robson (1996a) considers …nite populations and derives conditions under which the population does not go extinct with probability one. We consider continuum populations and, by assumption, they never go extinct. The assumption regarding the continuum population enables us to integrate out the idiosyncratic risk and sidestep problems related to extinction. We note that if a population grows at a constant rate asymptotically, it either goes extinct or it becomes very large. The former case is uninteresting from the evolutionary viewpoint and in the context of the latter one, the continuum assumption does not seem far-fetched. Robson (1996b) analyses a strategic biological model. First, each male chooses a lottery. The outcome of the lottery is the male’s income. After the outcomes of the lotteries are realized, each female decides which male to mate with. A male can mate with multiple females but not vice versa. The number of o¤spring of a female is a function of the male’s income per females who chose him. This model provides a rationale for a particular form of risk taking. Males are willing to take fair bets which result in small losses with large probability and large gains with small probability. The intuition is that a large gain might enable the male to mate with more females. Small losses, on the other hand, might reduce slightly the number of o¤spring per female but unlikely to reduce the number of females the male can mate with. Under reasonable assumptions on the reproduction function, the positive e¤ect on the number of females dominates the negative e¤ect on the number of o¤spring per female. Dekel and Scotchmer (1999) also analyze evolutionary optimal choices over lotteries in a strategic model. In their model, individuals participate in contests within groups. The outcome of a contest is determined by the realizations of the lotteries picked by the group members. The realizations of the lotteries are independent across individuals and the winner is the individual with the largest realization. The key assumption is that only the winner reproduces. Since the winner takes all, the evolutionary optimal behaviour will necessarily be risk-loving in some sense. However, the authors show that the evolutionary optimal behaviour cannot be represented by rational preferences. Unlike Robson (1996b) and Dekel and Scotchmer (1999), our model does not involves strategic interactions; individuals only solve decision problems. Indeed, it is not clear to us how one might identify preferences and strategic components from equilibrium behaviour. It is, of course, not to
6
say that the strategic nature of the environment does not in‡uence the evolution of preferences. Finally, the di¤erences between the e¤ects of idiosyncratic and aggregate risks on preferences are also emphasized by Robson and Samuelson (2009). The authors consider the evolution of intertemporal preferences in an age-structured biological model. Among other things, they show that if the e¤ects of aggregate shocks on an individual’s survival probability do not depend on the age of the individual, then aggregate risk slows down population growth. As we mentioned above, we examine a continuous time version of this model where the risk is de…ned over death-rates. We show that the population growth is fully determined by the expected death-rates and there is no distinction between the aggregate and idiosyncratic components.
2
Model
Time is continuous and is indexed by t 2 R+ . At each moment, a continuum of individuals make
up the population. The population dynamics are governed by the lottery L=
; G; f (!)g!2 ; fFe ( j!)g!2 ; fFn ( j!)g!2
:
(3)
R is the set of possible states of the world. Let ! t denote the state of the world at time t. The dynamic process f! t gt2R is a Markov process with unique ergodic distribution G.8 We assume that ! t is almost surely continuous in t almost everywhere.
Environmental shocks.— The net birth-rate of an individual at time t is "t = bt denotes her birth rate and
t
t;
where bt
denotes her death rate. The variable "t 2 R is distributed according
to the conditional c.d.f. Fe ("t j! t ) and it is measurable in t almost surely. We allow the net birth
rate of an individual, "t , to change over time even if ! t stays constant. We assume that E ("j!) is a bounded function of !, and Fe ("j!) is uniformly continuous in !. The realizations of "0t s are assumed to be independent across individuals conditional on ! t . We assume that the Law of Large Numbers holds, and hence, at time t the population grows at rate Z r (! t ) "dFe ("j! t ) :
(4)
Natural disasters.— Natural disasters hit the population stochastically according to a nonhomogenous Poisson process. The arrival rate of the process at time t is :
1
! R+ is a bounded, measurable function in L ( ; G). If a natural disaster occurs at time
t, an individual survives with probability 8 In
(! t ) 2 [0; 1) where
particular, whenever
R
t
2 [0; 1]9 , and Fn ( j! t ) denotes the distribution of
t
0 0 A 1dG (!) > 0 then for each ! 2 ; conditional on ! 0 = ! : Z I (! t 2 A) dt = +1;
almost surely. That is, conditional on any initial condition, the process spends an in…nite amount of time in every positive G-measure set with probability one (see e.g. Glynn (1994) and Du¢ e and Glynn (2004)). 9 For our mathematical results to hold, we do not need t to be weakly less than one. We only make this assumption for the sake of interpretation.
7
conditional on ! t . We assume that Fn ( j!) is uniformly continuous in !. We also assume that
E ( j!) is uniformly bounded away from zero, that is, the population never goes extinct. Again, the realization of
t
is independent across individuals conditional on ! t . This assumption essentially
implies that the survival probability hence, the distribution of
t
is re-drawn at each moment a natural disaster occurs, and
is indeed independent of the previous history of disasters. We appeal
t
to the Law of Large Numbers once more and assume that the fraction of the population that survives a natural disaster is j (! t )
Z
dFn ( j! t ) :
(5)
We further assume that for each individual, the shocks "t and
t
are independent at each time
t conditional on ! t . In what follows, we describe the simple case in which each random variable is binary and all changes in the environment arrive according to Poisson distributions. = !H ; !L
Example. Suppose that
Poisson process with arrival rate
and the state of the world switches according to a
> 0. This process is assumed to be independent of all other
random variables. So, the ergodic distribution of ! is given by Pr ! H = Pr ! L = 1=2. by
The net birth rate of an individual is also binary, "t 2 "l ; "h and its distribution is described "t =
8 <"h :"l
with probability pe (! t ) ; with probability 1
pe (! t ) ;
where "h > "l and pe ! H > pe ! L . These inequalities imply that if the state is ! H , an individual reproduces at a higher rate in expectation. The net birth rate of an individual is constant over time unless the state of the world changes. When the state changes, the net birth rate of each individual is instantaneously redrawn according to the distribution described above. Natural disasters arrive according to a Poisson process with rate one, that is, 1. The survival probability of an individual,
t,
can take one of two values from
!H =
!L =
l
and its
;
h
distribution is described by
t
where
h
>
l
and pn ! H
=
8 < :
h l
with probability pn (! t ) ; with probability 1
pn (! t ) ;
> pn ! L . These inequalities imply that if the state is ! H , each
individual is more likely to survive a natural disaster. At the moment when the state, ! t , changes or a natural disaster occurs, the survival probability of each individual is redrawn according to the distribution described above. Otherwise, the survival probability of each individual is constant. Finally, the aggregate net birth rates and the average survival rates in disasters are described by r (!) = "h pe (!) + "l [1
8
pe (!)] ;
(6)
j (!) =
h
pn (!) +
l
[1
pn (!)] ;
(7)
respectively.
3
Results
In this section we analyze the speed of population growth. Let yt denote the size of the population at time t and let y0 be normalized to one. If there is a natural disaster at time t, yt denotes the size of the surviving population. The basic di¢ culty is that, due to aggregate shocks, the population does not grow at a steady state rate. Nevertheless, it is possible to characterize a growth rate, g, such that if t is large enough, the size of the population is approximately the same as if it were growing at a constant rate g, that is, egt :
yt
(8)
Such a growth rate g is called the continuously compounded growth rate of the population and is formally de…ned in Section 3.2. Next, we derive an expression for the law of motion of the population along a realized path of the random variables. In Section 3.2, we use this expression to prove our key result, which is a characterization of the continuously compounded growth rate in terms of the lottery (3). Finally, in Section 3.3, we introduce stochastically arriving choice problems. Each choice problem is a set of lotteries, and each lottery is just like L in (3). We show that only those genes which behave as if they were maximizing the continuously compounded growth rate will survive.
3.1
Law of motion of the population
Let N (t) denote the number of natural disasters which occurred between time zero and time t. In addition, let
i
2 [0; t], i 2 f1; :::; N (t)g ; denote the arrival time of the ith natural disaster.
Proposition 1 The size of the population at time t is given by: yt = exp
Z
N (t)
t
r (! s ) ds
0
where
QN (t) i=1
Y
j (! i ) ;
(9)
i=1
j (! i ) is de…ned to be one if N (t) = 0.
A notable property of the expression on the right-hand side of (9) is that it is multiplicatively separable in natural disasters and environmental shocks. To illustrate this observation, let us consider a population that grows at a constant rate for a unit amount of time. Suppose that this population is hit by a disaster and half of the population dies. Then, irrespective of the exact time of the disaster, the population at the end of the time period will be half as large as it would have been if the disaster had not occurred. Of course, this argument presumes that the disaster leaves
9
the net birth rate of the surviving population una¤ected. In our model, this assumption is satis…ed because "t and
t
are independent conditional on ! t .
Proof. We prove the proposition by induction with respect to the number of natural disasters, N (t). Suppose …rst that N (t) = 0. Then, by (4), the law of motion of the population is described by the following di¤erential equation between time zero and time t: :
y = y r (! ) .
(10)
The solution of this di¤erential equation is10 Z
yt = exp
t
r (! s ) ds ;
0
which is just the statement of the proposition for N (t) = 0. Suppose that the statement of the proposition is true for all t whenever N (t)
k and let us
assume that N (t) = k + 1. By the inductive hypothesis, lim !
y = exp
N (t)
<
At time
N (t)
Z
N (t) 1
Y
Nt
r (! s ) ds
0
N (t)
j (! i ) .
i=1
there is a natural disaster and, by (5), only a fraction j !
survives. Hence, y
N (t)
= exp
Z
2[
of the population
N (t)
Y
Nt
r (! s ) ds
0
The law of motion of the population for
N (t)
j (! i ) :
(11)
i=1 N (t) ; t]
is again described by the di¤erential equation
(10) and initial condition (11). The solution is yt = y
exp N (t)
Z
!
t
r (! s ) ds N (t)
= exp
Z
N (t)
t
0
r (! s ) ds
Y
j (! i ) ;
i=1
where the second equality follows from (11).
3.2
Continuously compounded growth rate
Motivated by (8), we …rst provide a formal de…nition for the continuously compounded growth rate. De…nition 1 We call the number g 2 R the continuously compounded growth rate of the population
if
lim
t!1
log yt =g t
almost surely. 1 0 The
solution exists because r is continuous and bounded, see Coddington and Levinson (1955), Chapter 2.
10
In order to see that the continuously compounded growth rate is indeed a useful object in the evolutionary context, consider two populations, yt1 and yt2 , with corresponding compounded growth rates g1 and g2 . We show that if g1 > g2 then, asymptotically and with probability one, yt1 is going to be in…nitely large relative to yt2 . To see this, note that, by De…nition 1, log lim
t!1
t
yt1 yt2
log yt1 t!1 t
log yt2 = g1 t!1 t
= lim
lim
g2 > 0
almost surely. But this can only be the case if yt1 =yt2 converges to in…nity as t goes to in…nity with probability one. Next, we show that the continuously compounded growth rate exists and characterize it in terms of the lottery, L. This is our main result. Theorem 1 For almost all (f! t g ; fN (t)g)t2R ; Z Z Z log yt = "dFe ("j!) dG (!) + lim t!1 t
(!) log
Z
dFn ( j!) dG (!) :
Let us explain the basic idea of the proof. By Proposition 1, Rt PN (t) r (! s ) ds log yt 0 i=1 log j (! i ) = lim + lim ; lim t!1 t!1 t!1 t t t
(12)
(13)
where the second term in the right-hand side is de…ned to be zero if N (t) = 0. Since ! t is an ergodic process, both r (!) and log j (!) are also ergodic. As a consequence, the right-hand side of (12) is the sum of the time averages of two ergodic variables. Birkho¤’s Ergodic Theorem states that, under certain conditions, the time average of the realization of an ergodic variable converges to the expected value of the variable with probability one, where the expectations are formed according to the ergodic distribution. In the proof, we argue that Birkho¤’s Ergodic Theorem is applicable, and show that the time average of r (!) and log j (!) converge to the …rst and second terms in the right-hand side of (12), respectively. Proof. First, consider the time average of r (! t ). Since the state of the world has an ergodic distribution and r is continuous, Birkho¤’s Ergodic Theorem implies that Rt Z r (! s ) ds 0 lim = r (!) dG (!) t!1 t almost surely11 . Substituting the de…nition of the function r in (4), we obtain Rt Z Z r (! s ) ds lim 0 = "dFe ("j!) dG (!) : t!1 t
(14)
We now turn to the second expression on the right-hand side of (13) and rewrite it as PN (t) N (t) i=1 log j (! i ) : t N (t) 1 1 See,
for instance, Doob (1953), Chapter XI, for a version of Birkho¤’s Ergodic Theorem for continuous-time
processes.
11
If
R
(!) dG (!) = 0 then
= N (t) = 0 almost surely, so (14) implies (12). In what follows we R restrict attention to the case when (!) dG (!) > 0. We …rst show that Z N (t) lim = (!) dG (!) (15) t!1 t almost surely. To this end, assume that t 2 N for simplicity and de…ne Xi = N (i)
N (i
1) for
all i 2 N. Since N (t) is a non-homogenous Poisson process, the random variables X1 ; X2 ; ::: are Ri independent conditional on the realization of f! t gt2R+ , and EXi = V ar (Xi ) = i 1 (! ) d . By Kolmogorov’s Strong Law of Large Numbers12 lim
Rt
N (t)
0
(! ) d
t
t!1
= lim
t!1
Pt
i=1
h
Xi
Ri
i 1
(! ) d
t
i
=0
(16)
almost surely conditional on f! t gt2R+ . Finally, notice that Birkho¤’s Ergodic Theorem implies
that
1 t!1 t lim
Z
t
(! s ) ds =
0
Z
(!) dG (!)
(17)
almost surely. From (16) and (17) the equation in (15) follows. Next, we show that lim
t!1
PN (t)
almost surely. Recall that
i=1
log j (! i ) = N (t)
Z
log j (!) R
(!) dG (!) (! 0 ) dG (! 0 )
(18)
denotes the time of the ith natural disaster. For all i 2 N, de…ne R Zi = log j (! i ). By (15) and (!) dG (!) > 0 it follows that there are in…nitely many disasters i
with probability one, so these variables are well-de…ned. The discrete time process fZi gi2N is also ^ is given by13 an ergodic Markov process and its ergodic distribution, G, R Z (!) dG (!) ^ (!) = f!:logRj(!)2Bg dG (! 0 ) dG (! 0 ) f!:log j(!)2Bg
for all Borel subset B of ( 1; 0]. Therefore, Birkho¤’s Ergodic Theorem implies that14 Pn Z i=1 Zi ^ (!) lim = log j (!) dG n!1 n
^ into the previous equation yields with probability one. Substituting the de…nitions of Zi and G (18). Finally, notice that from (15) and (18) it follows that
lim
t!1 1 2 This
1 3 See 1 4 See
PN (t) i=1
log j (! i ) = t
Z
(!) log j (!) dG (!) =
Z
(!) log
Z
dFn ( j! t ) dG (!)
theorem is indeed applicable since (See e.g. Feller (1950), Chapter X.7): Z i Z 1 X V ar [N (si ) N (si 1 )] 1 1 = lim 2 (! ) d = lim (!) dG (!) = 0 < 1: 2 i!1 i i!1 i i 0 i=1
Proposition 2 in Du¢ e and Glynn (2004). Corollary 1 of Proposition 2 of Du¢ e and Glynn (2004).
12
(19)
almost surely, where the second equality is just (5). From equations (14) and (19) the statement of the theorem follows. Example (Continued). We apply Theorem 1 to the example presented in Section 2. Recall that in this example, the ergodic distribution on
= ! H ; ! L is given by Pr ! H = Pr ! L = 1=2.
Then substituting (6) and (7) into (12) we get lim
t!1
log yt t
= =
1 1 r !H + r !L + log j ! H + log j ! L 2 2 X 1 "h pe (!) + "l (1 pe (!)) 2 H L
(20)
!2f! ;! g
1 + 2
X
log
h
pn (!) +
l
(1
pn (!))
!2f! H ;! L g
almost surely.
3.3
Endogenous growth and utility representation
As previously discussed, if the population corresponding to a particular gene has a higher continuously compounded growth rate than that of another population, it will eventually grow to be in…nitely larger than that other population. Therefore, if the level of environmental risk is determined by choices made by individuals, and that choice behaviour is genetic, the continuously compounded growth rate is the evolutionary optimal decision criterion. That is, only those genes which generate the largest continuously compounded growth rate survive in the long run. This leads us to interpret U (L) =
Z Z
"dFe ("j!) dG (!) +
Z
log
Z
dFn ( j!) (!) dG (!)
(21)
as the utility representation of the evolutionary optimal choice behaviour. One concern which might arise with respect to our analysis thus far pertains to our implicit assumption that the choice of a lottery is made once and for all, and determines the growth rate of the gene forever. We have not yet demonstrated that the same utility criterion is used to solve individual choice problems if the overall risk is determined by a combination of various decisions. Next, we take our analysis one step further and formalize the claim that the function U can indeed be interpreted as a utility function of the evolutionary optimal choice behaviour in this context. To this end, we introduce the stochastic arrival of choice problems. Each choice problem is a set of lotteries. Each of these lotteries is like L de…ned in (3). We de…ne a gene to be a choice behaviour, that is, a mapping from choice sets into choices. We then consider the dynamics of the population corresponding to a certain gene. At each moment in time, the dynamics of the population are determined by the lottery chosen in the most recent choice problem. We show that only those genes which behave as if they were maximizing the function U de…ned in (21) will survive.
13
n
Let L denote an arbitrary set of lotteries of the form (3). The pair (flk gk=1 ; f
an environment where lk
L, jlk j < 1 and
A gene, c, in the environment
n (flk gk=1
;f
n k gk=1 )
is called
k (> 0) is the Poisson arrival rate of lk (k = 1; :::; n). n k gk=1 ) is a mapping from choice sets to elements of
these sets, that is, c : fl1 ; ::; ln g ! L, such that c (lk ) 2 lk for all k = 1; :::; n. Note that a gene is de…ned to be simply a choice behaviour on a restricted domain.
In what follows, we investigate the population growth corresponding to a certain gene, c. The n
population dynamics in the environment (flk gk=1 ; f
n k gk=1 )
are de…ned as follows. At t = 0, the
population is normalized to one and a choice problem is drawn, lk with probability
k=
n i=1 i .
If
t > 0 and there is an s < t, such that (i) no choice problem arrived during the time interval (s; t] and (ii) a choice problem l arrived at s, and c (l) = L then the dynamics of the population at time t are determined by the lottery L. We say that a gene survives evolution if its induced continuously compounded growth factor is at least as large as that of any other gene. n
Theorem 2 For each set of lotteries L and for each environment (flk gk=1 ; f ing gene, c , satis…es
c (lk ) 2 arg max U (L) , L2lk
n k gk=1 ) ;
the surviv(22)
for all k = 1; ::; n. Proof. See the Appendix. Let us emphasize that the decision criterion in (22) is robust in the sense that the utility of a lottery is independent of the environment. That is, it depends neither on the other potential choice problems, nor on their frequencies of arrival. We have not allowed for idiosyncratic uncertainty in the arrival of choice problems. However, it would not be di¢ cult to introduce some such heterogeneity, and allow for the possibility that at some points in time di¤erent individuals of the same population face di¤erent problems. We might allow, for example, the population to consist of many colonies and the choice problems to arrive independently across colonies. In this case the size of the total population is simply the sum of the sizes of the colonies, so Theorem 2 remains valid. Interpretation The two terms on the right-hand side of (21) have very di¤erent behavioural implications. Observe that the lottery L can be viewed as the combination of two compound lotteries: (fFe ( j!)g! ; G) and (fFn ( j!)g! ; G), which are associated with environmental shocks and natural disasters respec-
tively. Equation (21) implies that a decision maker who maximizes U reduces the compound lottery
(fFe ( j!)g! ; G) to a simple one. That is, conditional on the risk associated with natural disasters,
choices will be based only on the expected rate of reproduction. Whether environmental shocks
are aggregate or idiosyncratic is irrelevant. Therefore, the …rst term is simply a standard von Neumann-Morgenstern representation. In sharp contrast, the second term does not correspond to expected utility maximization. In particular, the compound lottery (fFn ( j!)g! ; G) is not reduced to a simple one in the utility
14
function U . This term is formally identical to the representation of smooth ambiguity-averse preferences. In the context of ambiguity ( ; G) corresponds to the subjective state space and beliefs, Fn ( j!) describes uncertainty, and ambiguity-aversion is determined by the logarithmic
function. Since the logarithmic function is concave, this representation implies that the decision maker with utility function U is less tolerant towards aggregate risk than towards idiosyncratic risk. One notable feature of the function U is its additive separability in the risks due to environ-
mental shocks and natural disasters. This arises from the fact that yt is multiplicatively separable in these two types of aggregate risk. As previously mentioned, this separability is due to our assumption that, for each individual, "t and
t
are independent conditional on ! t . We could relax
this independence assumption and still obtain an additive representation similar to (21). However, the c.d.f. Fe would be replaced by an ergodic distribution of the birth rates which would depend on, among other objects, Fn and . So, while the utility representation would still be additive, the term corresponding to environmental shocks would depend on the risks faced due to disasters.
4
On the di¤erence between environmental shocks and natural disasters
Theorem 1 implies that environmental shocks and natural disasters a¤ect growth in very di¤erent ways. We should perhaps shed some additional light on this observation, and we shall make it the objective of this section. To this end, we revisit Robson’s (1996a) discrete-time model with two goals in mind. First, we investigate the limit of a discrete time model as the length of the time intervals shrinks to zero. Indeed, it seems quite reasonable to suspect that environmental shocks might be approximated arbitrarily well with a sequence of small natural disasters. In the context of a binary example, we show that, in the limit, the growth rate induced by a lottery does not depend on the decomposition of the risk. In this sense, there is no con‡ict between our results related to continuous-time models and the limiting behaviour of discrete-time models. Second, we establish a mapping from our continuous-time models with only environmental shocks to Robson’s (1996a) discrete-time models. We make use of this mapping to provide a clear explanation for the divergence between the continuous and discrete models with respect to di¤erent types of risk: in the continuous model, individuals make no distinction between aggregate and idiosyncratic risks, while in the corresponding discrete model they do.
4.1
On the limit of the model of Robson (1996a)
Let us consider discrete populations facing two di¤erent lotteries; one which is purely idiosyncratic and the other purely aggregate. Robson (1996a) shows that the population facing the idiosyncratic lottery grows faster. In what follows, we show that as the length of a time interval approaches
15
zero, the growth rates of the two populations converge.15 Suppose that time is discrete and indexed by t 2 N. Assume that
ergodic distribution of ! is given by Pr !
H
= Pr !
L
=
!H ; !L
and the
= 1=2. Each individual lives for one
period, and her number of o¤spring is the realization of a lottery. In each period t, the realization of the lottery is independent across individuals conditional on ! t . We consider the following two lotteries LA =
(
H
if ! = ! H ;
L
if ! = ! L
and LI =
(
H
with probability 1=2;
L
with probability 1=2,
where 1 < L < H. Note that the lottery LA involves only aggregate risk and the lottery LI involves only idiosyncratic risk. Also note that the risks induced by these two lotteries are the same from the viewpoint of a single individual; with probability one-half she produces H o¤spring, and with probability one-half she produces L. Let g (L) denote the compounded growth factor of the population corresponding to the lottery L (2 fLA ; LI g). The main result from Robson (1996a) implies
log g (LA ) =
log H + log L < log 2
H +L 2
= log g (LI ) ;
(23)
and hence, the population choosing LI grows faster than the other. In what follows, we shrink the length of the time intervals from one to consequences on the speed of population growth as
, and examine the
approaches zero. If the time intervals are
downscaled, and in each period individuals reproduce according to the lotteries LA and LI , the populations grow faster, and they explode as rates across di¤erent
goes to zero. Therefore, in order to compare growth
’s, the lotteries governing population dynamics must be rede…ned. We
consider two ways of doing so. First, we shrink the intensity of the shocks speci…ed by the lotteries as
goes to zero. The idea of taking the limit this way is to spread the e¤ect of the original lottery
over many smaller time periods. Second, we change the frequency of reproduction. That is, if the length of a period is
, then only with probability
does an individual have the opportunity
to reproduce. However, conditional on the opportunity, individuals reproduce according to the original lotteries, LA or LI . Taking the limit this way means we spread the number of reproducing individuals uniformly across the smaller time periods. In each of these cases, we show that the perunit-period growth rates of the populations corresponding the idiosyncratic and aggregate lotteries converge to the same value as
goes to zero. In other words, the distinction between idiosyncratic
and aggregate risk disappears. The intuition behind these observations can be explained as follows. If in the size of the population is also small within a
is small, the change
-long period. This means that the logarithmic
function in Robson’s (1996a) utility function, (1), can be approximated by a linear function arbitrarily well as
goes to zero. Therefore, the logarithmic function can be replaced by the linear
one in the limit. 1 5 We
are able to prove the same result in the general model of Robson (1996a) but we believe that the binary
example is su¢ cient to illustrate our point.
16
Shrinking the intensity of the shocks.— For each , we de…ne the following two lotteries: ( ( H if ! = ! H ; H with probability 1=2; LA = and LI = L L if ! = ! L with probability 1=2. The idea behind this rescaling is as follows. If the realization of the lottery is constant within a unit of time, the induced growth factor does not depend on produces H
o¤spring in each
. To see why, note that if an individual
-long period, she will have H
1=
= H genetic copies after
one unit of time. We maintain the assumption that the ergodic distribution of ! is given by Pr ! H = Pr ! L = 1=2.16 Let g corresponding to L
2 LA ; LI
that log g Next, we take
LA =
L
denote the compounded growth factor of the population
per
-long time period. Again, from Robson (1996a) it follows
log H
+ log L 2
< log
H
+L 2
= log g
LI .
(24)
to zero and compare the growth factors per unit interval of the populations
corresponding to LA and LI . Note that if the growth factor per 1=
growth factor per unit interval is (g )
-long period is g
then the
. Consider …rst the population governed by the aggregate
lottery LA . By (24), lim log g
1=
LA = lim
!0
log H
!0
+ log L 2
=
log H + log L : 2
(25)
Comparing this expression with the left-hand side of (23), we conclude that the population facing the aggregate lottery is una¤ected by the time scaling. Consider now the population governed by the idiosyncratic lottery. Again by (24),
lim log g
1=
!0
log LI
= lim
H +L 2
!0
:
On the right-hand side, both the numerator and the denominator converge to zero. We apply L’Hopital’s rule to obtain lim log g !0
1=
LI
= lim
!1
1 H +L 2
H log H + L log L 2
=
where the second equality follows from the observation that both H goes to zero. From (25) and (26), we conclude that, as
1 (log H + log L) , 2 and L
(26)
converge to one as
goes to zero, the population facing
aggregate risk grows just as fast as the population facing only idiosyncratic risk. In the limit, do these discrete models become continuous-time models with only environmental shocks? De…ne h and l such that H = eh and L = el and note that h and l are the rates corresponding to the factors H and L, respectively. The continuously compounded growth rates 1 6 We
emphasize that this does not mean that the state of the world switches more and more frequently as
to zero. One can assume, for example, that ! switches with only probability
in each
the state of the world switches once per unit-period in expectation irrespective of
17
.
goes
-long time period. Then
in (25) and (26) can then be written as (h + l) =2. In other words, the limits of LA and LI are lotteries involving only environmental shocks according to which the net birth rate of an individual is h or l with equal probability. The limit of LA induces only aggregate risk and the limit of LI generates only idiosyncratic risk. By Theorem 1, this distiction, however, has no impact on the growth rate. Shrinking the 8 > >H > < e LA = L > > > :1
In each
frequency of reproduction.— De…ne: 8 > > H > < e L with probability if ! = ! ; and LI = L > > > :1 with probability 1 ; if ! = ! H ;
with probability
with probability
=2;
with probability
=2;
with probability 1
:
-long period, the probability that a certain individual does not reproduce is 1
,
irrespective of the lottery she faces. We capture this by de…ning the lotteries such that they each result in one descendant (presumably the individual herself) with probability 1 with probability
. Otherwise,
, an individual reproduces according to the original lottery. Therefore, in each
-long period, only a fraction of
of the total population will reproduce. e yields e and L Applying the main result of Robson (1996a) to lotteries L I A e A = log [ H + (1 L
log g
)] + log [ L + (1 2
)]
< log
H +L +1 2
Let us now compute the unit-period growth factors in the limit as e . Then, the aggregate lottery L
= log g
e L I
goes to zero. Consider …rst
A
lim log g
eA L
1=
!0
=
lim
=
lim
!0
!0
log [ H + (1 1 2
H 1 H +1
)] + log [ L + (1 )] 2 1 L 1 H +L + = 2 L+1 2
(27) 1,
where the second equality follows from L’Hospital’s rule. Let us now turn our attention to the e . Then, idiosyncratic lottery L I
lim log g
1=
!0
eI L
= lim
log
H+L 2
+1
= lim
!0
!0
H+L 2 H+L + 2
1
1
=
H +L 2
1;
(28)
where the second equality follows from the L’Hospital rule again. Comparing (27) and (28), we e and L e converge to the conclude that the growth rates of the populations corresponding to L A
same value as 1 7 The
I
goes to zero.17
limit of the examples of this section does not correspond to our model with only environmental shocks.
The reason is that, in our model, an individual has exactly one o¤spring at reproduction. In contrast, the number of o¤spring at reproduction is a random variable in these examples. We believe that it is not hard to generalize our model to allow the number of o¤spring to be a random variable.
18
.
4.2
Mapping continuous models into discrete models
There is a natural mapping from continuous-time models with only environmental shocks into Robson’s (1996a) models. Indeed, one might simply evaluate the size of a continuously growing population at integer moments, and assume that all changes in population size happen at those moments. In the continuous model, in order to make the evolutionary optimal choice, an individual only needs to know the expected net birth rates generated by the lotteries. In contrast, in the discrete model the growth rate of the population is not determined by the expected number of o¤spring; before making her choice, an individual needs to know whether the risk she faces is aggregate or idiosyncratic. It is straightforward to map a continuous model into a discrete model with similar aggregate population dynamics, but it is less clear how one might de…ne a discrete-time lottery which individuals will view as similar to continuous-time lotteries. One notable feature of our continuous model is the fact that, within a given time period, an individual might produce not only o¤spring but also grandchildren. Since children and grandchildren are identical, population growth is determined by the distribution of an individual’s descendants (as opposed to her o¤spring). Therefore, the distribution of o¤spring speci…ed by the discrete lottery should be equal to the one-period-ahead distribution of descendants in the continuous model. There is clearly no distinction between aggregate and idiosyncratic risks in the continuous model. The fact that such a distinction does exist in the discrete model might at …rst seem puzzling. One might argue, naively, that the expected number of descendants induced by a discrete lottery is pinned down by the expected net birth rate generated by the corresponding continuous lottery. It therefore does not seem possible that, in the continuous model, this expected value provides an individual with enough information to make a choice, while in the discrete model it does not. This reasoning is false. An individual’s net birth rate determines only the distribution of her number of o¤spring, and does not in fact determine her expected number of descendants. The distribution of her number of descendants also depends on whether the risk she faces is aggregate or idiosyncratic. In particular, if an individual’s birth rate is high, then her o¤spring’s birth rate is more likely to be high under aggregate risk than under idiosyncratic risk. Therefore, an individual with a high birth rate expects to have relatively more descendants if the risk she faces is aggregate. To resolve the puzzle, let us consider two continuous lotteries with the same expected net birth rate. On the one hand, the corresponding discrete time lotteries may have di¤erent expected values. In fact, a discrete lottery involving more aggregate risk will generate a higher expected number of descendants, which, in turn, promotes population growth. On the other hand, as Robson (1996a) showed, the larger the aggregate part of a discrete lottery, the slower the population will grow. It turns out that these two e¤ects cancel each other out, and the two discrete lotteries, with their di¤erent expected values, will generate the same growth rate. Indeed, each of these discrete-time lotteries yields the same utility a’la Robson (1996a) de…ned by (1). Let us now illustrate this argument with an example.
19
Example. Suppose that
= ! H ; ! L and the state of the world can change only at integer
moments. Suppose that ! is redrawn at each t 2 N according to an i.i.d process de…ned by Pr ! H = Pr ! L = 1=2. Assume further that the net birth rate of an individual is constant between integer moments and is determined by one of the following two lotteries: 8 8
Note that LcA involves only aggregate risk while LcI involves only idiosyncratic risk. In addition, the
expected net birth rate generated by both lotteries is log 2, and hence, each period, the populations corresponding to these lotteries double in expectation. The continuously compounded growth rate of both populations is log 2. In what follows, we characterize the discrete-time lotteries, LdA and LdI , corresponding to LcA and LcI . We show that the expected number of o¤spring induced by LdA is larger than that generated by LdI . However, we also show that the growth rates generated by these lotteries are the same, that is, u LdA = u LdI , where u is de…ned by (1). Consider …rst the case involving aggregate risk, that is, the lottery LcA . At each integer moment, the state of the world switches to ! L with probability one-half, and an individual’s net birth rate is set to zero. So, after one unit of time, her expected number of descendants (including herself) is one. With equal probability, ! switches to ! H , and the individual’s net birth rate becomes log 4. Because the risk is aggregate, within this time period the net birth rate of each of her descendants is also log 4. Therefore, her expected number of descendants is elog 4 = 4. To summarize, the discrete time lottery, LdA , corresponding to LcA is de…ned by ( 4 if ! = ! H , d LA = 1 if ! = ! L . Since the two possible states are equally likely, the expected number of descendants is 2:5. Let us next consider the case involving idiosyncratic risk, that is, the lottery LcI . At each integer moment, an individual’s net birth rate switches to zero with probability one-half, so after one unit of time her expected number of descendants is one (including herself). With the same probability, her net birth rate is set to log 4 at the beginning of the period. In addition, the net birth rate of each of her descendants born within that period is log 4 with probability one-half and zero otherwise. We do not characterize the distribution of the number of descendents but we observe that the expected number must equal three.18 Therefore, we conclude that the expected number of descendants generated by LdI is 2. Let us now examine the choice between LcA and LcI from a forward-looking perspective. Note that the lotteries LdA and LdI can be viewed as the summaries of the one-period-ahead consequences 1 8 The
reason is that an individual’s ex ante expected number of descendants is 2 because the population doubles
in each period and the risk is idiosyncratic. Since the expected number of descendant of an individual with birth rate zero is one, the expected number of descendants of an individual with birth rate log 2 must be 3.
20
of LcA and LcI , respectively. On the one hand, an individual will have more descendants after one unit of time if she chooses the aggregate lottery LcA . Indeed, we have shown that ELdA = 2:5 > 2 = ELdI . On the other hand, aggregate risk slows down population growth (by the main result of Robson (1996a)). It turns out that these two e¤ects cancel each other out. Indeed, the two discrete lotteries corresponding to LcA and LcI induce the same growth rate. By the main result of Robson (1996a) (and our Theorem 2) the continuously compounded growth rates generated by LdI and LdA are both
1 1 log 4 + log 1 = log 2 = log ELdI = u LdI . 2 2 In other words, the long term consequences of a continuous lottery depend on the decomposition u LdA =
of the risk into idiosyncratic and aggregate parts. Nevertheless, an individual does not need to know these consequences. The expected net birth rate captures all the information she needs to make an optimal choice.
5
Age-structured Population
The discrete model of Robson (1996a) and our continuous model both involve populations which are made up of identical adults. In particular, neither model involves individuals whose reproductive values vary with their age. The objective of this section is to examine how the results described above generalize to arbitrary age-structured populations. First, we generalize the discrete time model of Robson (1996a) to an arbitrary age-structured population. We derive the evolutionary optimal criterion and show that it does not have an expected utility representation. In particular, the logarithmic function appears in it just as in Robson’s (1996a) criterion. We then consider the age-structured version of our continuous-time model without natural disasters. We show that the evolutionary optimal behaviour maximizes the expected value of the average net birth-rate in the population and, in this sense, it has an expected utility representation. In this section, we abstract from natural disasters. Nevertheless, it is not hard to show that the conclusion of Robson (1996a) holds for natural disasters even if the population is age-structured, that is, the logarithmic function would appear in the optimal criterion.
5.1
The Generalization of Robson’s (1996a) Discrete Model
Time is discrete and is indexed by t 2 f0; 1; :::g. At each moment, a continuum of individuals make up the population. Each individual can live at most
dynamics are governed by the lottery L=
; G; fF ( j!; )g
!2 2f1;:::; g
2 N[ f1g periods. The population
; fH ( j!; )g
!2 2f1;:::; g
:
(29)
R is the set of possible states of the world. Let ! t denote the state of the world at time t. The dynamic process f! t gt2N is an ergodic Markov process with ergodic distribution G.
21
Reproduction.— The number of o¤spring of an individual that is "t 2 N+ . 0-year old individuals are newborns, so "t 19
surely survive until the age of one.
periods old at time t is
0. For simplicity, we assume that newborns
We refer to individuals who are not newborns as adults. The
variable "t is distributed according to the conditional c.d.f. F ( j! t ; ). We assume that E ("j!; )
is a bounded function of !, and F ( j!; ) is uniformly continuous in !. The realizations of "t are
assumed to be independent across individuals conditional on ! t . We assume that the Law of Large
Numbers holds, so that at time t the number of o¤spring produced by -year old individuals is Z r (! t ) "dF ("j! t ; ) : (30) Death.— A
-year old individual dies at time t with probability
t
survived until this age). Death occurs after reproduction. The variable
(conditional on having t
2 [0; 1] is distributed
according to the conditional c.d.f. H ( j! t ; ). Since individuals cannot live longer than
we assume that
t
1. The realizations of
t
periods,
are assumed to be independent across individuals
conditional on ! t . We assume that the Law of Large Numbers holds, so that at time t the fraction of -year old individuals who dies is d (! t )
Z
dH ( j! t ; ) :
(31)
One might want to restrict attention to models where young individuals cannot produce o¤spring. Such a requirement is a restriction on the possible lotteries. For example, if individuals cannot reproduce prior to age , then only those lotteries which specify "t
0 for all
should
be considered. The model of Robson (1996a) can be viewed as a special case of this model in the following two ways. First, suppose that individuals live for one period and then die, that is, 1 t
= 1 and
1. Second, suppose that individuals can live forever but adults are identical, that is,
and F ( j!; )
F ( j!), H ( j!; )
=1
H ( j!). Robson (1996a) interprets his model according to
(1) but the two models are mathematically identical.
The Evolutionary Optimal Criterion in the Discrete Model In general, population growth in a given time period depends not only on the state of the world !, but also on the age-distribution of the population in that period. If there is aggregate risk, the age-distribution might change over time. We show that the age-distribution and the state of the world in a given period fully determine the growth factor. We rede…ne the state space and include the age-distribution as a new state variable. Observe that the age distribution and the realization of the state of the world in a given period determine the age distribution in the next period via the lottery L. In other words, the lottery L de…nes a Markov process over the set of possible age-distributions. In what follows, we characterize population growth in terms of the ergodic distribution over the new state space. 1 9 Alternatively,
"t could denote the surviving o¤spring.
22
To this end, let
denote the set of possible age-distributions of the adult population, that is, ( ) X = = f g 2f1;:::; g : =1 ; =1
where
denotes the fraction of
-year old individuals (the fraction of newborns is uniquely determined by the distribution of the adult population). We de…ne e = as the new augmented e denote the ergodic distribution over e induced by the lottery L. state space. Let G The overall aggregate risk in an age structured population is captured by
e; G e and not by
( ; G). Indeed, given fF ( j!; )g and fH ( j!; )g, the correlation between the reproductive values of two randomly selected adults is fully determined by (!; ) but is not determined by !. The space ( ; G) determines only the correlation between the reproductive values of two randomly e is in‡uenced by selected individuals of the same age. We emphasize that the distribution G
the idiosyncratic component of the lottery, fF ( j!; )g and fH ( j!; )g, because the evolution of
the age distribution is a function of these objects. In the model of Robson (1996) without agestructure, the aggregate properties of population growth are determined by the state space, ( ; G), and the expected reproductive values, fE ("j!)g! . Similarly, in the age structured population, the
idiosyncratic components, F ( j!; ) and H ( j!; ), a¤ect the age distribution only through their expectations, E ("j!; ) and E ( j!; ).
Theorem 3 The continuously compounded growth rate of the population is # " Z Z Z X e (!; ) : "dF ("j!; ) + 1 dH ( j!; ) dG log
(32)
=1
Proof. See the Appendix.
First, observe that this formula is a generalization of Robson’s (1996a) criterion, (1). We show this in both interpretations of Robson’s (1996a) model described above. Consider the …rst interpretation where individuals live for one period, that is, adults is degenerate:
1
1 t
1. Then the age distribution of
1. So, the expression (32) simpli…es to Z Z log "dF 1 ("j!) dG (!) ;
which is exactly the criterion in Robson (1996a). Consider now the second interpretation where = 1 and F ( j!; ) =
= =
F ( j!), H ( j!; ) = H ( j!). Since does not a¤ect reproductive value, " # Z Z Z X e (!; ) log "dF ("j!) + 1 dH ( j!) dG (33) Z Z
log log
"
=1
Z
X
=1
! Z
"dF ("j!) +
"dF ("j!) +
Z
1
Z
1
dH ( j!)
dH ( j!) dG (!) ;
23
#
e (!; ) dG
which is again the criterion in Robson (1996a). Note that the argument of the logarithmic function in (32) is the sum of the average number of o¤spring and the average survival probability of the population conditional on (!; ). Indeed, this quantity determines population growth conditional on (!; ). Observe that the continuously compounded growth rate is not just the expectation of this quantity. The criterion requires the computation of this quantity for each (!; ) and then integrates the logarithmic function of this according to the ergodic distribution. To summarize, criterion (32) does not simply maximize the sum of the expected number of o¤spring and survival probabilities and, in this sense, it is inconsistent with expected utility maximization.
5.2
The Age-Structured Continuous Model
Time is continuous and is indexed by t 2 R+ . At each moment, a continuum of individuals make 2 R+ [f1g. The population
up the population. Each individual can live at most until the age of dynamics are governed by the lottery L=
; G; fF ( j!; )g
!2 2[0; ]
; fH ( j!; )g
!2 2[0; ]
:
(34)
R is the set of possible states of the world. Let ! t denote the state of the world at time t. The dynamic process f! t gt2R is an ergodic Markov process with ergodic distribution G. We assume that ! t is almost surely continuous in t almost everywhere.
Reproduction.— The birth-rate of a -year old individual at time t is "t . The variable "t 2 R+
is distributed according to the conditional c.d.f. F ("j! t ; ) and it is measurable in t almost surely and continuous in
. We allow the birth rate of an individual to change over time even if ! t
stays constant. We assume that E ("j!; ) is a bounded function of !, and F ("j!; ) is uniformly continuous in !. The realizations of "t are assumed to be independent across individuals conditional on ! t . We assume that the Law of Large Numbers holds, so that at time t, -year old individuals reproduce at rate r (! t )
Z
"dF ("j! t ; ) :
Death.— The death rate of a -year old individual at time t is
(35) t.
The variable
t
2 R+ [f1g is
distributed according to the conditional c.d.f. H ( j! t ; ) and is measurable in t almost surely and continuous in . Since individuals cannot live longer than assume that E ( j!; ) is a bounded function of ! for in ! and continuous in . The realizations of
t
periods, we assume that
t
1.20 We
< , and H ( j!; ) is uniformly continuous
are assumed to be independent across individuals
conditional on ! t . We assume that the Law of Large Numbers holds, so that at time t, -year old individuals die at rate d (! t ) 2 0 Alternatively,
Z
dH ("j! t ; ) :
we can just assume that individuals older than
24
cannot reproduce.
(36)
We highlight that our baseline model without natural disasters is a special case of this model where
= 1 and F ( j!; )
F ( j!) and H ( j!; )
H ( j!) for all . That is, individuals can
reach any age and their reproductive value is not a¤ected by their age. Note that this corresponds to the second interpretation of the discrete model of Robson (1996a). The Evolutionary Optimal Criterion in the Continuous Model
As in the discrete model, population growth at a given moment depends not only on the state of the world !, but also on the age-distribution of the population. Again, we rede…ne the state space and include the age-distribution. The lottery L de…nes a Markov process over the set of possible age-distributions. Again, we characterize population growth in terms of the ergodic distribution over the new state space. To this end, let
denote the set of possible age-distributions, that is, ) ( Z 0; d =1 ; = = f g 2[0; ] : 0
denotes the fraction of -year old individuals. We de…ne e = as the new augmented e denote the ergodic distribution over e induced by the lottery L. state space. Let G
where
Theorem 4 The continuously compounded growth rate of the population is Z Z Z Z e (!; ) : "dF ("j!; ) dH ( j!; ) d dG
(37)
0
Proof. See the Appendix.
We show that this is a generalization of the part of our criterion which corresponds to environmental shocks (the …rst term in (2)), where
= 1 and F ("j!; )
H ( j!) for all . Indeed, since Z
= =
Z Z
F ("j!) and H ( j!; )
does not a¤ect reproductive values, (37) simpli…es to ! Z Z Z e (!; ) "dF ("j!) dH ( j!) d dG
Z Z
"dF (" j!) "dF ("j!)
Z
Z
(38)
0
e (!; ) dH ( j!) dG
dH ( j!) dG (!) :
We next consider the extent to which criterion (37) is consistent with expected utility maximization. De…ne the myopic reproductive value of an individual at a given moment to be her net birth rate at that moment. These quantities are myopic because they do not take the individual’s and her descendants’futures into account. We show that, in the continuous model, the evolutionary optimal criterion, (37), maximizes the expected population average of the myopic reproductive values. To this end, note that for a given (!; ), the average reproductive value of the population is
Z
0
Z
"dF ("j!; )
25
Z
dH ( j!; ) d :
e Therefore, the expected myopic Furthermore, the state (!; ) is distributed according to G. reproductive value is just (37). In other words, this criterion maximizes the expected myopic
reproductive value of the population, and it does not matter whether this expectation is based on aggregate risk (!; ) or idiosyncratic risk. In contrast, as explained above, the optimal criterion in
the discrete model, (32), di¤ers from the expected myopic reproductive value, and it is important whether the myopic reproductive value is in‡uenced by aggregate or idiosyncratic risk. Finally, we show that (37) can be interpreted as the expected myopic reproductive value of e (!; ) a randomly selected individual from the population at a random moment. Note that G
is a distribution over age distributions for each !. Since a distribution over distributions is a e (!; ) generates a distribution over [0; ] for each !. Let G (!; ) denote this distribution, G distribution of the age of a randomly selected individual conditional on !. Then (37) can be rewritten as
Z Z
Z
"dF ("j!; )
Z
dH ( j!; ) dG (!; ) ,
which is the myopic reproductive value of a randomly selected individual.
5.3
Intertemporal E¤ects
In the context of age-structured populations, lotteries specify risk over streams of reproductive values. As a consequence, the preferences over lotteries de…ned by (29) in the discrete model and by (34) in the continuous model re‡ect not only attitudes towards risk, but also attitudes towards intertemporal trade-o¤s. To elaborate on the lotteries’intertemporal e¤ects, note that the realization of the state of the world at time t, ! t , in‡uences not only the population growth rate at time t, but also the growth rates in future periods. This is because in general ! t a¤ects the age distribution of the population in future periods. In turn, as we have shown, the age-distribution a¤ects the growth rate. Therefore, by including the potential age distributions in the state space, we restricted our attention to risk and abstracted from intertemporal trade-o¤s. Indeed, the growth rate of the population at time t is fully determined by the realization (! t ;
t ).
Previous
realizations of the augmented state a¤ect the growth rate at time t only indirectly, through their e We explained stochastic e¤ects on (! t ; t ). This risk is captured by the ergodic distribution G. above that (37) can be interpreted as the expected net birth-rate of a randomly selected individual
at a random moment in time. In other words, augmenting the state space allows one to collapse the intertemporal trade-o¤s to a risk about age. Another, and perhaps more natural, way to eliminate time preferences from our analysis is to restrict attention to choices over a set of lotteries which do not involve intertemportal trade-o¤s. An example of such a set is lotteries which only di¤er in age-independent survival probabilities (in the discrete model) or death-rates (in the continuous model). The reason is that the risk over age-independent survival probabilities and death-rates has no impact on the age distribution in the population, and hence choices re‡ect preferences over age-independent risk. As we mentioned
26
in the Introduction, Robson and Samuelson (2009) examine choices over this set of lotteries in a discrete model. They show that adding aggregate risk without changing expected individual survival probabilities slows down population growth. In other words, optimal choices depend not only on expected age-independent survival probabilities, but also on the decomposition of risk into aggregate and idiosyncratic components. In contrast, we now show that, in our continuous model, population growth is fully determined by expected death-rates and there is no distinction between the aggregate and idiosyncratic components. In other words, adding aggregate risk to death-rates has no impact as long as it does not change expected death-rates. Proposition 2 Consider two lotteries Li = such that
R
; G; fF ( j!; )g
dH1 ( j!; ) =
R
!2 2[0; ]
; fHi ( j!; )g
!2 2[0; ]
dH2 ( j!; ) + d (!) for some d :
the two lotteries induce the same growth rate.
; i = f1; 2g ;
! R. If
R
d (!) dG (!) = 0, then
Proof. Since age-independent death rates have no e¤ect on the age distribution, both lotteries generate the same Markov process over age distributions. This implies that the ergordic distributions generated by L1 and L2 over the augmented state space are the same. Therefore, by (37), L1 R induces a larger continuous compounded growth rate than does L2 if and only if d (!) dG (!) 0. R In particular, if d (!) dG (!) = 0, then the two lotteries induce the same growth rate. In the next section, we take this analysis one step further and consider an arbitrary set of
lotteries which generate the same Markov process over age distributions. We show that the optimal choice behaviour over this set satis…es the independence axiom in the continuous model.
5.4
The Independence Axiom
We now discuss the di¤erences in choice behaviours between the discrete and continuous models. We show that the independence axiom is violated according to the discrete criterion but satis…ed according to the continuous criterion. We revisit the framework of subjective utility theory in Anscombe and Aumann (1963). We associate the aggregate state space in our biological model with the subjective state space of Anscombe and Aumann (1963), and the idiosyncratic risk in our model with known uncertainty in Anscombe and Aumann (1963). First, we explain the di¤erences in choice behaviours between the original model of Robson (1996a) and our baseline model without age-structure. We state the independence axiom for lotteries with the same state space ( ; G) and argue that it is satis…ed by criterion (38) and violated by Robson’s (1996a) criterion (1). We then turn our attention to age-structured populations. As explained above, in general lotteries have intertemporal e¤ects. In order to isolate risk preferences from time preferences, we state the e . We then independence axiom for lotteries which generate the same aggregate state space e ; G
show that the continuous criterion, (37), satis…es the independence axiom whereas the discrete one,
27
(32), violates it. Finally, we show that the violation of the independence axiom when it is stated e , can be solely due to intertemporal for lotteries with di¤erent augmented state spaces, e ; G trade-o¤s and might not re‡ect risk preferences. Indeed, we show that the independence axiom can be violated by lotteries even if they do not involve any risk. Without age structure.— Consider the lottery L =
; G; fF ( j!)g!2 ; fH ( j!)g!2
model where adult individuals are identical. The myopic reproductive values, "
in a
, correspond to
the prizes in Anscombe and Aumann (1963). The aggregate state space ( ; G) corresponds to the subjective state space and beliefs in Anscombe and Aumann (1963). Finally, the idiosyncratic risk, fF ( j!)g!2 ; fH ( j!)g!2 , corresponds to objective randomization conditional on the realization
of the state. As in Anscombe and Aumann (1963), we de…ne the convex combination of two lotteries, L1 =
; L2 =
; G; fF1 ( j!)g!2 ; fH1 ( j!)g!2
; G; fF2 ( j!)g!2 ; fH2 ( j!)g!2
;
as follows: L1
(1
) L2 =
; G; f F1 ( j!) + (1
) F2 ( j!)g!2 ; f H1 ( j!) + (1
) H2 ( j!)g!2
.
That is, for each !, an individual’s myopic reproductive value is determined by F1 ( j!) and H1 ( j!) with probability
and it is determined by F2 ( j!) and H2 ( j!) with probability (1
).21
The Independence Axiom: Let L1 ; L2 and L3 be lotteries corresponding to the same state space ( ; G). Then for all 2 [0; 1] ;
L1
L2 , L1
(1
) L3
L2
Note that our continuous criterion, (38), is linear in
(1
) L3 :
(39)
and hence this criterion satis…es this
axiom. In contrast, the discrete criterion of Robson (1996), (33), violates this criterion. Indeed, the discrete criterion corresponds to a smooth ambiguity averse preference where the ambiguity aversion is determined by the logarithmic function. Age Structure.— As in the models without age structure, we state the independence axiom for lotteries which correspond to the same aggregate risk. As explained above, the overall aggregate risk e , so we require the independence axiom to hold for lotteries corresponding is determined by e ; G
to the same augmented state space. To this end, …x ( ; G) and let D denote the set of Markov processes on
. For each d 2 D, let L (d) denote the set of those lotteries which specify
( ; G) and generate the Markov process d over
. Note that if L1 and L2 correspond to the
same state space ( ; G) and EL1 ("j!; ) = EL2 ("j!; ) and EL1 ( j!; ) = EL2 ( j!; ) then they de…ne the same Markov process over 2 1 In
, that is, L1 ; L2 2 L (d) for some d. Two lotteries
Anscombe and Aumann (1963), the parameter
represents an objective uncertainty; hence we assume that
it corresponds to idiosyncratic risk in our biological model.
28
also de…ne the same dynamic process over the augmented state space if they di¤er only in terms of state-dependent but age-independent death rates. The independence axiom for age structured populations can then be stated as follows: For each d 2 D, and for all L1 ; L2 ; L3 2 L (d), (39)
holds.
Theorem 5 The continuous criterion (37) satis…es the Independence Axiom (39). Proof. See the Appendix. In contrast, the discrete criterion, (32), violates the independence axiom. To see this, recall that we have shown that the discrete criterion (32) is a generalization of Robson’s (1996a) criterion, (1), and that this latter criterion violates (39). Finally, we show that, in an age structured population, the violation of the independence axiom e might for lotteries with the same state space ( ; G) but di¤erent augmented state spaces e ; G have nothing to do with risk preferences. In the following example, we consider lotteries which do not involve any risk and show that, due to the intertemporal trade-o¤ described in the previous section, the independence axiom is violated. Example. Consider the following two discrete time lotteries: Li =
; G; fFi ( j!; )g
2f1;2g
; fHi ( j!; )g
2f1;2g
i 2 f1; 2g :
Neither of the lotteries involve any risk and, according to each lottery, individuals die at age 2, that is,
= 2.
L1 : individuals surely survive until the age of two. They reproduce once at each age. Formally, for each ! 2 = 1; 2.
, H1 ( j!; 1) = 1 for all
0, F1 ("j!; ) = 0 if " < 1 and F1 ("j!; ) = 1 if "
1 for
L2 : individuals reproduce once at age 1 and then die. Had they survived until the age of 2, they would have produced four o¤spring. Formally, for each ! 2 F2 ("j!; 1) = 0 if " < 1 and F2 ("j!; 1) = 1 if " if "
, H2 ( j!; 1) = 0 for all
1,
1, and F2 ("j!; 2) = 0 if " < 4 and F2 ("j!; 2) = 1
4. The advantage of the population governed by L1 is that individuals reproduce at age two.
Therefore, the population evolving according to L1 grows faster than the population evolving according to L2 , that is, L1
L2 . Next, we argue that L1
1 L1 2
1 L1 2
1 L2 2
1 L1 ; 2
that is, the independence axiom is violated. To see this, note that according to 21 L2
(40) 1 2 L1 ,
each
individual survives until age two with probability one half and produces four or one o¤spring with equal probabilities. Therefore, from the perspective of a newborn, each individual reproduces 1:25 o¤spring at age 2 in expectation. According to L1 , each individual has only one o¤spring at age 2. Since this is the only di¤erence between L1 and 12 L2
29
1 2 L1 ,
we conclude that (40) holds.
Note that neither of the individual lotteries above involve any risk. Why then is the independence axiom violated? Combining L1 with L2 has the following intertemporal e¤ect on an individual. If an individual at age 1 is lucky, her survival probability is determined by L1 , so she survives until the age of 2 (she would die if her survival probability were determined by L2 .) If at age two she is lucky again, the number of o¤spring she has is determined by L2 and so she produces four o¤spring (as opposed to just one according to L1 ). In other words, an individual might have high reproductive value at age two because her survival probability was determined by L1 in the previous period. To put it di¤erently, note that the reproductive value of exactly half of the population is determined by L2 in each period. However, the age distribution of this part of the population is di¤erent from any age distribution which can be generated by L2 alone. In this extreme example, the age distribution generated by L2 is degenerate,
1
1 and
2
0. The
reproductive value of an individual at age 2 is irrelevant because individuals never survive until the age of two. On the other hand, when combining L2 with L1 , a signi…cant fraction of the population survives until age two and reproduces according to L2 . L1 should therefore not be considered an irrelevant alternative when combining it with L2 . The alternative L1 is very relevant because it a¤ects the age distribution on which L2 operates. In general, the combination of two lotteries can result in age distributions which never arise according to the individual lotteries. Population growth is determined by evaluating the lotteries on these new age distributions. To summarize, the violation of the independence axiom of arbitrary lotteries can be due to the intertemporal e¤ects of the lotteries. In contrast, the combination of two lotteries, L1 and L2 , such that L1 ; L2 2 L (d), does not require the consideration of intertemporal
e¤ects. Indeed, consider the combination of these two lotteries, L1 a fraction
(1
) L2 . In each period,
of the population evolves according to L1 . In addition, the age distribution of the
population is exactly the same as it would have been if the population had evolved according to L1 . Therefore, the growth rate of this population at time t is the same as it would have been if the population had evolved according to L1 before t, and the reproductive values are determined by L1
6
(1
) L2 at time t. In this sense, L2 has no intertemporal e¤ect on L1 .
Conclusion
We believe endogenizing preferences to be an important research agenda, and that adopting the biological approach, as we have done, might prove to be a fruitful enterprise. It seems reasonable to hypothesize that evolution did not in‡uence only physical traits but also shaped choice behaviours. An advantage of this approach is that it has strong predictions about the relationship between fertility and choices which, in principle, can be tested empirically. We do recognize that many choice problems faced by individuals in modern times were unlikely to be faced in evolutionary times. Yet, we hypothesize that preferences, at least in part, are hardwired and that many choices made today re‡ect the evolutionary optimal behaviour.
30
References [1] Anscombe, F. J. and R. J. Aumann (1963): “A De…nition of Subjective Probability,” The Annals of Mathematical Statistics, 34, 199-205. [2] Becker, G. (1976): “Altruism, Egoism, and Genetic Fitness: Economics and Sociobiology,” Journal of Economic Literature, 14, 817-826. [3] Coddington, E.A. and N. Levinson (1955): Theory of Ordinary Di¤ erential Equations, McGraw-Hill. [4] Cohen, D (1966): “Optimizing Reproduction in a Randomly Varying Environment,” Journal of Theoretical Biology, 12, 119-129. [5] Dekel, E. and S. Scotchmer (1999): “On the Evolution of Attitudes towards Risk in WinnerTake-All Games,” Journal of Economic Theory, 87, 1999, 125–143. [6] Doob, J.L. (1953): Stochastic Processes, John Wiley and Sons. [7] Dreze, J.H. and F. Modigliani (1970): “Consumption Decisions under Uncertainty,” Journal of Economic Theory, 5, 308-335. [8] Du¢ e, D. and P. Glynn (2004): “Estimation of Continuous-Time Markov Processes Sampled at Random Time Intervals,” Econometrica, 72, 1773-1808. [9] Feller, W. (1957): An Introduction to Probability Theory and Its Applications, Volume 1, John Wiley and Sons. [10] Galor, O and S. Michalopoulos (2012): “Evolution and the growth process: Natural selection of entrepreneurial traits,” Journal of Economic Theory, 147, 759-780. [11] Glynn, P. (1994): “Some Topics in Regenerative Steady-State Simulation,”Acta Applicandae Mathematicae, 34, 225-236. [12] Klibano¤, P., M. Marinacci and Sujoy Mukerji (2005): “A smooth model of decision making under ambiguity”, Econometrica, 73, 1849-1892. [13] Kreps, D.M. and E.L. Porteus (1978): “Temporal Resoution of Uncertainty and Dynamic Choice Theory,” Econometrica, 46(1), 185-200. [14] Kreps, D.M. and E.L. Porteus (1979): “Temporal von Neumann-Morgenstern and Induced Preferences,” Journal of Economic Theory, 20, 81-109. [15] Machina, M.J. (1984): “Temporal Risk and the Nature of Induced Preferences,” Journal of Economic Theory, 33, 199-231.
31
[16] Markowitz, H (1959): Portfolio Selection: E¢ cient Diversi…cation of Investments, Yale Univ. Press, New Haven, Conn. [17] Mossin, J. (1969): “A Note on Uncertainty and Preferences in a Temporal Context,” The American Economic Review, 59(1), 172-174. [18] Robson, A. (1996b): “The Evolution of Attitudes to Risk: Lottery Tickets and Relative Wealth”, Games and Economic Behavior, 14, 190-207. [19] Robson, A. (1996a): “A Biological Basis for Expected and Non-Expected Utility”, Journal of Economic Theory, 68, 397-424. [20] Robson, A. (2001): “The Biological Basis of Economic Behavior,” Journal of Economic Literature, 39, 11-33. [21] Robson, A. (2002): “Evolution and Human Nature,”Journal of Economic Perspectives, 16(2), 89-106. [22] Robson, A. and L. Samuelson (2009): “The Evolution of Time Preference with Aggregate Uncertainty,” The American Economic Review, 99(5), 1925-1953. [23] Spence, M. and R. Zeckhauser (1972): “The E¤ect of the Timing of Consumption Decisions and the Resolution of of Lotteries on the Choice of Lotteries,”Econometrica, 40(2), 401-403.
7
Appendix n
Proof of Theorem 2. Fix an environment, (flk gk=1 ; f Lk =
k
; Gk ;
n
k
o (!)
!2
k
; Fek ( j!)
denote c (lk ). For notational convenience, assume that continuously compounded growth rate of the gene is n X
k 1
k=1
+ ::: +
k
n k gk=1 ) ;
; Fnk ( j!)
!2
\
and an arbitrary gene c. Let
m
U (Lk ) :
!2
= f;g if k 6= m. We show that the
(41)
n
In other words, the asymptotic growth rate of the gene is the average of the continuously compounded growth rates induced by the chosen lotteries weighted by the frequencies of these choices. Note that the statement of the theorem follows from (41), because the fastest growing gene maximizes the summation in (41) pointwise, that is, it satis…es (22). We show that the dynamics of this population can be described by a lottery of the form (3). The set of possible states of the world,
c
, is [nk=1
k
. Since choice problems arrive independently
of ! and according to constant arrival rates, the lottery Lk determines the dynamics during a
32
fraction of
k= ( 1
+ ::: +
n)
of the times. Therefore, the stationary distribution on
c
, Gc , is
given by k
Gc (!) = 1
if ! 2
k
k
. Similarly, conditional on ! 2
Gk (!)
+ ::: +
(42)
n
, the distribution of net birth rates, the arrival rate of
a natural disaster, and the distribution of survival probabilities are given by c
Fec ("j!) = Fek ("j!) ;
k
(!) =
(!) , and Fnc ( j!) = Fnk ( j!) ;
(43)
respectively. By (21), the continuously compounded growth rate of the population is given by Z Z Z Z c c "dFe ("j!) dG (!) + log dFnc ( j!) c (!) dGc (!) = =
Z n X
k=1 n X
k=1
!2
k
Z
"dFec
("j!) dG (!) +
Z
log
!2
Z
k 1
c
+ ::: +
k
!2
n
Z
k
Z
c
dFnc ( j!)
"dFek ("j!) dGk (!) +
Z
log
!2
k
(!) dGc (!)
Z
dFnk ( j!)
k
(!) dGk (!) ;
where the second equality follows from (42) and (43). Finally, notice that the last line of the previous equality chain is just (41). Let yt and yt denote the size of the total adult population and the
Proof of Theorem 3.
size of the -old population at time t. The number yt does not include the newborns at time t. We normalize y0 to be one. In addition, let that is, yt =
t
denote the fraction of -old individuals at time t,
t yt . Next, we establish the relationship between yt and yt
individuals at time t is the number newborns at t X
yt1 =
yt
1r
(! t
1)
= yt
1
1
= yt
1 t 1.
weakly older than 2 are those adults at time t
=2
X
yt
1
X
1) ;
(1
d (! t
1 who survived that period, that is 1 ))
= yt
1
X
t 1
(1
d (! t
1 )) ,
=1
where the equality follows from yt X
(! t
The number of individuals at time t who are
=1
yt =
t 1r
=1
where the equality follows from yt
yt =
The number of age-1
1; which is
=1
X
1.
1
= yt
yt = yt
=1
1
1 t 1.
X
t 1
To sum up the previous two displayed equalities: (r (! t
1)
+1
d (! t
=1
Using this formula recursively, yt =
tY 1
v=0
"
X
v
(r (! v ) + 1
=1
33
#
d (! v )) .
1 )) .
Hence, the continuously compounded growth rate of the population is Pt 1 P d (! v )) log yt =1 v (r (! v ) + 1 v=0 log lim = lim t!1 t!1 t t Using Birkho¤’s Theorem, the right-hand side of the previous equality can be rewritten as " # Z X e (!; ) . log (r (!) + 1 d (!)) dG =1
Plugging (30) and (31) into the previous expression yields (32).
Proof of Theorem 4. Let yt and yt denote the size of the total population and the size of the -old population at time t. We normalize y0 to be one. Using equations (35) and (36), the average net-birth rate in the population is Z d (! t )) d : t (r (! t ) =0
Therefore, similarly to (10), the population dynamics is governed by the following di¤erential equation: :
y t = yt The solution to this equation is yt = e
Z Rt
t
(r (! t )
d (! t )) d .
=0
z=0
R
=0
z (r
(! z ) d (! z ))d dz
:
Therefore, the continuously compounded growth rate can be written as Rt R (r (! z ) d d (! z )) d dz log yt = lim z=0 =0 z : lim t!1 t!1 t t The right-hand side is the time average of the net birth rate of the population. By Birkho¤’s Theorem, and plugging in (35) and (36), it is just (37). Proof of Theorem 5. Fix lotteries L1 ; L2 and L3 such that they all specify ( ; G) and generate the Markov process d over Z Z Z Z
. Suppose that L1 L2 , that is, Z Z e (!; ) "dF1 ("j!; ) dH1 ( j!; ) d dG
0
0
This inequality is equivalent to Z Z Z 0
+ (1 Z Z + (1
)
Z Z
0
Z
0
)
Z Z
0
Z
"dF2 ("j!; )
"dF1 ("j!; ) Z
Z
Z
e (!; ) dH3 ( j!; ) d dG
e (!; ) dH2 ( j!; ) d dG
"dF3 ("j!; )
34
e (!; ) . dH2 ( j!; ) d dG
e (!; ) dH1 ( j!; ) d dG
"dF3 ("j!; )
"dF2 ("j!; ) Z
Z
Z
Z
e (!; ) . dH3 ( j!; ) d dG
Since L1 ; L2 ; L3 2 L (d) ; the previous inequality is equivalent to Z Z
Z Z
0
0
Z Z
"d [ F1 + (1
) F3 ] ("j!; )
"d [ F2 + (1
This last inequality is just L1
Z Z
) F3 ] ("j!; ) (1
) L3
L2
35
d [ H1 + (1 d [ H2 + (1 (1
e (!; ) )] H3 ( j!; ) d dG
e (!; ) : )] H3 ( j!; ) d dG
) L3 , hence, (39) indeed holds.
