Natural Science Vol.07 No.03(2015), Article ID:54398,13 pages
10.4236/ns.2015.73012
Equilibrium Allele Distribution in Trading Populations
Gilles Saint-Paul
Paris School of Economics, New York University, Abu Dhabi, UAE
Email: gstpaulmail@gmail.com
Copyright © 2015 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 13 February 2015; accepted 3 March 2015; published 4 March 2015
ABSTRACT
This paper extends the results of Saint-Paul (2007) regarding the long-run survival rates of alleles in trading populations, to a more general context where the number of loci is arbitrarily large under general assumptions about sexual reproduction. The central result is that fitness-reducing alleles can survive in a trading population, provided their frequency is not too large. However, the greater the number of loci that matter for fitness, the more stringent the conditions under which these alleles can survive.
Keywords:
Gene-Culture Coevolution, Markets, Division of Labor, Population Genetics
1. Introduction
Can culture affect the genetic makeup of a population? While this question has been dealt with some detail regarding cultural institutions such as cooperation and social norms,1 there is much less work dealing with a key component of culture: markets.2 Do we expect populations who trade for long enough to develop a different distribution of alleles compared with population where individuals remain in relative autarky?
In Saint-Paul [8] ,3 I consider the evolution of the gene pool in a population under alternative economic institutions, and show that alleles that cannot survive natural selection under autarky can survive under trade, because individuals can specialize in activities so as to avoid the fitness disadvantages associated with these alleles. The results are based on a very simplified representation of sexual reproduction, with only one chromo- some (instead of pairs of chromosomes), and only two loci that determine the individual’s productivity at two activities that affect fitness.
This paper generalizes these results for a more general system of sexual reproduction, with an arbitrary number of chromosomes and loci. Its contribution is twofold. First, it provides a set of assumptions under which one can meaningfully state that some alleles dominate their alternatives and eventually eliminate them in the long run. Second, it extends the results in Saint-Paul [8] by characterizing the distribution of alleles for a trading population in a long-run equilibrium (LRE), defined as a stationary distribution of alleles which is also an equilibrium in an economic sense.
The central result is that fitness-reducing alleles can survive in a trading population, provided their frequency is not too large. However, the greater the number of loci that matter for fitness, the more stringent the conditions under which these alleles can survive. That means that in the long run, we expect low alleles to survive only at a relatively small number of loci. Knowing more about the long-run distribution of alleles when their initial distribution does not satisfy the conditions for an LRE would involve analyzing the dynamics, which I do not do here but is an interesting topic for further research.
2. Notations and Genetic Properties of Stationary Populations
A genotype consists of an
-tuple
, where
denotes a particular locus, and
is interpreted as the number of alleles of the “high type” at locus
(in the actual world where chromosomes come by pairs, one has
). Therefore, there are
alleles of the “low type” at locus
. The set of possible genotypes is denoted by
. We will also denote by
the
th element
of
.
2.1. The Survival Function
The survival rate of an individual only depends on its genotype, and is denoted
by. Note that
the
function is not independent of culture. The opportunity to trade and specialize
will dramatically change the
mapping. It
is useful to introduce the genetic improvement operators
, which, for any genotype
such that
,
maps it into another genotype
,
defined by
,
and
.
Note that
.
The survival function is monotonic at locus
if it satisfies
(1)
Thus, having more of a high allele at locus
cannot increase mortality, everything else equal. Note that this assumes that the
role played by an allele in mortality has the same sign regardless of what other
alleles are present.
We will say that a locus
is selective if
(2)
2.2. The Distribution of Offsprings
We assume a quite general process for transmitting genes to offsprings, which in
particular is compatible with real-world genetics. When genotypes
and
mate, the fraction of their offsprings with genotype
is given by a probability distribution function
.
We shall assume that it satisfies the following properties:
1. Gene conservation
(3)
This says that on average, the number of high alleles at locus
among offsprings, denoted by
,
is equal to the its average between the two parents. For a given pair of parents,
the average among actual off- springs will be different from the parental average.
However, with a continuum of individuals, the law of large number will apply, and
will be equal to the population average of the number of H-alleles at
among all offsprings of all couples with genotypes
and
.
2. Allele independence
(4)
This assumption tells us that, among offsprings with the same parental genotypes,
the distribution of other genes among those who have the same number of high alleles
at locus,
does not depend on that particular number. If that property did not hold, having
many good alleles at one locus could in principle be systematically correlated with
having many bad alleles at another locus, and this complementarity could sustain
a positive amount of mortality-increasing alleles in the long-run, or, conversely,
eliminate mortality-reducing ones.
3. Mixing
For any,
,
for
all
such that
(5)
there exists
such that
,
for
,
and
.
The RHS of (5) is the maximum number of H-alleles at locus
if one inherits
alleles from each parent; the LHS is the minimum number of H-alleles. That assumptions
says that for any number between these two bounds, there is a positive probability
for a couple
,
to
have an offspring with exactly that number. Furthermore, we can pick up that offspring
such that at all other loci, its number of high alleles is between that of its two
parents. Loosely speaking, that means that the distribution of offsprings spans
all possible cases.
4. Symmetry
(6)
5. Monotonicity
For any,
any
,
any
such that
,
and any
such that
(7)
(8)
This assumption says that if instead of,
a genetically improved genotype at locus
mates with
,
then holding the alleles at other loci constant, the proportion of H-alleles at
locus
improves in a first-order stochastic dominance sense: offsprings are more likely
to have a higher umber of H-alleles at
.
Formally, applying the
operators starting from an initial genotype
such that
,
allows to compute the marginal distribution of
among offsprings holding other alleles constant. The last equality says that the
partial distribution of the genotype at all other loci except
is invariant when one mates with a genetic improvement of
at
instead of
.
2.3. Demographics
These assumptions allow to write down the demographic evolution equations of each
genotype. We denote by
total population at date
and by
the fraction of people with genotype
.
People mate randomly.
There are
matches of types
and
at date
.
They produce
offpsrings, and a frac- tion
of offsprings with genotype
reach maturity. Consequently,
evolves
according to
Adding all these equations across all possible genotypes we get that
It is also useful to define the population frequency of high alleles at locus:
Note that if the gene conservation law holds, then one also has
(9)
3. Elimination of Less Fit Alleles
In this section, I provide the basic results regarding the elimination of less fit
alleles. A first lemma, which derives from the random mating and mixing properties,
states that if a genotype exists and if a high allele exists in the population at
locus,
then we can find another genotype that differs from it only in that it is “improved”
at locus
,
unless, of course, the initial genotype has the maximum number of H-alleles at
.
LEMMA 1―Assume the mixing property holds. Assume there exists a steady state, a
locus
and a genotype
such that in that steady state,
,
,
and
.
Then
.
PROOF―First note that because of random mating there exists a positive measure of matches between two arbitrary genotypes, provided these genotypes are in positive measure in the parent population.
If,
the mixing property applied at locus
implies that offsprings of
with itself include
with positive probability. Assume
.
Since
,
there exists
such that
and
.
We can then iterate the mixing property, by looking at stage
at the mates between
and
,
starting with
.
If at stage
,
there exists
such that
,
say
,
by applying
the mixing property at locus
we know that among the offsprings between
and
,
there exists one
such
that
―implying
in steady state―
,
and
.
In other words, the “genetic distance” between
and
strictly goes down with
.
Once we have reached the stage where
for all
,
we apply the same procedure to locus
,
until we have produced an offspring such that
,
and
.
At that stage.
Q.E.D.
The following key result tells us that genes which increase mortality eventually disappear:
PROPOSITION 1―Assume that one of these two conditions holds:
(i) locus
is selective, OR
(ii)
is
monotonic at
and there exists one genotype
such that
in steady state,
,
and
.
Assume (A3) and (A4) holds. Then in any steady state with,
one must have
.
PROOF―The frequency of the high allele at
evolves according to
In steady state, we have that,
(10)
and
(11)
The term
can be rewritten as follows:
That can be rewritten as:
This formula rests on the fact that all the genotypes such that
can be deducted by applying the transform
to all genotypes such that
.
Furthermore, the allele independence property implies that for
such that
,
(12)
where
is the total fraction of genotypes with
among the offsprings of
and
.4
Note that one must have
(13)
Hence:
Now, if locus
is selective, then
is strictly increasing in
.
Consequently we have
(14)
This inequality rests on the fact that.
It holds with a strict inequality unless all the
but one are equal to zero.
We now show that unless
or
,
there exists a pair of genotypes
such that
,
,
and (14) strictly holds. First note that if
,
there exists a genotype
such that
,
and
.5
Next, note that if there exists
such that
,
the mixing property implies that for two parents of the same genotype
, there is a
positive probability of having an offpsring
such that
,
for any
between
and
.
As long as
and
,
there are more than two values of
that satisfy that property. Consequently, there are at least two strictly positive
values of
,
and one can take
.
Thus, if,
it must be that there exists a pair
such that
,
,
and (14) strictly holds.
Alternatively, consider the case where
is monotonic. Then (14) also holds. Furthermore, assume there exists
such that
,
,
and
.
Let
.
Then (14) will hold with strict inequality for
,
such
that
,
,
and
.
If
,
taking
and applying the mixing property to locus
,
generates both offsprings with
and
implying
that
and
.
If
,
the Lemma implies that
.
Taking
and
then generates both offsprings with
and
implying that
and
.
Thus, we can again pick up a genotype
and a pair
such that
,
and
(14) strictly holds.
From (14), we get that
(15)
Once again, there exists a pair
such that
,
,
and (15) strictly holds. The RHS can be rewritten
where the first step derives from (13) and the second one from (12).
Inequality (15) means that the fitness of the high alleles in the gene pool of the
offsprings of
and
is higher than the average fitness of the offsprings as individuals, because those
with more H-alleles at
live longer. In order to get that, the allele invariance property is needed. Otherwise,
it could be that the offsprings of
,
that
have a high
have a lower fitness than the others because they are systematically poorly endowed
at other loci.
Going back to (11), we see that
(16)
where the strict inequality comes from the fact that
for at least one pair
such that
.
We now have
(17)
where we have applied gene conservation and
is defined as
Observe that
can be rewritten as
Furthermore, one can write
where
.
Iterating the monotonicity property, we find that
is nonincreasing in
,
while
does not depend on
.
We then have that
Since
is monotonic, the term in brackets is nonpositive. Thus, the sum is nondecreasing
in
,
while the last term is constant in
.
Therefore, the LHS is nondecreasing in
,
for any
such that
.
Summing this property across these
’s,
we also find that
is nondecreasing in
.
Roughly, that property means that the average mortality of offsprings improves when
one parent is genetically enhanced at locus
.
The monotonicity property is needed to get that. Otherwise, it could be that parents
with more H-alleles at
,
everything else equal, have an
systematically biased toward high-mortality genotypes.
Let us now go back to (17), which we can rewrite
For a given,
we have that
,
that
increases with
and that
weakly increases with
.
Thus, once again, we have the following inequality:
Consequently,
where the steady-state condition (10) has been used to derive the first term.
By virtue of (16), (3) and (6), the last term in that formula must be equal to,
so that
.
(16) then implies that
,
which is a contradiction. Hence, it must be that either
or
.
Q.E.D.
The last set of inequalities tell us that since parents who have a greater
have children with a higher fitness, these parents’ children tend to increase the
survival rate of the high allele at
relative to average. Since, in addition, the survival rate of the high allele at
among their children is greater than their children’s average survival rate, these
two effects together imply that the fitness of the high allele at
is strictly higher than average. But that cannot be in steady state, unless
or
.
4. Autarky
We now describe how an individual’s genotype
affects his/her productivity at various activities, depending on the ecomic setting.
The alleles present at a given locus
determine the individual’s productivity at a corresponding activity denoted by the
same index
.
This productivity is a strictly increasing function
of
,
the number of H-alleles at locus
.
Any individual has a total time endowment equal to 1. The time allocation constraint
of genotype
is therefore given by
(18)
where
is the individual’s output in activity
.
Finally the individual’s fitness is
where
is the individual’s consumption of activity
,
and
is the “utility function”, which is concave in each argument, and satisfies the
“Inada conditions”:
,
.
Under autarky, we have,
and the following result holds:
PROPOSITION 2―Under autarky, all loci are selective. Therefore, in any steady state
such that,
,
all individuals are of genotype
,
i.e. the H-allele is fixed at all locations.
Proof―Type
has a more favorable time budget constraint than type
.
Therefore, it achieves a higher fitness. The rest follows from the previous subsection.
Q.E.D.
Note that the case
is not of interest: it means that the high allele does not exist at that locus.
5. Trade
Let us now look at the trade case. Each good
is traded at price
.
We assume the following normalization for the price vector
(19)
People allocate their time between the various activities so as to maximize their
income,
subject to the time allocation constraint (18). Their demand vector is the one which
maximizes
subject to their budget constraint:
Types with lower incomes must achieve lower fitness and therefore disappear in the long run.
Furthermore,
must
be monotonic at all loci. The reason is that the vector
supplied by a geno- type
can also be supplied by genotype
.
On the other hand, all loci need not be selective, as genotypes with fewer H-alleles
at locus
may achieve the same income as fitter genotypes, by just specializing.
Define a long-run equilibrium (LRE), as a stationary state such that the economy is in equilibrium, i.e. each genotype sets its supply and demand as just described, and markets clear for each good. The following proposition generalizes the results derived for the two-loci case in Saint-Paul (2007).
PROPOSITION 3―(i) In any LRE such that,
,
a given type only supplies goods corresponding to loci in their genotype where they
have the highest number of
-alleles:
(ii) In any LRE such that,
the price vector is
such that
(20)
(iii) In any LRE, there exists a locus
such that
,
i.e. allele H is fixed at locus
.
Proof of (i)―Iterating the mixing property with appropriately chosen parents, one
can easily show that if,
,
in steady state there exists a strictly positive supply of genotypes with a arbitrary,
strictly positive number of H-alleles
at each locus
.
In particular, there exists a strictly positive mass of the best genotype
. Next, note
that if
,
then genotype
achieves higher fitness than
,
and hence
.
Assume there exists a genotype
such that
for
such that
Clearly, the plan
achieves a strictly higher income level and is feasible (i.e. satisfies (18)) for
.
Consequently,
,
implying
.
But, given that
is monotonic, Proposition 1, under assumption (ii), would then imply that
, which makes
it impossible for
to exist. Consequently, any type
only supplies goods where it has an
-allele.
Proof of (ii)―The price vector defined by (20) is the one which makes type
indifferent between all activities. Assume there exists an LRE with a different
price vector. Then there exists a pair of goods
such that
(21)
and
since more income is yielded for type
by offering good
than good
.
Since
satisfies the Inada conditions, the demand for good
is strictly positive; since
does not
supply good,
there exists
such that
and
.
By virtue of (i),
.
Further- more,
,
otherwise
would prefer to supply
instead of
as well.
The income of type
is
.
The supply vector
is feasible for type
,
since
is more productive than
at all activities. The supply vector
defined by
,
,
,
,
also
satisfies (18) for
.
Therefore,
where the last inequality comes from (21). But, this cannot hold since it again
implies
Consequently, there exists
such that
Furthermore, as
,
iterating Lemma 1 implies that.
Monotonicity of
then implies that (ii) in Proposition 1 is satisfied. Consequently,
. But that contradicts
the requirement that
.
Q.E.D.
Proof of (iii)―Suppose not; then by iterating the mixing property with appropriately
chosen parents, one can prove that.
But that contradicts (i). Q.E.D.
The preceding proposition tells us what properties an LRE must necessarily have,
but does not tell us whether an LRE exists and whether, as in the preceding analysis,
one can construct equilibria with a positive level of some
-alleles.
We now establish a result which tells us that an LRE exists with a strictly positive
proportion of L-alleles, provided these alleles are not too frequent.
To do so, for any subset
of
we define
as
is the set of all
genotypes such that their loci saturated with
-alleles
(which define the activities at which they can possibly specialize) are all in
.
PROPOSITION 4―Let
be the inverse demand function for the fitness maximization problem of an individual
with income R facing price vector
Let
Then there exists an LRE with a distribution
of genotypes if and only if this distribution satisfies the following property:
(22)
Proof―We first prove that this condition is necessary. The RHS of (22) is the total
time supplied by genotypes in
(relative to the population's total time). Proposition 3, (i) implies that it must
be allocated among goods
such that
,
i.e. among goods in
.
It also implies that in any candidate equilibrium, income per
capita (equal to the income of any genotype) must be equal to
Thus,
is
the per capita amount of good
consumed and produced in any candidate equilibrium. The LHS of (22) is therefore
the total time input needed to produce all the goods in
.
It must be greater than or equal to its RHS, since genotypes in
cannot produce any other good. Otherwise, supply would exceed demand. Note that
(22) applied to
implies that one
-allele
is fixed (the RHS is then the total supply of all genotypes
such that
).
Also, (22) applied to
boils down to Walras’ law, since it is equivalent to
,
and by Walras law
.
Let us now prove sufficiency. In order to do so, we construct a set of functions,
representing the share of time of genotype
devoted to activity
,
such that:
(23)
(24)
(25)
6One can trivially check that such an allocation exists, since one
-allele
is fixed, all genotypes have at least one locus where
.
If we are able to construct such functions, then this is indeed an equilibrium,
since supply equals demand for all goods, and since the price vector in (20) implies
that a genotype is indifferent between supplying all the goods at which it has
-alleles.
To construct the,
we use the following algorithm. We start from any arbitrary allocation
satisfying (23) and (25). This defines the initial stage.6 Then we move
from stage
to stage
as follows. At the beginning of stage
,
the set
can be partitioned into three subsets:
That is, those goods for which supply equals demand, those for which there is excess
demand, and those for which there is excess supply. Note that since,
is
empty if and only if
is empty. If
,
then we have an equilibrium, and the algorithm stops.
Assume therefore that it is not the case. Then neither
nor
is empty. We now distinguish two cases.
Case A. Assume there exists a partition
of
such that
and:
(P)
That is, people who do produce goods in
cannot produce goods in
.
For any good,
let
Clearly, one has,
for all
.
We then have
This strict inequality comes from the fact that
and from the fact that
is non-empty.
Furthermore,
This is because if
and
,
then
,
implying
.
Therefore
.
Interverting, we get
Now, note that,
,
:
if I produce one good in
,
all my loci with
-alleles
are also in
.
Consequently,
which clearly violates assumption (22). Case A is therefore ruled out.
Case B. Assume then that there exists no such partition. We can construct a chain
of
goods
such that the following property holds:
PROPERTY Q:
(a)
(b)
(c),
for
(d),
,
and
To construct such a chain, proceed as follows. We will write
if (
,
and
).
In this case, we call
the set of genotypes that satisfy this property:
.
Property (d) implies that the chain we want to construct is such that
Start from a set.
As property (P) is violated, there exists
,
,
such that
If,
stop the procedure there, and take
.
If not, then.
Add
to
:
.
Since
,
it must be that
,
.
Use again the fact that (P) is violated. There exists
,
,
such that
.
Given that either
or
,
there exists a chain of length
,
such
that
,
,
,
and
.
If
,
we use that chain and stop the procedure. Otherwise, we add
to
,
and iterate again.
More generally, at each iteration,
there is a set
such that
,
and such that for all
,
there exists a chain
such that
,
,
and
(The chain property)
Because (P) is violated, there exists
and
,
such that
.
Let
be the chain corresponding to
.
If
,
we use the chain
and stop the procedure. If
,
we use
and iterate the procedure. As the new member
is connected to
via the chain
,
still
satisfies the chain property. As
,
it also satisfies
.
As the number of elements in
goes up by one unit at each iteration, one must find an
in
a finite number of iterations.
Next, we can use such a chain to construct a new allocation of labor for stage.
Let
and
Define the new allocation as follows:
The new allocation clearly still satisfies (23) (as),
and (25) (as
).
Futhermore, one has
,
as
.
Finally, for
,
.
Hence, all markets that were in equilibrium remain so. Furthermore, as
, market
weakly remains in excess supply, and similarly market
weakly remains in excess demand. Therefore:
Finally, we note that either
(i),
which will be true provided
or
.
In such cases the new allocation restores equilibrium in market
(resp.
).
(ii) Or, the chain
and its associated chain of genotypes
no longer satisfy Q; that is the case if
for some
,
in which case
.
In such a case, we have constructed a new allocation such that
has increased by at least one unit, and which satisfies (23) and (25).
Thus, at each stage, the quantity
strictly increases. As it is bounded, the proce- dure cannot go on forever, and
the only case in which one cannot iterate it is if
.
This proves
the existence of equilibrium. Q.E.D.
Clearly, conditions (22) are pretty stringent, so that it is not straightforward to construct an equilibrium.
However for
close enough to 1, i.e.
small
enough when
,
they are clearly satisfied, since
appears on the RHS only for
,
in which case (22) is always satisfied with equality, due to Walras’ law:
. Therefore there
always exist equilibria with a strictly positive fraction of genotypes with
-alleles,
provided this fraction is small enough.
Note that the greater the number of loci, the greater the number of conditions that
must hold. Intuitively, it suggests that the equilibrium fraction of
-alleles
must become smaller. Intuitively, if the initial distribution of alleles in the
population is such that (22) is violated, we expect a number of H-alleles to eventually
become fixed, which is equivalent to a reduction in
.
The process would continue until
is small enough for the number of relevant activities not to be too large, so that
(22) holds.
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NOTES
1In this line of work, grouyp selection often plays an important role. See for example Cavalli-Sforza and Feldman [1] ; Lumsden and Wilson [2] ; Gintis [3] , Boyd and Richerson [4] .
2Interesting surveys on interactions between the economic and biological spheres include Hirshleifer [5] , Robson [6] , and Seabright [7] .
3I addition to this, the most closely related paper is Horan et al. [9] . A related literature (see Hammerstein [10] , and in particular Bowles and Hammerstein [11] ), studies the rise of markets and specialization in animal societies, but does not draw this paper’s implications for the gene pool.
4If,
we can write down the same steps using the smallest value of
such that
as a benchmark.
5The only other possibility is to only have genotypes such
and such that
,
but random mating and mixing imply that they will produce offsprings such that
.