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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.

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 [^{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 [

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.

A genotype consists of an

where chromosomes come by pairs, one has

The survival rate of an individual only depends on its genotype, and is denoted by

The survival function is monotonic at locus

Thus, having more of a high allele at locus

We will say that a locus

We assume a quite general process for transmitting genes to offsprings, which in particular is compatible with real-world genetics. When genotypes

1. Gene conservation

This says that on average, the number of high alleles at locus

2. Allele independence

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

3. Mixing

For any

there exists

The RHS of (5) is the maximum number of H-alleles at locus

4. Symmetry

5. Monotonicity

For any

This assumption says that if instead of

applying the

partial distribution of the genotype at all other loci except

These assumptions allow to write down the demographic evolution equations of each genotype. We denote by

There are

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

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

LEMMA 1―Assume the mixing property holds. Assume there exists a steady state, a locus

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 at locus

one

down with

procedure to locus

At that stage

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

(ii)

Assume (A3) and (A4) holds. Then in any steady state with

PROOF―The frequency of the high allele at

In steady state, we have that

and

The term

That can be rewritten as:

This formula rests on the fact that all the genotypes such that

Furthermore, the allele independence property implies that for

^{4}If

^{5}The only other possibility is to only have genotypes such

where ^{4} Note that one must have

Hence:

Now, if locus

This inequality rests on the fact that

We now show that unless ^{5} Next, note that if there exists

that

Thus, if

Alternatively, consider the case where

exists

taking

From (14), we get that

Once again, there exists a pair

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

Going back to (11), we see that

where the strict inequality comes from the fact that

We now have

where we have applied gene conservation and

Observe that

Furthermore, one can write

Since

last term is constant in

property means that the average mortality of offsprings improves when one parent is genetically enhanced at locus

Let us now go back to (17), which we can rewrite

For a given

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

The last set of inequalities tell us that since parents who have a greater

We now describe how an individual’s genotype

The alleles present at a given locus

where

Finally the individual’s fitness is

where

Under autarky, we have

PROPOSITION 2―Under autarky, all loci are selective. Therefore, in any steady state such that

Proof―Type

Note that the case

Let us now look at the trade case. Each good

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

Types with lower incomes must achieve lower fitness and therefore disappear in the long run.

Furthermore,

with fewer H-alleles at locus

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

(ii) In any LRE such that

(iii) In any LRE, there exists a locus

Proof of (i)―Iterating the mixing property with appropriately chosen parents, one can easily show that if

genotype

Assume there exists a genotype

Proposition 1, under assumption (ii), would then imply that

Proof of (ii)―The price vector defined by (20) is the one which makes type

and

Since

supply good

The income of type

where the last inequality comes from (21). But, this cannot hold since it again implies

iterating Lemma 1 implies that

Proof of (iii)―Suppose not; then by iterating the mixing property with appropriately chosen parents, one can prove that

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

To do so, for any subset

genotypes such that their loci saturated with

PROPOSITION 4―Let

Then there exists an LRE with a distribution

Proof―We first prove that this condition is necessary. The RHS of (22) is the total time supplied by genotypes in

capita (equal to the income of any genotype) must be equal to

amount of good

implies that one

Let us now prove sufficiency. In order to do so, we construct a set of functions

^{6}One can trivially check that such an allocation exists, since one

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

To construct the^{6} Then we move from stage

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

Assume therefore that it is not the case. Then neither

Case A. Assume there exists a partition

and:

That is, people who do produce goods in

For any good

Clearly, one has

This strict inequality comes from the fact that

Furthermore,

This is because if

Interverting, we get

Now, note that,

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

PROPERTY Q:

(c)

(d)

To construct such a chain, proceed as follows. We will write

Start from a set

If

If not, then

More generally, at each iteration

Next, we can use such a chain to construct a new allocation of labor for stage

and

Define the new allocation as follows:

The new allocation clearly still satisfies (23) (as

Finally, we note that either

(i)

(ii) Or, the chain

Thus, at each stage, the quantity

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

positive fraction of genotypes with

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