Status competition, inequality, and fertility
Shenk, Kaplan & Hooper
Electronic Supplementary Materials (ESM)
Status competition, inequality, and fertility:
Implications for the demographic transition
Mary K. Shenk, Hillard S. Kaplan & Paul L. Hooper
Contents
ESM Section 1. Additional Model Details
ESM Section 2. Additional Figure
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Status competition, inequality, and fertility
Shenk, Kaplan & Hooper
ESM Section 1. Additional Model Details
Fitness maximization
Each parents attempts to maximize fitness with respect to two decision variables: a and s. The
parent’s fertility f is implied by any given values of a and s. Isolating f in eq. (3) in the main text yields 𝑓 =
𝑔Vp ⁄(𝑎 + 𝑠). Substituting this expression into eq. (2) in the main text yields the following fitness
maximand:
W(𝑎, 𝑠) =
𝑔Vp A(𝑎)R(𝑠̇ )
𝑎+𝑠
(E1)
General simulation details
The simulation was implemented in the R programming language (R Core Team 2015). Each run
of the simulation consists of multiple generations that reproduce, invest in their offspring, and die.
Ecological parameters are held constant within a run, but vary across runs. Each runs continues for an
average of 50 generations, a length of time selected to allow the expression of a representative sample of
the dynamics of a given ecology.
In each generation, the decision variables a and s (and thus f) are optimized to maximize expected
fitness (eq. (E1)) for each parent given the distribution of ṡ (i.e. post-shock status investments) observed
in their own generation using the optim() function. To yield 𝑠̇ , the value of s chosen by a parent is then
multiplied by the shock value 𝜀, sampled from a gamma distribution with mean 1 and shape parameter
exp(Q) using the rgamma() function. Inequality in shocks is greater for smaller values of exp(Q). For ease
of interpretation, inequality in shocks is reported in terms of their coefficient of variation, CV(𝜀).
As noted in the discussion, the simulation does not represent variation across parents in the
payoffs to PI. As a result, the optimal investment is invariant across parents (i.e. a single scalar value
within each generation), despite variation in budgets (Kaplan 1996). As a result, the multiplication of this
value of s by a gamma distribution with a fixed shape parameter yields roughly equal degrees of
inequality across generations, without a build-up of inequality.
Simulation details underlying Fig. 1
Two populations were simulated with two different levels of inequality in shocks to social capital.
For the low-inequality condition, exp(Q) was set to 5, resulting in CV(𝜀) = 0.4 and gini(𝜀)=0.24; in the
high-inequality condition, exp(Q) was set to 1, resulting in CV(𝜀) = 1.0 and gini(𝜀) = 0.50. To illustrate the
effects of changing the shape of the effect of social capital function, the fitness-investment relationship
was plotted for five values of 𝑟̂ (0.25, 0.375, 0.5, 0.625, and 0.75) and relatively lower and higher values 𝜌
(2.5 and 10). (Given the illustrative intent of this figure, the range of values it explores is narrower than
that of Fig. 2, which reports a wider, more systematic exploration.)
Simulation details underlying Fig. 2 and ESM Fig. E1
A total of 1400 runs were simulated. In each run, the inflection point for the effect of social capital
function 𝑟̂ was sampled from the set (0.25, 0.375, 0.5, 0.625, 0.75). The slope parameter 𝜌 was sampled
from the set (0, 1.25, 2.5, 5, 10, 20). These parameter ranges produce reasonably wide variation in the
intensity and steepness of the effect of social capital function. For each run, the value of Q determining the
shape parameter for inequality in social capital was sampled from a uniform distribution ranging from
ln(1) to ln(15). These shape values yield gini coefficients in the range 0.14—0.51, similar to the range
observed for socioeconomic variables in small- and large-scale human societies (Borgerhoff Mulder et al.
2009).
The inflection point 𝑎̂ and slope parameter 𝛼 for the effect of survival-and-productivity capital
were held constant at 0.5 and 10, respectively. The growth constant g was held fixed at 12. The number of
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Status competition, inequality, and fertility
Shenk, Kaplan & Hooper
generations in each run was determined stochastically as 1/T, with T sampled from a normal distribution
with mean 1/50 and coefficient of variation 1/3. The size of the population in each run was held constant
at a value sampled from a normal distribution with mean 500 and coefficient of variation 1/3. The mean
value of 𝑠̇ in the first generation of each run was sampled from a normal distribution with mean 0.1 and
coefficient of variation 1/3. To summarize the results, the loess() function was used with default settings
to estimate local polynomial fits for fertility (Fig. 2) and social capital (ESM Fig. E1) as a function of CV(𝜀)
for each unique combination of 𝑟̂ and 𝜌.
Simulation details underlying Fig. 3
One run was simulated for each of the following four conditions: (a) exp(Q) = 5 and 𝑟̂ = 0.25; (b)
exp(Q) = 1 and 𝑟̂ = 0.25; (c) exp(Q) = 5 and 𝑟̂ = 0.75; (d) exp(Q) = 1 and 𝑟̂ = 0.75. 𝜌 was held constant at
2.5 across conditions. The inflection point 𝑎̂ and slope parameter 𝛼 for the effect of survival-andproductivity capital were held constant at 0.5 and 10, respectively. The growth constant g was held fixed
at 12. Each run consisted of 50 generations each with 500 individuals. The mean value of 𝑠̇ in the first
generation of each run was sampled from a normal distribution with mean 0.1 and coefficient of variation
1/3.
Simulation details underlying Fig. 4
A total of 60 runs were simulated. 20 runs sampled uniformly from a gradient in which 𝑎̂ varied
linearly from 0.25 to 0.75. 20 runs sampled from a gradient in which 𝑎̂ varied across this range, and
exp(Q) varied linearly from ln(1) to ln(15). 20 runs sampled from a gradient in which 𝑎̂ and exp(Q)
varied, and 𝑟̂ also varied linearly from 0.25 to 0.75.
The slope parameters 𝜌 were 𝛼 were held constant at 2.5 and 10, respectively. The growth
constant g was held fixed at 12. The number of generations in each run was determined stochastically as
1/T, with T sampled from a normal distribution with mean 1/50 and coefficient of variation 1/3. The size
of the population in each run was held constant at a value sampled from a normal distribution with mean
500 and coefficient of variation 1/3. The mean value of 𝑠̇ in the first generation of each run was sampled
from a normal distribution with mean 0.1 and coefficient of variation 1/3. The loess() function was used
with smoothing parameter 0.9 to estimate local polynomial fits for fertility as a function of the ecological
gradient for each of the three gradients.
References
Borgerhoff Mulder, Monique, et al. (2009) Intergenerational wealth transmission and the dynamics of
inequality in small-scale societies. Science 326: 682-688.
Kaplan, Hillard. (1996) A theory of fertility and parental investment in traditional and modern human
societies. American Journal of Physical Anthropology 101: 91-135.
R Core Team (2015) R: A Language and Environment for Statistical Computing. Vienna: R Foundation for
Statistical Computing.
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Status competition, inequality, and fertility
Shenk, Kaplan & Hooper
0.0 0.2 0.4 0.6 0.8 1.0
Social capital (s× )
(a)
r=0
^r = 0.25
^r = 0.375
^r = 0.5
^r = 0.625
^r = 0.75
0.3 0.4 0.5 0.6 0.7 0.8 0.9
ESM Section 2. Additional Figure
ESM Fig. E1. Mean optimal investment in social capital (s) as a function of
inequality in the inputs to status (CV(𝜀), varying across the horizontal axis) and
the inflection point (𝑟̂ , varying across the dashed and solid lines) and slope (𝜌,
varying across panels) of the effect of social capital function. The simulations
underlying this figure are identical to those underlying Fig. 2 in the main text.
Inequality (CV(e))
r = 1.25
0.0 0.2 0.4 0.6 0.8 1.0
Social capital (s× )
(b)
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Inequality (CV(e))
r = 2.5
0.0 0.2 0.4 0.6 0.8 1.0
Social capital (s× )
(c)
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Inequality (CV(e))
r=5
0.0 0.2 0.4 0.6 0.8 1.0
Social capital (s× )
(d)
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Inequality (CV(e))
r = 10
0.0 0.2 0.4 0.6 0.8 1.0
Social capital (s× )
(e)
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Inequality (CV(e))
r = 20
0.0 0.2 0.4 0.6 0.8 1.0
Social capital (s× )
(f)
4
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Inequality (CV(e))