141 lines
3.7 KiB
Text
141 lines
3.7 KiB
Text
---
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title: "Chapter 8"
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---
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# Chapter 8
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## Population size and highly skilled individuals
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We can use some theoretical distribution to represent the skills of people in a population
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Using a gamma distribution here we create a scenario where most people are concentrated at lower skill levels, while a longer tail extends towards high skill.
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The high-skill threshold is fixed, so increasing population size.
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If people copu only the 2% most skliied individuals, then a bigger population wil mathematically increases the number of skilled individual
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```{r skill-population-calculation}
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population_sizes <- c(500, 1000)
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skill_shape <- 2
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skill_scale <- 1.5
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high_skill_cutoff <- 6
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high_skill_proportion <- pgamma(
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high_skill_cutoff,
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shape = skill_shape,
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scale = skill_scale,
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lower.tail = FALSE
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)
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high_skill_counts <- round(population_sizes * high_skill_proportion)
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data.frame(
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Population = population_sizes,
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`High-skill proportion` = round(high_skill_proportion, 3),
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`Expected high-skill individuals` = high_skill_counts,
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check.names = FALSE
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)
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```
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```{r skill-population-figure, fig.width=9, fig.height=4.5, fig.cap="The same skill distribution in two populations. The blue tail marks individuals above the high-skill threshold."}
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skill_x <- seq(
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0,
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qgamma(0.999, shape = skill_shape, scale = skill_scale),
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length.out = 600
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)
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tail_x <- skill_x[skill_x >= high_skill_cutoff]
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skill_density <- dgamma(
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skill_x,
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shape = skill_shape,
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scale = skill_scale
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)
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tail_density <- dgamma(
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tail_x,
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shape = skill_shape,
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scale = skill_scale
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)
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common_y_max <- max(population_sizes) * max(skill_density) * 1.18
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line_cols <- c("#244A68", "#355E3B")
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body_cols <- c("#DCE8F0", "#D4DFC8")
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tail_cols <- c("#1769AA", "#3E713F")
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old_par <- par(
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mfrow = c(1, 2),
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mar = c(4.5, 4.5, 3, 1),
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oma = c(0, 1.5, 0, 0)
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)
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for (i in seq_along(population_sizes)) {
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people_density <- population_sizes[i] * skill_density
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high_skill_density <- population_sizes[i] * tail_density
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plot(
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skill_x,
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people_density,
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type = "n",
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xlim = range(skill_x),
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ylim = c(0, common_y_max),
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axes = FALSE,ylab="",xlab = ""
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)
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polygon(
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c(min(skill_x), skill_x, max(skill_x)),
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c(0, people_density, 0),
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col = body_cols[i],
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border = NA
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)
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polygon(
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c(high_skill_cutoff, tail_x, max(tail_x)),
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c(0, high_skill_density, 0),
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col = tail_cols[i],
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border = NA
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)
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lines(skill_x, people_density, lwd = 2.2, col = line_cols[i])
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abline(v = high_skill_cutoff, lty = 3, col = tail_cols[i])
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axis(
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1,
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at = c(
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qgamma(0.1, shape = skill_shape, scale = skill_scale),
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qgamma(0.93, shape = skill_shape, scale = skill_scale)
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),
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labels = c("Low skill", "High skill"),
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lwd = 0,
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lwd.ticks = 1
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)
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axis(2, las = 1, lwd = 0, lwd.ticks = 1)
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mtext("Skill level", side = 1, line = 2.5)
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arrows(
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min(skill_x), 0,
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max(skill_x), 0,
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length = 0.08,
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lwd = 1.2,
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xpd = NA
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)
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arrows(
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min(skill_x), 0,
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min(skill_x), common_y_max,
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length = 0.08,
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lwd = 1.2,
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xpd = NA
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)
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title(main = paste(
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"Population:",
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format(population_sizes[i], big.mark = ",")
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))
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text(
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high_skill_cutoff + 0.38 * (max(skill_x) - high_skill_cutoff),
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people_density[which.max(skill_density)] * 0.62,
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labels = paste(
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format(high_skill_counts[i], big.mark = ","),
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"high-skill\nindividuals"
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),
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col = tail_cols[i],
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font = 2
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)
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}
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mtext("Number of people", side = 2, outer = TRUE, line = 0.2)
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par(old_par)
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```
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If the skill level is transmitted culturally from these skilled individual to the next generation, we can then add different type of social transmission process to then get prediction about on the spread (and retention) of skills within the population.
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