---
title: "Chapter 8"
---

# Chapter 8

## Population size and highly skilled individuals

We can use some theoretical distribution to represent the skills of people in a population
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. 
The high-skill threshold is fixed, so increasing population size.
If people copu only the 2% most skliied individuals, then a bigger population wil mathematically increases the number of skilled individual
```{r skill-population-calculation}
population_sizes <- c(500, 1000)
skill_shape <- 2
skill_scale <- 1.5
high_skill_cutoff <- 6

high_skill_proportion <- pgamma(
  high_skill_cutoff,
  shape = skill_shape,
  scale = skill_scale,
  lower.tail = FALSE
)

high_skill_counts <- round(population_sizes * high_skill_proportion)

data.frame(
  Population = population_sizes,
  `High-skill proportion` = round(high_skill_proportion, 3),
  `Expected high-skill individuals` = high_skill_counts,
  check.names = FALSE
)
```

```{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."}
skill_x <- seq(
  0,
  qgamma(0.999, shape = skill_shape, scale = skill_scale),
  length.out = 600
)
tail_x <- skill_x[skill_x >= high_skill_cutoff]

skill_density <- dgamma(
  skill_x,
  shape = skill_shape,
  scale = skill_scale
)
tail_density <- dgamma(
  tail_x,
  shape = skill_shape,
  scale = skill_scale
)
common_y_max <- max(population_sizes) * max(skill_density) * 1.18

line_cols <- c("#244A68", "#355E3B")
body_cols <- c("#DCE8F0", "#D4DFC8")
tail_cols <- c("#1769AA", "#3E713F")

old_par <- par(
  mfrow = c(1, 2),
  mar = c(4.5, 4.5, 3, 1),
  oma = c(0, 1.5, 0, 0)
)

for (i in seq_along(population_sizes)) {
  people_density <- population_sizes[i] * skill_density
  high_skill_density <- population_sizes[i] * tail_density

  plot(
    skill_x,
    people_density,
    type = "n",
    xlim = range(skill_x),
    ylim = c(0, common_y_max),
    axes = FALSE,ylab="",xlab = ""
  )

  polygon(
    c(min(skill_x), skill_x, max(skill_x)),
    c(0, people_density, 0),
    col = body_cols[i],
    border = NA
  )
  polygon(
    c(high_skill_cutoff, tail_x, max(tail_x)),
    c(0, high_skill_density, 0),
    col = tail_cols[i],
    border = NA
  )
  lines(skill_x, people_density, lwd = 2.2, col = line_cols[i])
  abline(v = high_skill_cutoff, lty = 3, col = tail_cols[i])

  axis(
    1,
    at = c(
      qgamma(0.1, shape = skill_shape, scale = skill_scale),
      qgamma(0.93, shape = skill_shape, scale = skill_scale)
    ),
    labels = c("Low skill", "High skill"),
    lwd = 0,
    lwd.ticks = 1
  )
  axis(2, las = 1, lwd = 0, lwd.ticks = 1)
  mtext("Skill level", side = 1, line = 2.5)
  arrows(
    min(skill_x), 0,
    max(skill_x), 0,
    length = 0.08,
    lwd = 1.2,
    xpd = NA
  )
  arrows(
    min(skill_x), 0,
    min(skill_x), common_y_max,
    length = 0.08,
    lwd = 1.2,
    xpd = NA
  )

  title(main = paste(
    "Population:",
    format(population_sizes[i], big.mark = ",")
  ))
  text(
    high_skill_cutoff + 0.38 * (max(skill_x) - high_skill_cutoff),
    people_density[which.max(skill_density)] * 0.62,
    labels = paste(
      format(high_skill_counts[i], big.mark = ","),
      "high-skill\nindividuals"
    ),
    col = tail_cols[i],
    font = 2
  )
}

mtext("Number of people", side = 2, outer = TRUE, line = 0.2)
par(old_par)
```