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+---
+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)
+```