You are called upon to advise an underdeveloped country on methods for increasing per-capita income. This problem briefly considers two difficulties you may encounter. It is an economics theory result that per-capita income is greater when accumulated capital per capita is greater. The idea is that, under suitable assumptions, since more capital is available it is used to help improve production. Do you think this applies to underdeveloped countries? What happens if capital is invested abroad or foreign capital is brought in? Let's assume that the economies theory result still applies. By definition, the capital accumulated in a year equals income (i.e., production) minus consumption.
(a) Fractional rates of growth are defined in the same way as net growth rates in biology: $x^{\prime}(t) / x(t)$. We denote the fractional rate of growth of $x$ by $x^*$. Let $K$ stand for total capital and $P$ for total population. Show that per-capita income is increasing if and only if $K^*>P^*$.
(b) One could suppose that $P^*$ and $K^*$ depend on per-capita income. Argue this point. Supposing it to be true, plot $P^*$ and $K^*$ as functions of per-capita income and show that intersections of the curves correspond to equilibria. How can you determine stability?
(c) In each of the following cases, discuss the shape of the $K^*$ and $P^*$ curves near the given income level and use (b) to explain why these effects can keep per-capita income from increasing.
(i) Rising expectations: At a certain income level, savings decrease because people try to mimic more affluent societies.
(ii) Population explosion: At a certain income level, improved sanitation and diet reduce the death rate, but the birth rate takes much longer to fall because it is the result of custom.
(d) That's the background for showing the ministers of the country some of the problems they face and what is going on. Now, advise them.