Malthusian Growth
Overview
Thomas Malthus was one of the first economists to theorize about economic growth and create a somewhat formal model.
Malthus' conclusions were pessimistic and earned economics the moniker of the "dismal science". Even though his conclusions
have proven to incorrect, his reasoning is still important for its assumptions and reasoning.
Diminishing Returns
One of Malthus' key assumptions was the assumption of diminishing returns to the factors of production.
That is, keeping everything else the same, increasing the amount of a factor of production by a single unit,
will return less than the prior unit.
In Malthus time, the primary factors of production was land and labor. Assuming that the amount of farmable
land is held constant, as you add additional workers, you get less production for each laborer. This is
because it is assumed that the most fertile land is farmed first, so the additional laborers can only farm
less fertile land.
Cobb Douglas Production Function
{% Y = AL^{\alpha}K^{1-\alpha} %}
{% \partial{Y}/\partial{L} = (1-\alpha)AK^{\alpha}L^{-\alpha} > 0 %}
The Malthasian Trap
A key element of the Malthusian trap is that the birth and death rate are a function of the economies
production. That is, as the economy produces more, people have more children, and the death rate goes down.
Assuming discrete time, the equation for the current amount of population can be written as:
{% L(t) = L(t-1) + B(t,y_{t-1}) - D(t, y_{t-1}) %}
