The Malthusian Trap is a mechanism that describes how population growth suppresses the average income growth in the pre-industrial human history. Studies of the pre-industrial world demonstrated the staggering is worldwide. From the original Malthusian literature, “The perpetual tendency in the race of man to increase beyond the means of subsistence is one of the general laws of animated nature which we can have no reason to expect will change.”, the tendency to “increase beyond the means of subsistence” is an axiom of the Malthusian Trap mechanism. The tendency to “increase beyond the means of subsistence” can also be understand as increase the birth rate as high as possible. On the other hand, the birth rate is observed to be decreasing drastically in modern society. The Demographic transition theory demonstrate this transition from high-birth-high-death to low-birthlow-death over course of economic development. This paper developed a comprehensive mathematical model that tries explaining the high birth tendency in term of natural selection and the transition from high-birth-high-death states to low-birth-low death states in term of whether or not natural selection functions. In the model, three fundamental elimination processes are demonstrated.1) The Low-Average-and-High-Birth-Rate elimination; 2) the High-QuotaElimination based on extra resource; and 3) the High-Quota-Elimination based pure resource distribution. It is also demonstrated that how phenotypes co-exist when they are advantage in different elimination mechanism. Whether or not the elimination process function demonstrated two drastically different situations. Under Malthusian trap, the average income decreased as low as possible while birth rate is pushed as high as possible. Out of Malthusian trap, the average income growth rate is pushed high by decreasing birth rate. The model draws a necessary and sufficient condition between the average income stagger and the elimination processes, thus explained the transition from high-birth-high-death to low-birth-low-death based on shutting off elimination processes. Further analysis on the model demonstrated that the average income under Malthusian trap is related to marginal productivity similar to how wage is decided by the marginal productivity.
