The Dynamics of Population Growth, Differential Fertility, and Inequality: Comment
通过矩阵代数模型证明，即使某收入阶层稳态比例下降，其出生人口比例仍会上升，且上升幅度最大；同时反驳了“儿童比例上升则人口比例下降”的假设。
It has been hypothesized that when the decrease in the steady-state proportion of income class i is large enough the percentage of children in the class will fall despite increased fertility. However the mathematical model developed in this article demonstrates that in spite of a potential reduction in the relative proportion of steady-state population born to that class will always increase and moreover will increase the most. As a direct consequence the possible decrease in the steady-state proportion of income class i is limited. Through the use of matrix algebra rather than differential calculus upper and lower bounds for the relative change in the proportion of the steady-state population of any income class can be derived. The diagonal matrix F gives the income-specific net reproductive rates while M represents the intergenerational mobility matrix. The analysis also refutes the hypothesis that if the percent children in any income class increases then the proportion of the population after intervention of mobility decreases. Apparently decreased inflows can never outweigh increased inflows. It should be noted that the results of this analysis hold without any restrictions on the mobility matrix; thus they are simply properties of the model.