This paper analyzes and predicts the changes of relationship between income and fertility rate of cross-countries using a bivariate mixture model and a latent change score model. This paper has shown that there is a negative relationship between income and fertility rate, which is presented in the form of inverted S-shaped curve which shows the three regimes of demographic transition. Some developed countries have completed their demographic transition in fertility rate, and in developing countries, the demographic transition in fertility rate is still in progress. This paper has also shown that the number of peaks of income distribution has increased in recent years comparing to 1960s and the number won't decrease in the future. However, the number of peaks of fertility rate distribution hasn't changed from 1960s to recent years but due to the shift, finally, the distribution will change to a uni-modal distribution in the future. The income will be applied to the conditional convergence and the fertility rate will be applied to the absolute convergence. The fertility gap among cross-countries will disappear, but the income gap won't. Although the population conditions in developing countries will improve, income inequality in cross-country may not be improved after all.
income distribution fertility rate mixture distribution latent change score model Bayesian estimation