Macroeconomists, development scholars, and policy makers have long recognized the importance of poverty traps as a mayor cause of persistent inequality and a serious limitation to growth. A poverty trap may be defined as a threshold level below which individuals or households will not increase their well-being despite the conditions of the economy. While the importance of poverty traps is widely accepted, their microfoundations (the rationality) behind them are not very well understood. Under the Mexican setting, this paper contributes in two ways. First, we assume that income depends on the capital (both physical and human) that a household posses. Hence, if a household is poor and it is not able to accumulate capital it will remain poor (unless there is a sudden increase to the returns of its existing capital). Thus a poverty trap will be generated. Following Chavas (2004, 2005) we explicitly model the preferences, consumption, and the physical and human capital accumulation of Mexican households. We argue that the typical dynamic model with additive utilities and constant discount rates will not be able to capture poverty traps. The reason is that survival motives are involved (endogenous discounting is needed). Second, employing the same model, we test the impact of the Mexican government most important social policy program (Progresa-Oportunidades), in alleviating poverty traps. In the case of households with youngsters, this program can provide funds conditioned on kids attending school. This will somehow, force the participants to increase their human capital. A comparison between households in the programs versus non participants should shed some light in the effectiveness of the program and the sensitivity of persistent poverty to cash transfers.