Analyzing the sustainable decision-making mechanism between household consumption and education investment can theoretically develop education. This study uses the continuous-time utility model to demonstrate the independent characteristics of consumption and education investment, as well as the principle of decision incompatibility in the decision-making process of the utility maximization problem. Then, we establish a three-phase logarithmic utility model to obtain the intertemporal decision-making path of a family. The analysis shows that the investment allocation ratio between the two phases depends on the expected and discounted level of the offsprings' abilities, while the total investment level is related to parental altruism. When parents, with foresight, factor in prospective transfer payments from progeny, the optimal decision is to maximize their children's ultimate human capital within a given total investment. Education investment not only squeezes out consumption but also promotes consumption in various periods due to future transfer payments. The decision-making process of three typical growth stages indicates that as offspring mature and their human capital increases, parents' willingness to invest in education decreases while self-consumption escalates. This study provides a new perspective and theoretical basis for studying household education expenditure, motivation, and related policy formulation.
Keywords: continuous investment decision; human capital; maximum principle; transfer payment; utility model.