预算约束下人力资本积累的平均场博弈模型
Mean field game model for human capital accumulation under budget constraints
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中文总结 AI 辅助
该研究构建预算约束下人力资本积累的平均场博弈模型，将个体最短时间问题转化为状态约束最优控制问题，通过不动点定理证明有限时间范围内至少存在一个均衡。
中文摘要 AI 辅助
我们研究一种确定性一阶平均场博弈模型，用于描述人力资本积累，其中连续统的智能体通过投资教育与培训，以最短时间达到某一资格水平。学习的生产率取决于个体当前的人力资本水平和人口分布，以此建模知识溢出与社会互动。学习由现有人力资本产生的收入提供资金，由此产生状态约束，要求演化过程中财务资源始终非负。对于固定的人口流，我们将个体问题表述为具有时变动力学的状态约束最短时间最优控制问题。我们在适当的正则性和凸性假设下，证明了最优控制与轨迹的存在性，通过动态规划原理和哈密顿-雅可比方程分析相关值函数，并刻画最优学习努力。最优反馈诱导出描述人口分布演化的连续性方程，所得平均场博弈被表述为人口流的不动点问题。通过构造紧不变集并应用绍德尔不动点定理，我们证明在每个有限人口时间范围内至少存在一个均衡，而个体最短时间问题仍在无限时间范围内构建。
英文摘要
We study a deterministic first-order mean field game model for human capital accumulation in which a continuum of agents invest in education and training to reach a certain qualification level in minimum time. The productivity of learning depends on both the individual's current level of human capital and the distribution of the population, modeling knowledge spillovers and social interactions. Learning is financed by the income generated by existing human capital, leading to a state constraint requiring the financial resources to remain nonnegative throughout the evolution. For a fixed population flow, we formulate the individual problem as a state-constrained minimum-time optimal control problem with time-dependent dynamics. We establish the existence of optimal controls and trajectories, analyze the associated value function through the Dynamic Programming Principle and a Hamilton--Jacobi equation, and characterize the optimal learning effort under suitable regularity and convexity assumptions. The optimal feedback induces a continuity equation describing the evolution of the population distribution. The resulting mean field game is formulated as a fixed-point problem for the population flow. By constructing a compact invariant set and applying Schauder's fixed-point theorem, we prove the existence of at least one equilibrium on every finite population horizon, while the individual minimum-time problem remains formulated on the infinite time horizon.
