跨时消费、收入与闲暇选择的理论与数理分析

Theoretical and Mathematical Analysis of Intertemporal Consumption, Income, and Leisure Choice

摘要: 本文基于数理经济学分析方法，构建了一个同时包含跨时消费选择、收入与闲暇决策的理论模型。通过引入动态优化框架，探讨消费者在生命周期内如何最优分配消费、闲暇和储蓄，以实现新古典经济学所提出的终身效用最大化。模型综合考虑预算约束、时间约束、利率变动、工资率变化及通货膨胀等因素的影响。理论推导结果显示：1) 消费者实现经济学上的跨期均衡条件取决于时间偏好率与市场利率的对比关系，而消费–闲暇替代弹性则决定了劳动供给对工资变动的反应灵敏度。进一步，通过将生产函数引入分析框架，揭示了生产技术特征如何通过影响要素报酬来间接塑造消费者的决策模式。数值模拟分析表明：2) 财富积累阶段、通胀环境以及风险厌恶程度都会对最优决策路径产生系统性影响，利率上升显著抑制当期消费并促进储蓄，工资上涨通过替代效应减少闲暇并增加劳动供给，通胀则通过侵蚀实际利率降低未来消费。本研究为个体跨期决策行为提供了严谨的数理经济学分析框架，对解释现实经济现象和制定社保、货币等政策具有一定的参考价值。

Abstract: Based on the analytical methods of mathematical economics, this paper constructs a theoretical model that simultaneously incorporates intertemporal consumption choice and income-leisure decisions. By introducing a dynamic optimization framework, it explores how consumers optimally allocate consumption, leisure, and savings over the life cycle to maximize lifetime utility, as proposed by neoclassical economics. The model comprehensively accounts for the effects of budget constraints, time constraints, interest rate changes, wage rate variations, and inflation. The theoretical derivations yield the following findings: 1) The consumer’s intertemporal equilibrium condition depends on the relationship between the rate of time preference and the market interest rate, while the consumption-leisure substitution elasticity determines the sensitivity of labor supply to wage changes. Furthermore, by incorporating a production function into the analytical framework, the study reveals how the characteristics of production technology indirectly shape consumer decision patterns through their influence on factor returns. Numerical simulation analysis demonstrates that: 2) The wealth accumulation phase, inflationary environment, and degree of risk aversion all exert systematic impacts on the optimal decision path. Rising interest rates significantly dampen current consumption and promote savings; wage increases reduce leisure and raise labor supply through the substitution effect; and inflation lowers future consumption by eroding the real interest rate. This study provides a rigorous mathematical economics framework for analyzing individual intertemporal decision-making and offers valuable insights for explaining real-world economic phenomena and for formulating social security, monetary, and other related policies.