Stochastic Problem

Consumption-Saving Problem

The problem is

\begin{aligned} max_{{c_{t}, k_{t + 1}}_{t = 0}^{\infty}} & E_{0} [\sum_{t = 0}^{\infty} β^{t} ε_{t} u (c_{t})] \\ s.t. & k_{t + 1} = (1 + r) k_{t} - c_{t} \\ k_{t + 1} \geq 0 \\ k_{0} = k \end{aligned}

假设由一个偏好冲击序列 ${ε_{t}}_{t = 0}^{\infty}$ ，其中 $ε_{t}$ 服从 Markov process，也就是说 $ε_{t}$ 的分布只取决于 $ε_{t - 1}$ 的值。注意 $E_{0}$ 指的是 conditional on $0$ ，而不是 $0$ 期的期望。

此时有两个状态变量 $k_{t}$ 和 $ε_{t}$ ，Bellman Equation 为

\begin{aligned} V (k_{0}, ε_{0}) & = max_{{k_{t + 1}}_{t = 0}^{\infty}} \sum_{t = 0}^{\infty} β^{t} E_{t} [ε_{t} u (c_{t})] \\ = max_{{k_{t + 1}}_{t = 0}^{\infty}} ε_{0} u (c_{0}) + \sum_{t = 1}^{\infty} β^{t} E_{t} [ε_{t} u (c_{t})] \\ = max_{{k_{t + 1}}_{t = 0}^{\infty}} ε_{0} u (c_{0}) + β \sum_{t = 1}^{\infty} β^{t - 1} E_{t} [ε_{t} u (c_{t})] \\ = max_{{k_{t + 1}}_{t = 0}^{\infty}} ε_{0} u (c_{0}) + β E_{1} V (k_{1}, ε_{1}) \end{aligned}

F.O.C.

\frac{\partial V (k_{0}, ε_{0})}{\partial k_{1}} = - ε_{0} \frac{\partial u (c_{0})}{\partial c_{0}} + β E_{1} \frac{\partial V (k_{1}, ε_{1})}{\partial k_{1}} = 0

根据包络定理

\frac{\partial V (k_{0}, ε_{0})}{\partial k_{0}} = ε_{0} \frac{\partial u (c_{0})}{\partial c_{0}} (1 + r)

one step forward

\frac{\partial V (k_{1}, ε_{1})}{\partial k_{1}} = ε_{1} \frac{\partial u (c_{1})}{\partial c_{1}} (1 + r)

代回F.O.C.可得

ε_{0} \frac{\partial u (c_{0})}{\partial c_{0}} = β (1 + r) E_{1} (ε_{1} \frac{\partial u (c_{1})}{\partial c_{1}})

代入政策函数 $h (k_{0}, ε_{0}) = c_{0}$ 可得显式解