Overlapping Generations Model

分散经济

个体：

\begin{aligned} R (t) & = f^{'} (k (t)) \\ w (t) & = f (k (t)) - k f^{'} (k (t)) \end{aligned}

市场出清：

1 + r (t) R (t)

个体效用最大化问题为

\begin{aligned} max_{{c_{1} (t), c_{2} (t + 1), s (t)}} & u (c_{1} (t)) + β u (c_{2} (t + 1)) \\ s.t. & c_{1} (t) + s (t) \leq w (t) \\ c_{2} (t + 1) \leq [1 + r (t + 1)] s_{t} \end{aligned}

$u (\cdot)$ 关于 $c$ 是严格递增的，因此约束条件取等号，问题等价于

\begin{aligned} max_{{c_{1} (t), c_{2} (t + 1), s (t)}} & u (c_{1} (t)) + β u (c_{2} (t + 1)) \\ s.t. & c_{1} (t) + \frac{c_{2} (t + 1)}{1 + r (t + 1)} = w (t) \end{aligned}

Lagrangian 函数为

L = u (c_{1} (t)) + β u (c_{2} (t + 1)) + λ [w (t) - c_{1} (t) - \frac{c_{2} (t + 1)}{1 + r (t + 1)}]

F.O.C.

\begin{aligned} \frac{\partial L}{\partial c_{1} (t)} & = u^{'} (c_{1} (t)) - λ = 0 \\ \frac{\partial L}{\partial c_{2} (t + 1)} & = β u^{'} (c_{2} (t + 1)) - λ \frac{1}{1 + r (t + 1)} = 0 \end{aligned}

整理得欧拉方程

\frac{u^{'} (c_{1} (t))}{u^{'} (c_{2} (t + 1))} = β [1 + r (t + 1)]

从而解得

s (t) = s (w (t), R (t + 1))

注意：可以证明，稳态时储蓄率 $s^{*} = \frac{\partial s (t)}{\partial w (t)}$ 和 Solow Model 一样为常数。

资本的动态方程为

K (t + 1) = L (t) s (t) = L (t) s (w (t), R (t + 1))

其中 $L (t) \equiv (1 + n)^{t} L (0)$ ，等式两边除以 $L (t + 1) = (1 + n) L (t)$ 得

k (t + 1) = \frac{s (w (t), R (t + 1))}{1 + n}

Note

如果考虑技术进步，资本动态方程为

K (t + 1) = A (t) L (t) s (t)

等式两边除以 $A (t + 1) = (1 + g) A (t)$ 得

k (t + 1) = \frac{s (w (t), R (t + 1))}{(1 + n) (1 + g)}

代入厂商最优条件以及稳态条件 $k^{*} = k (t + 1) = k (t)$ 得

k^{*} = \frac{s (f (k^{*}) - k^{*} f^{'} (k^{*}), f^{'} (k^{*}))}{1 + n}

假设效用函数形式为

U = \frac{c_{1} (t)^{1 - θ} - 1}{1 - θ} + β \frac{c_{2} (t + 1)^{1 - θ} - 1}{1 - θ}

欧拉方程为

{[\frac{c_{1} (t)}{c_{2} (t + 1)}]}^{- θ} = β R (t + 1)

代入约束条件求解 $s (t)$

\begin{aligned} {[\frac{w (t) - s (t)}{R (t + 1) s (t)}]}^{- θ} & = β R (t + 1) \\ s (t) & = \frac{w (t)}{ϕ (t + 1)} \end{aligned}

其中 $ϕ (t + 1) \equiv [1 + β^{- \frac{1}{θ}} R (t + 1)^{\frac{θ - 1}{θ}}] > 1$

假设生产函数形式为 $F (K, L) = K^{α} L^{1 - α} ⟹ f (k) = k^{α}$ ，最优化条件为

\begin{aligned} w (t) & = (1 - α) k^{α} \\ R (t) & = α k^{α - 1} \end{aligned}

资本动态方程为

k (t + 1) = \frac{w (t)}{(1 + n) ϕ (t + 1)}

代入厂商最优条件以及稳态条件得

k^{*} = \frac{(1 - α) (k^{*})^{α}}{(1 + n) [1 + β^{- \frac{1}{θ}} α^{\frac{θ - 1}{θ}} (k^{*})^{(α - 1) \frac{θ - 1}{θ}}]}

假设效用函数形式为

U = \ln c_{1} (t) + β \ln c_{2} (t + 1)

欧拉方程为

\frac{c_{2} (t + 1)}{c_{1} (t)} = β R (t + 1)

代入约束条件求解 $s (t)$

\begin{aligned} \frac{R (t + 1) s (t)}{w (t) - s (t)} & = β R (t + 1) \\ s (t) & = \frac{β}{1 + β} w (t) \end{aligned}

假设生产函数形式为 $F (K, A L) = K^{α} (A L)^{1 - α} ⟹ f (k) = k^{α}$ ，最优化条件为

\begin{aligned} w (t) & = (1 - α) k^{α} \\ R (t) & = α k^{α - 1} \end{aligned}

资本动态方程为

k (t + 1) = \frac{\frac{β}{1 + β} w (t)}{(1 + n) (1 + g)}

代入厂商最优条件以及稳态条件得

k^{*} = \frac{\frac{β}{1 + β} (1 - α) (k^{*})^{α}}{(1 + n) (1 + g)} ⟹ k^{*} = {[\frac{β (1 - α)}{(1 + β) (1 + n) (1 + g)}]}^{\frac{1}{1 - α}}

Note

对比 Solow Model 的结果

s (k^{*})^{α} = (n + g + δ) k^{*} ⟹ k^{*} = {[\frac{s}{n + g + δ}]}^{\frac{1}{1 - α}}

这里的储蓄率 $s$ 对应实例二当中的 $\frac{β}{1 + β}$ ，可见二者非常相似；实际上实例二的结果和离散 Solow Model 完全相同。

社会计划者最优化问题为

\begin{aligned} max & \sum_{t = 1}^{\infty} β_{s}^{t} [u (c_{1} (t)) + β u (c_{2} (t + 1))] \\ s.t. & F (K (t), L (t)) = K (t + 1) + L (t) c_{1} (t) + L (t - 1) c_{2} (t) \end{aligned}

最终解得欧拉方程为

\frac{u^{'} (c_{1} (t))}{u^{'} (c_{2} (t + 1))} = β f^{'} (k (t + 1))

相当于将利润最大化条件代入分散经济的欧拉方程，因此分散经济并无效率损失。

在预算约束等式两边除以 $L (t) = (1 + n) L (t - 1)$ 后取稳态

\begin{aligned} \frac{F (K (t), L (t))}{L (t)} & = \frac{L (t + 1)}{L (t)} \frac{K (t + 1)}{L (t + 1)} + c_{1} (t) + \frac{c_{2} (t)}{1 + n} \\ f (k^{*}) - (n + 1) k^{*} & = c_{1}^{*} (t) + \frac{c_{2}^{*}}{1 + n} \equiv c^{*} \end{aligned}

我们将等式右侧定义为跨期总消费 $c^{*}$ ，为使其最大化

\frac{\partial c^{*}}{\partial k^{*}} = f^{'} (k^{*}) - (n + 1) = 0 ⟹ k_{g}^{*} = (f^{'})^{- 1} (n + 1)

因此存在一个资本黄金率 $k_{g}^{*}$ 以及对应储蓄率 $s_{g}^{*}$ 使得跨期总消费最大化。