Barro(1990) Model

代表性家庭

假设劳动供给是固定常数且标准化为 1，即 $L (t) = \bar{L} = 1$

瞬时效用函数为

u (c (t)) = \frac{c (t)^{1 - σ} - 1}{1 - σ}

目标函数为

U (0) = \int_{0}^{\infty} e^{- ρ t} u (c (t)) \bar{L} d t (ρ > 0)

资产的动态方程为

\dot{A} (t) \equiv (1 - τ) [r (t) A (t) + w (t) \bar{L}] - c (t) \bar{L}

其中，人均资产定义为

a (t) \equiv \frac{A (t)}{\bar{L}} ⟹ \dot{A (t)} = \frac{d a (t) \bar{L}}{d t} = \dot{a} (t) \bar{L}

从而得到：

\dot{a} (t) = (1 - τ) [r (t) a (t) + w (t)] - c (t)

综上所属，家庭最优化问题为

\begin{aligned} max_{c (t)} & \int_{0}^{\infty} \frac{c (t)^{1 - σ} - 1}{1 - σ} e^{- ρ t} d t \\ s.t. & \dot{a} (t) = (1 - τ) [r (t) a (t) + w (t)] - c (t) \\ lim_{t \to \infty} a (t) \exp [- \int_{0}^{t} (r (s) - n) d s] \geq 0 \end{aligned}

Set up the current-value Hamiltonian function

\hat{H} (a, c, μ) = \frac{c (t)^{1 - σ} - 1}{1 - σ} + μ (t) {(1 - τ) [r (t) a (t) + w (t)] - c (t)}

Maximum Principle

\begin{aligned} (1) & \frac{\partial \hat{H} (a, c, μ)}{\partial c} & = 0 & ⟹ & c (t)^{- σ} = μ (t) \\ (2) & \frac{\partial \hat{H} (a, c, μ)}{\partial a} & = - \dot{μ} (t) + ρ μ (t) & ⟹ & \frac{\dot{μ} (t)}{μ (t)} = ρ - (1 - τ) r (t) \\ (3) & \frac{\partial \hat{H} (a, c, μ)}{\partial μ} & = \dot{a} (t) \\ (4) & lim_{t \to \infty} [e^{- ρ t} μ (t) a (t)] & = 0 \end{aligned}

根据(1)可得

- σ \ln [c (t)] = \ln [μ (t)] ⟹ - σ \frac{\dot{c} (t)}{c (t)} = \frac{\dot{μ} (t)}{μ (t)}

结合(2)得到欧拉方程

\frac{\dot{c} (t)}{c (t)} = \frac{(1 - τ) r (t) - ρ}{σ}

代表性企业生产函数为

y (t) = A k (t)^{α} g (t)^{1 - α}

其中，政府支出为

g (t) = τ y (t)

利润最大化问题为

max A k (t)^{α} g (t)^{1 - α} - R (t) k (t) - w (t)

F.O.C.

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

根据 Cobb-Douglas函数的性质可知

\frac{R (t) k (t)}{w (t)} = \frac{α}{1 - α}

以及

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

现在我们已经有资产动态恒等式和边际产出恒等式，此外：

政府平衡预算恒等式为

τ (r (t) a (t) + w (t)) = g (t)

市场出清恒等式为

\begin{aligned} a (t) & = k (t) \\ r (t) & = R (t) \end{aligned}

利用恒等式消去资本动态方程中的变量得到变量关系式

\begin{aligned} \dot{k} (t) & = (1 - τ) [r (t) k (t) + w (t)] - c (t) \\ = (1 - τ) [α y (t) + (1 - α) y (t)] - c (t) \\ = y (t) - g (t) - c (t) \end{aligned}

利用恒等式消去欧拉方程中的变量得到数值解

\begin{aligned} \frac{\dot{c} (t)}{c (t)} & = \frac{(1 - τ) r (t) - ρ}{σ} \\ = \frac{(1 - τ) A α k (t)^{α - 1} g (t)^{1 - α} - ρ}{σ} \end{aligned}

其中 $\frac{g (t)}{k (t)}$ 可以通过消去生产函数中的产量得到

\begin{aligned} g (t) & = τ A k (t)^{α} g (t)^{1 - α} \\ {[\frac{g (t)}{k (t)}]}^{α} & = τ A \end{aligned} ⟹ {[\frac{g (t)}{k (t)}]}^{1 - α} = (τ A)^{\frac{1 - α}{α}}

代入欧拉方程得到数值解

\frac{\dot{c} (t)}{c (t)} = \frac{(1 - τ) A α (τ A)^{\frac{1 - α}{α}} - ρ}{σ} = \frac{α (1 - τ) A^{\frac{1}{α}} τ^{\frac{1 - α}{α}} - ρ}{σ}

求使得增长率最大化的税率 $τ$

\begin{aligned} \frac{\partial \cdot}{\partial τ} & = \frac{α A^{\frac{1}{α}}}{σ} [- τ^{\frac{1 - α}{α}} + \frac{1 - α}{α} (1 - τ) τ^{\frac{1 - α}{α} - 1}] \\ = \frac{α A^{\frac{1}{α}}}{σ} τ^{\frac{1 - α}{α}} [- 1 + \frac{1 - α}{α} \frac{1 - τ}{τ}] \\ = \frac{α A^{\frac{1}{α}}}{σ} τ^{\frac{1 - α}{α}} \frac{1}{α τ} [- α τ + (1 - τ - α + α τ)] \\ = \frac{α A^{\frac{1}{α}}}{σ} τ^{\frac{1 - α}{α}} \frac{1 - τ - α}{α τ} \end{aligned}

令上式等于零得到最优税率 $τ^{*} = 1 - α$ ，且

{\begin{cases} \frac{\partial \cdot}{\partial τ} > 0 & if τ < 1 - α \\ \frac{\partial \cdot}{\partial τ} < 0 & if τ > 1 - α \end{cases}

所以这是唯一的最优税率。

我们回顾一下背后的机制：

在分散经济模型中，厂商自认为无法影响政府投资，因此私人边际产出为

M P P_{k} = {\frac{\partial y (t)}{\partial k (t)} |}_{g (t) = g (t)} = A α k (t)^{α - 1} g (t)^{1 - α} = α (1 - τ) A^{\frac{1}{α}} τ^{\frac{1 - α}{α}}

因此社会边际产出为

M S P_{k} = {\frac{\partial y (t)}{\partial k (t)} |}_{g (t) = τ y (t)} = A^{\frac{1}{α}} τ^{\frac{1 - α}{α}}

Note

在生产函数中消去政府支出得到的就是社会生产函数（形式类似 AK Model）

\begin{aligned} y (t) & = A k (t)^{α} τ^{1 - α} y (t)^{1 - α} \\ y (t)^{α} & = A k (t)^{α} τ^{1 - α} \\ y (t) & = A^{\frac{1}{α}} τ^{\frac{1 - α}{α}} k (t) \end{aligned}

显然 MPP 低于 MSP，相应地分散经济增长率低于社会计划者经济。