Real-Business-Cycle (RBC) Model

消费者

偏好集

效用函数为

u (c_{t}, 1 - h_{t})

其中 $c_{t}$ 表示消费， $h_{t}$ 表示劳动

劳动有两种调整的边际：

内边际（Intensive margin）：选择劳动时间，即劳动是可分的
外边际（Extensive margin）：选择是否工作，即劳动是不可分的

这里考虑劳动不可分的情形。假设企业与同质的劳动者签订同样的劳动合同，使得劳动者以 $α_{t}$ 的概率选择 $h_{0}$ 的劳动水平，以 $1 - α_{t}$ 的概率选择不工作。无论劳动者事后选择工作与否都会得到工资，即完全失业保险制度。

定义 $h_{t} \equiv α_{t} h_{0} + (1 - α) \cdot 0$ ，此时效用函数变为

\begin{aligned} u (c_{t}, 1 - h_{t}) & = [γ \ln c_{t} + (1 - γ) \ln (1 - h_{0})] α_{t} + [γ \ln c_{t} + (1 - γ) \ln (1 - 0)] (1 - α_{t}) \\ = γ \ln c_{t} + (1 - γ) α_{t} \ln (1 - h_{0}) \\ = γ \ln c_{t} + (1 - γ) \frac{\ln (1 - h_{0})}{h_{0}} h_{t} \end{aligned}

为方便起见，单调变换后得到最终的效用函数为

u (c_{t}, 1 - h_{t}) \equiv \ln c_{t} + B h_{t}

其中， $B = \frac{1 - γ}{γ} \frac{\ln (1 - h_{0})}{h_{0}}$

选择集

消费者的一般预算约束为

c_{t} + \underset{i_{t}}{\underset{⏟}{[k_{t + 1} - (1 - δ) k_{t}]}} = w_{t} h_{t} + r_{t} k_{t}

此外考虑货币先行约束（Cash-in-Advance Constraint, CIA），即使用名义货币消费

p_{t} c_{t} \leq m_{t - 1} + T

其中 $p_{t}$ 是价格水平， $m_{t - 1}$ 是上期留存至当期的货币， $T$ 是政府转移支付的货币。

假设

T \equiv M_{t} - M_{t - 1} = (\frac{M_{t}}{M_{t - 1}} - 1) M_{t - 1} = (g_{t} - 1) M_{t - 1}

其中 $M_{t}$ 是人均货币存量（实际上也是货币总量）， $g_{t} \equiv \frac{M_{t}}{M_{t - 1}}$ 是货币增长率。

此时，消费者一般预算约束改写为

p_{t} {c_{t} + [k_{t + 1} - (1 - δ) k_{t}]} + m_{t} = p_{t} (w_{t} h_{t} + r_{t} k_{t}) + m_{t - 1} + (g_{t} - 1) M_{t - 1}

可以考虑两种货币供应法则：

$g_{t} = \bar{g}$ ，固定货币增长率
$g_{t}$ 服从某个随机过程

一般地，当 $\bar{g} \neq 1$ 时，一般找不到 $p_{t}, m_{t}, M_{t}$ 的稳态值，因此需要标准化相关变量。

定义 ${\hat{p}}_{t} = \frac{p_{t}}{M_{t}}, {\hat{m}}_{t} = \frac{m_{t}}{M_{t}}$ ，从而将预算约束改写为

\begin{aligned} {\hat{p}}_{t} {c_{t} + [k_{t + 1} - (1 - δ) k_{t}]} + {\hat{m}}_{t} & = {\hat{p}}_{t} (w_{t} h_{t} + r_{t} k_{t}) + \frac{m_{t - 1} + (g_{t} - 1) M_{t - 1}}{M_{t}} \\ {\hat{p}}_{t} {c_{t} + [k_{t + 1} - (1 - δ) k_{t}]} + {\hat{m}}_{t} & = {\hat{p}}_{t} (w_{t} h_{t} + r_{t} k_{t}) + \frac{{\hat{m}}_{t - 1} + (g_{t} - 1)}{g_{t}} \\ c_{t} + [k_{t + 1} - (1 - δ) k_{t}] + \frac{{\hat{m}}_{t}}{{\hat{p}}_{t}} & = w_{t} h_{t} + r_{t} k_{t} + \frac{{\hat{m}}_{t - 1} + (g_{t} - 1)}{g_{t} {\hat{p}}_{t}} \end{aligned}

CIA 约束改写为

\begin{aligned} {\hat{p}}_{t} c_{t} & = \frac{m_{t - 1} + (g_{t} - 1) M_{t - 1}}{M_{t}} \\ c_{t} & = \frac{{\hat{m}}_{t - 1} + (g_{t} - 1)}{g_{t} {\hat{p}}_{t}} \end{aligned}

最优化

消费者最优化问题为

\begin{aligned} max & E_{0} \sum_{0}^{\infty} β^{t} (\ln c_{t} + B h_{t}) \\ s.t. & k_{t + 1} - (1 - δ) k_{t} + \frac{{\hat{m}}_{t}}{{\hat{p}}_{t}} = w_{t} h_{t} + r_{t} k_{t} \\ c_{t} = \frac{{\hat{m}}_{t - 1} + (g_{t} - 1)}{g_{t} {\hat{p}}_{t}} \end{aligned}

注意：CIA 约束将预算约束的货币部分消去了

Lagrangian is

\begin{aligned} L = & E_{0} \sum_{t = 0}^{\infty} β^{t} {[\ln c_{t} + B h_{t}] \\ + χ_{t}^{1} [w_{t} h_{t} + (1 + r_{t} - δ) k_{t} - \frac{{\hat{m}}_{t}}{{\hat{p}}_{t}} - k_{t + 1}] \\ + χ_{t}^{2} [\frac{{\hat{m}}_{t - 1} + (g_{t} - 1)}{g_{t} {\hat{p}}_{t}} - c_{t}]} \end{aligned}

F.O.C.

注意凡是要递推到 $t + 1$ 的部分都要 conditional on $t$ ，这是因为包含随机变量 $g_{t}$

\begin{aligned} (1) & \frac{\partial L}{\partial c_{t}} & = β^{t} [\frac{1}{c_{t}} - χ_{t}^{2}] = 0 \\ (2) & \frac{\partial L}{\partial h_{t}} & = β^{t} [B + χ_{t}^{1} w_{t}] = 0 \\ (3) & \frac{\partial L}{\partial k_{t + 1}} & = - β^{t} χ_{t}^{1} + β^{t + 1} E_{t} [χ_{t + 1}^{1} (1 + r_{t + 1} + δ)] = 0 \\ (4) & \frac{\partial L}{\partial {\hat{m}}_{t}} & = - β^{t} \frac{χ_{t}^{1}}{{\hat{p}}_{t}} + β^{t + 1} E_{t} [\frac{χ_{t + 1}^{2}}{g_{t + 1} {\hat{p}}_{t + 1}}] = 0 \end{aligned}

解得

\begin{aligned} (1) & χ_{t}^{1} & = - \frac{B}{w_{t}} \\ (2) & χ_{t}^{2} & = \frac{1}{c_{t}} \\ (3) & χ_{t}^{1} & = β E_{t} [χ_{t + 1}^{1} (1 + r_{t + 1} + δ)] \\ (4) & \frac{χ_{t}^{1}}{{\hat{p}}_{t}} & = β E_{t} [\frac{χ_{t + 1}^{2}}{g_{t + 1} {\hat{p}}_{t + 1}}] \end{aligned}

也即

\begin{aligned} \frac{w_{t + 1}}{w_{t}} & = β E_{t} (1 + r_{t + 1} + δ) \\ \frac{c_{t + 1}}{w_{t}} & = - \frac{β}{B} E_{t} [\frac{{\hat{p}}_{t}}{g_{t + 1} {\hat{p}}_{t + 1}}] \end{aligned}

厂商

\begin{aligned} max & Π_{t} = Y_{t} - w_{t} H_{t} - r_{t} K_{t} \\ s.t. & Y_{t} = λ_{t} K_{t}^{θ} L_{t}^{1 - θ} \end{aligned}

F.O.C.

\begin{aligned} r_{t} & = θ λ_{t} K_{t}^{θ - 1} H_{t}^{1 - θ} = θ \frac{Y_{t}}{K_{t}} \\ w_{t} & = (1 - θ) λ_{t} K_{t}^{θ} H_{t}^{- θ} = (1 - θ) \frac{Y_{t}}{H_{t}} \end{aligned}

一般均衡

对于代表性劳动者而言有

\begin{aligned} K_{t} & = \int_{0}^{1} k_{t} d i = k_{t} \\ H_{t} & = \int_{0}^{1} h_{t} d i = h_{t} \\ C_{t} & = \int_{0}^{1} c_{t} d i = c_{t} \\ M_{t} & = \int_{0}^{1} m_{t} d i = m_{t} \end{aligned}

最后一项意味着 ${\hat{M}}_{t} = {\hat{m}}_{t} = \frac{M_{t}}{M_{t}} = 1$

一般均衡模型为

\begin{aligned} k_{t + 1} + \frac{{\hat{m}}_{t}}{{\hat{p}}_{t}} & = w_{t} h_{t} + (1 + r_{t} - δ) k_{t} \\ c_{t} & = \frac{{\hat{m}}_{t - 1} + (g_{t} - 1)}{g_{t} {\hat{p}}_{t}} = \frac{1}{{\hat{p}}_{t}} \\ \frac{w_{t + 1}}{w_{t}} & = β E_{t} (1 + r_{t + 1} + δ) \\ \frac{c_{t + 1}}{w_{t}} & = - \frac{β}{B} E_{t} [\frac{{\hat{p}}_{t}}{g_{t + 1} {\hat{p}}_{t + 1}}] \\ r_{t} & = θ λ_{t} {(\frac{k_{t}}{h_{t}})}^{θ - 1} \\ w_{t} & = (1 - θ) λ_{t} {(\frac{k_{t}}{h_{t}})}^{θ} \end{aligned}

此外假设

\ln λ_{t + 1} = γ \ln λ_{t} + ε_{t + 1}, ε_{t + 1} \sim N (0, σ_{ε}^{2})

这意味着稳态时 $\ln \bar{λ} = 0 ⟹ \bar{λ} = 1$

稳态

\begin{aligned} \frac{1}{\hat{p}} & = \bar{w} \bar{H} + (\bar{r} - δ) \bar{K} \\ \bar{C} & = \frac{1}{\hat{p}} \\ 1 & = β (1 + \bar{r} + δ) \\ \frac{\bar{C}}{\bar{w}} & = - \frac{β}{B \bar{g}} \\ \bar{r} & = θ {(\frac{\bar{K}}{\bar{H}})}^{θ - 1} \\ \bar{w} & = (1 - θ) {(\frac{\bar{K}}{\bar{H}})}^{θ} \end{aligned}

对数线性化

对数线性化原则为

$x_{t} = \bar{x} \cdot \exp [{\tilde{x}}_{t}] \approx \bar{x} (1 + {\tilde{x}}_{t})$
$\frac{1}{x_{t}} = \frac{1}{\bar{x}} \cdot \exp [- {\tilde{x}}_{t}] \approx \bar{x} (1 - {\tilde{x}}_{t})$
$x_{t} y_{t} = \bar{x} \bar{y} \exp [{\tilde{x}}_{t} + {\tilde{y}}_{t}] \approx \bar{x} \bar{y} (1 + {\tilde{x}}_{t} + {\tilde{y}}_{t})$
$\frac{x_{t}}{y_{t}} = \frac{\bar{x}}{\bar{y}} \cdot \exp [{\tilde{x}}_{t} - {\tilde{y}}_{t}] \approx \frac{\bar{x}}{\bar{y}} (1 + {\tilde{x}}_{t} - {\tilde{y}}_{t})$

一般均衡模型对数线性化去稳态为

\begin{aligned} \bar{K} {\tilde{K}}_{t + 1} - \frac{1}{\bar{p}} {\tilde{p}}_{t} & = \bar{w} \bar{H} ({\tilde{w}}_{t} + {\tilde{H}}_{t}) + (1 - δ) \bar{K} {\tilde{K}}_{t} + \bar{r} \bar{K} ({\tilde{r}}_{t} + {\tilde{K}}_{t}) \\ {\tilde{C}}_{t} + {\tilde{p}}_{t} & = 0 \\ {\tilde{w}}_{t + 1} - {\tilde{w}}_{t} & = β E_{t} (1 + \bar{r} {\tilde{r}}_{t + 1} + δ) \\ . . . \end{aligned}

铸币税

假设政府增发货币全部用于政府购买而非转移支付。

重定义货币增长率为 $φ$ ，政府购买为 $g_{t}$

φ \equiv \frac{M_{t}}{M_{t - 1}} g_{t} \equiv \frac{M_{t} - M_{t - 1}}{p_{t}} = {\hat{g}}_{t} \bar{g}

其中 $\bar{g}$ 为平均政府购买， ${\hat{g}}_{t}$ 为随机过程，满足 $\ln {\hat{g}}_{t} = π \ln {\hat{g}}_{t - 1} + ε_{t}^{g}$

此时，CIA 约束为

p_{t} c_{t} \leq m_{t - 1} ⟹ {\hat{p}}_{t} c_{t} \leq \frac{m_{t - 1}}{M_{t - 1}} \frac{M_{t - 1}}{M_{t}} = \frac{{\hat{m}}_{t - 1}}{φ}

消费者一般预算约束为

\begin{aligned} p_{t} {c_{t} + [k_{t + 1} - (1 - δ) k_{t}]} + m_{t} & = p_{t} (w_{t} h_{t} + r_{t} k_{t}) + m_{t - 1} \\ \hat{p} {c_{t} + [k_{t + 1} - (1 - δ) k_{t}]} + {\hat{m}}_{t} & = {\hat{p}}_{t} (w_{t} h_{t} + r_{t} k_{t}) + \frac{m_{t - 1}}{M_{t - 1}} \frac{M_{t - 1}}{M_{t}} \\ c_{t} + k_{t + 1} - (1 - δ) k_{t} + \frac{{\hat{m}}_{t}}{{\hat{p}}_{t}} & = w_{t} h_{t} + r_{t} k_{t} + \frac{{\hat{m}}_{t - 1}}{φ {\hat{p}}_{t}} \end{aligned}

代入 CIA 紧约束可得

k_{t + 1} - (1 - δ) k_{t} + \frac{{\hat{m}}_{t}}{{\hat{p}}_{t}} = w_{t} h_{t} + r_{t} k_{t}

消费者最优化问题为

\begin{aligned} max & E_{0} \sum_{0}^{\infty} β^{t} (\ln c_{t} + B h_{t}) \\ s.t. & k_{t + 1} - (1 - δ) k_{t} + \frac{{\hat{m}}_{t}}{{\hat{p}}_{t}} = w_{t} h_{t} + r_{t} k_{t} \\ c_{t} = \frac{{\hat{m}}_{t - 1}}{φ {\hat{p}}_{t}} \end{aligned}

Bellman Equation 为

\begin{aligned} V (k_{t}, {\hat{m}}_{t - 1}, {\hat{g}}_{t}) = & max_{k_{t + 1}, {\hat{m}}_{t}} \ln (\frac{{\hat{m}}_{t - 1}}{φ {\hat{p}}_{t}}) + B (\frac{k_{t + 1} - (1 - δ) k_{t} + \frac{{\hat{m}}_{t}}{{\hat{p}}_{t}} - r_{t} k_{t}}{w_{t}}) \\ + β E_{t} V (k_{t + 1}, {\hat{m}}_{t}, {\hat{g}}_{t + 1}) \end{aligned}

F.O.C.

\begin{aligned} \frac{\partial V_{t}}{\partial k_{t + 1}} & = \frac{B}{w_{t}} + β E_{t} \frac{\partial V_{t + 1}}{\partial k_{t + 1}} = 0 \\ \frac{\partial V_{t}}{\partial {\hat{m}}_{t}} & = \frac{B}{{\hat{p}}_{t} w_{t}} + β E_{t} \frac{\partial V_{t + 1}}{\partial {\hat{m}}_{t}} = 0 \end{aligned}

根据包络定理

\begin{aligned} \frac{\partial V_{t}}{\partial k_{t}} & = - \frac{B (1 - δ + r_{t})}{w_{t}} & ⟹ \frac{\partial V_{t + 1}}{\partial k_{t + 1}} & = - \frac{B (1 - δ + r_{t + 1})}{w_{t + 1}} \\ \frac{\partial V_{t}}{\partial {\hat{m}}_{t - 1}} & = \frac{1}{{\hat{m}}_{t - 1}} & ⟹ \frac{\partial V_{t + 1}}{\partial {\hat{m}}_{t}} & = \frac{1}{{\hat{m}}_{t}} \end{aligned}

代入F.O.C.

\begin{aligned} \frac{1}{w_{t}} & = β E_{t} [\frac{1 - δ + r_{t + 1}}{w_{t + 1}}] \\ \frac{B}{{\hat{p}}_{t} w_{t}} & = - \frac{β}{{\hat{m}}_{t}} \end{aligned}