DSGE-税收

考虑一次性税收，预算约束为

C_{t} + S_{t} = Y_{t} - T_{t}

考虑各种比例税，预算约束为

\begin{aligned} C_{t} + S_{t} & = (1 - τ^{y}) Y_{t} - τ_{t}^{c} C_{t} - τ^{s} S_{t} \\ (1 + τ_{t}^{c}) C_{t} + (1 + τ^{s}) S_{t} & = (1 - τ^{y}) Y_{t} \end{aligned}

其中 $τ^{c, s, y}$ 分别为消费税率、储蓄税率、所得税率。

考虑消费税、劳动所得税、资本利得税，预算约束为

\begin{aligned} C_{t} + I_{t} & = w_{t} L_{t} + r_{t} K_{t} + G_{t} - τ_{t}^{l} w_{t} L_{t} - τ_{t}^{k} (r_{t} - δ) K_{t} \\ (1 + τ_{t}^{c}) C_{t} + I_{t} & = (1 - τ_{t}^{l}) w_{t} L_{t} + [(1 - τ_{t}^{k}) r_{t} + τ_{t}^{k} δ] K_{t} + G_{t} \end{aligned}

注意：资本利得税只对资本增值的部分而非全部收入征收。

消费者

效用最大化问题为

\begin{aligned} max & E_{t} \sum_{t = 0}^{\infty} β^{t} U (C_{t}, O_{t}) \\ s.t. & U (C_{t}, O_{t}) = γ \ln C_{t} + \ln (1 - γ) O_{t} \\ O_{t} + L_{t} \equiv 1 \\ (1 + τ_{t}^{c}) C_{t} + I_{t} = (1 - τ_{t}^{l}) w_{t} L_{t} + [(1 - τ_{t}^{k}) r_{t} + τ_{t}^{k} δ] K_{t} + G_{t} \\ I_{t} = K_{t + 1} - (1 - δ) K_{t} \end{aligned}

等价于

\begin{aligned} max & E_{t} \sum_{t = 0}^{\infty} β^{t} [γ \ln C_{t} + (1 - γ) \ln (1 - L_{t})] \\ s.t. & (1 + τ_{t}^{c}) C_{t} + K_{t + 1} = (1 - τ_{t}^{l}) w_{t} L_{t} + [1 + (1 - τ_{t}^{k}) (r_{t} - δ)] K_{t} + G_{t} \end{aligned}

Lagrangian is

\begin{aligned} L = & E_{t} {\sum_{t = 0}^{\infty} β^{t} [γ \ln C_{t} + (1 - γ) \ln (1 - L_{t})] \\ + λ_{t} [(1 - τ_{t}^{l}) w_{t} L_{t} + [1 + (1 - τ_{t}^{k}) (r_{t} - δ)] K_{t} + G_{t} - (1 + τ_{t}^{c}) C_{t} - K_{t + 1}]} \end{aligned}

F.O.C.

\begin{aligned} (1) & \frac{\partial L}{\partial C_{t}} & = β^{t} [\frac{γ}{C_{t}} - λ_{t} (1 + τ_{t}^{c})] = 0 \\ (2) & \frac{\partial L}{\partial L_{t}} & = β^{t} [- \frac{1 - γ}{1 - L_{t}} + λ_{t} (1 - τ_{t}^{l}) w_{t}] = 0 \\ (3) & \frac{\partial L}{\partial K_{t}} & = β^{t} λ_{t} [1 + (1 - τ_{t}^{k}) (r_{t} - δ)] + β^{t - 1} λ_{t - 1} (- 1) = 0 \end{aligned}

由 $(1) (2)$ 可得

\frac{\frac{1 - γ}{1 - L_{t}}}{\frac{γ}{C_{t}}} = \frac{(1 - τ_{t}^{l}) w_{t}}{1 + τ_{t}^{c}} ⟹ \frac{1 - γ}{γ} \frac{C_{t}}{1 - L_{t}} = \frac{(1 - τ_{t}^{l}) w_{t}}{1 + τ_{t}^{c}}

which in fact is in 效用最大化

\frac{U_{O}}{U_{C}} = \frac{P_{O}}{P_{C}} = \frac{(1 - τ_{t}^{l}) w_{t}}{1 + τ_{t}^{c}}

这表明劳动所得税使得闲暇的机会成本降低了，消费税使得消费的成本提高了。

由 $(1)$ 得到欧拉方程再利用 $(3)$ 消去 $λ$ 可得

\frac{(1 + τ_{t}^{c}) C_{t}}{(1 + τ_{t - 1}^{c}) C_{t - 1}} = β [1 + (1 - τ^{k} t) (r_{t} - δ)]

which exactly in 跨期效用最大化

\frac{U_{C, t - 1}}{U_{C, t}} = β [1 + (1 - τ_{t}^{k}) (r_{t} - δ)]

厂商

和 DSGE Model 完全一致

政府

假设收支平衡，税收全部用于转移支付。

τ_{t}^{c} C_{t} + τ_{t}^{l} w_{t} L_{t} + τ_{t}^{k} (r_{t} - δ) K_{t} = G_{t}

一般均衡

\begin{aligned} (M1) & \frac{1 - γ}{γ} \frac{C_{t}}{1 - L_{t}} & = \frac{(1 - τ_{t}^{l}) w_{t}}{1 + τ_{t}^{c}} \\ (M2) & \frac{(1 + τ_{t}^{c}) C_{t}}{(1 + τ_{t - 1}^{c}) C_{t - 1}} & = β [1 + (1 - τ_{t}^{k}) (r_{t} - δ)] \\ (M3) & r_{t} & = α A_{t} K_{t}^{α - 1} L_{t}^{1 - α} = \frac{α Y_{t}}{K_{t}} \\ (M4) & w_{t} & = (1 - α) A_{t} K_{t}^{α} L_{t}^{- α} = \frac{(1 - α) Y_{t}}{L_{t}} \\ (M5) & Y_{t} & = C_{t} + I_{t} \\ (M6) & Y_{t} & = A_{t} K_{t}^{α} L_{t}^{1 - α} \\ (M7) & I_{t} & = K_{t + 1} - (1 - δ) K_{t} \end{aligned}

此外假设

\ln A_{t} = (1 - ρ_{A}) \ln \bar{A} + ρ_{A} \ln A_{t - 1} + ε_{t}^{A}, ε_{t}^{A} \sim N (0, σ_{A}^{2})