这篇论文扩展了 neoclassical model，在其中加入了公共资本和公共投资，用来研究政府购买和投资对于经济增长的影响。模型中，政府可以通过 lump-sum tax 或者比例税收来资助政府购买与公共投资，政府也可以选择暂时增加支出（比如战争）或者永久增加支出。这篇论文非常重要的一个贡献在于将公共资本融入了生产函数，而量化模拟结果对公共资本的规模报酬变化十分敏感。本文中推导的 Eq (14)-(18) 主要集中关注政府购买的情况，因此公共资本在生产函数中没有出现。
Setup#
- Representative agents.
- Infinite horizon.
Preference#
\[U(\cdot) = \mathbb{E}_1\sum_{t=1}^\infty \beta^{(t-1)}\left[\log (C_t) + \theta_L \log (L_t) + \Gamma(G_t^B, K_t^G)\right]\]
- \(C_t\): 消费
- \(L_t\): leisure
- \(G_t^B\): 政府购买
- \(K_t^G\): 公共资本
Technology#
$$Y_t = F(K_t, K_t^G, N_t) = AK_t^{\theta_K}N_t^{\theta_N}\textcolor{blue}{(K_t^G)^{\theta_G}}$$
- \(K_t\): 私人资本
- \(\textcolor{blue}{K_t^G}\): 公共资本
- \(N_t\): 劳动投入
- 私人投入 Constant returns to scale: \(\theta_K + \theta_N = 1\)
Capital Evolution#
- \( K_{t+1} = \left[(1-\delta_K)K_t + I_t\right] \)
- \( \textcolor{blue}{K_{t+1}^G} \textcolor{blue}{= \left[(1-\delta_K)K_t^G + I_t^G\right]}\)
Resource Constraints#
- Household:
- \(L_t + N_t \leq 1\)
- \(C_t + I_t \leq (1-\tau_t)Y_t + \text{TR}_t\)
- Economy: \(C_t + I_t + G_t \leq Y_t\), where \(G_t := G_t^B + I_t^G\)
Public Finance Rules#
\[\tau_t Y_t = G_t + \text{TR}_t\]
- Lump-sum tax: \(\tau_t = 0\), 使用 \(\text{TR}_t\) 资助政府购买
- 比例税率:
- \(\tau_t = \frac{G_t + \text{TR}_t}{Y_t}\)
- 假设转移支付恒定：\(\Delta \text{TR} = 0\)
Characterizing the Equilibrium#
用 Lagrangian 和 First order conditions 很容易可以写出以下三个式子
- \(\frac{\partial U(C_t, L_t, G_t^B, K_t^G)}{\partial C_t} = \lambda_t\)
- \(\frac{\partial U(C_t, L_t, G_t^B, K_t^G)}{\partial L_t} = \lambda_t w_t,\) where \(\textcolor{blue}{w_t = (1-\tau_t)\frac{\partial F(\cdot)}{\partial N_t}}\)
- \(\beta\mathbb{E}_t\bigl[\lambda_{t+1}\bigl(1+q_{t+1}-\delta_K \bigr)\bigr] = \lambda_t,\) where \( \textcolor{blue}{q_t = (1-\tau_t)\frac{\partial F(\cdot)}{\partial K_t}}\)
以及 transversality condition
- \(\mathbb{E}_1\left[\lim_{t\rightarrow\infty} \beta^t\lambda_t K_{t+1}\right] = 0\)
为了简化式子，文章中定义了以下几个符号：
- 资本劳动比 \(\kappa = K/N\)
- Average product of labor \(\alpha = Y/N\)
- 资本收入占国民收入比例 \(s_K = qK/Y = q\kappa/\alpha \)
- 投资占国民收入比例 \(s_I = I/Y\)
- 劳动收入占国民收入比例 \(s_N = wN/Y = w/\alpha\)
- Share of leisure in full income: \(s_L^f = wL/Y^f\)
- Full-income elasticity of leisure \(\eta_L = \frac{\Delta L/L}{\Delta Y^f/Y^f}\)
Equation (14), (15)#
Eq (14): \(C_t +wL \leq Y^f\)
Eq (15): \(\frac{\Delta Y}{\Delta G} = = s_L^f\eta_L\)
这一部分考虑静态模型，假设用 lump-sum tax 来资助政府购买支出，即\(\tau_t = 0\)。假设 w 不变，我们可以定义 full income \(Y^f\)为一个人可以得到的收入上限\(Y^f := w + \Pi\)，而在 lump-sum tax 的情况下，\(\Pi = -G\)。由于\(C_t \leq w(1-L)+ \Pi\)，经过简单的替换，我们可以得到 Eq (14) \(C_t +wL \leq Y^f\)。
\(Y^f = w - G \Rightarrow \Delta Y^f = -\Delta G\)
\(N := 1-L \Rightarrow \Delta N = -\Delta L\)
\[\begin{aligned} \frac{\Delta Y}{\Delta G} = \frac{\Delta Y}{\Delta N}\frac{\Delta N}{\Delta Y^f}\frac{\Delta Y^f}{\Delta G} = w\frac{-\Delta L}{\Delta Y^f}(-1)= w\frac{L}{Y^f}\frac{Y^f}{L}\frac{\Delta L}{\Delta Y^f} = s_L^f\eta_L \end{aligned}\]
由此，我们也得到文章中的 Eq (15)。
Equation (16)#
Eq (16): \(\frac{\Delta Y}{\Delta G} = \frac{s_L^f \eta_L/s_N }{1+s_L^f \eta_L(s_K - s_I)/s_N }\)
这一部分考虑动态模型，因此我们考虑 Capital law of motion。在这种情况下，收入上限中除了劳动收入，还包括资本收入、投资支出和政府支出。因此\(Y^f = w +qK - I- G\)。如果我们集中关注稳态中的投资支出，那么我么可以得到\( K = (1-\delta_K)K + I \)，即\( I= \delta_K K \)。
我们也可以使用\(\kappa, \alpha, \delta_K\)来表示\(s_I\)
\[s_I = \frac{I}{Y} = \frac{I}{K}\frac{K}{N}\frac{N}{Y} = \delta_K \frac{\kappa}{\alpha}\]
因此,
\[\begin{aligned} Y^f &= w +qK - I- G = w +qK - \delta_K K- G = w +(q - \delta_K) K- G \\ &= w +(q - \delta_K) \kappa N \frac{\alpha}{\alpha} - G = w +\alpha(\frac{\kappa q}{\alpha} - \frac{\delta_K \kappa}{\alpha}) N - G\\ &= w +\alpha(s_K - s_I) N - G \end{aligned}\]
由全微分得到 \(\Delta Y^f= \alpha(s_K - s_I) \Delta N - \Delta G\)
由于 Eq (16) 计算的是\(G\)的变化和\(Y\)的变化之间的关系，而\(\Delta N, \Delta L\) 分别与\(\Delta Y^f， \Delta Y\)相关，我们可以从\(\Delta N=-\Delta L\)，或者\(w\Delta N=-w\Delta L\)入手进行以下推导
\[\begin{aligned} w\Delta N &=-w\Delta L = -w \Delta L \frac{L}{Y^f}\frac{L}{Y^f}\frac{\Delta Y^f}{\Delta Y^f} \\[0.6em] &= - \frac{wL}{Y^f}\frac{L}{Y^f}\frac{\Delta L}{\Delta Y^f}\Delta Y^f\\[0.6em] &= -s_L^f \eta_L \Delta Y^f \\[0.6em] &= -s_L^f \eta_L [\alpha(s_K - s_I) \Delta N - \Delta G]\\[0.6em] \Rightarrow s_L^f \eta_L\Delta G &= [w+s_L^f \eta_L\alpha(s_K - s_I)] \Delta N \end{aligned}\]
由于\(Y = \alpha N \Rightarrow \Delta Y = \alpha \Delta N\)，\(s_N = w/\alpha\)，再以及结合以上推导，我们可以得到 Eq (16):
\[\begin{aligned} s_L^f \eta_L\Delta G &= [w+s_L^f \eta_L\alpha(s_K - s_I)] \frac{\Delta Y}{\alpha} \\[0.6em] \Leftrightarrow s_L^f \eta_L\Delta G &= [s_N+s_L^f \eta_L(s_K - s_I)]\Delta Y \\[0.6em] \frac{\Delta Y}{\Delta G} &= \frac{s_L^f \eta_L }{s_N+s_L^f \eta_L(s_K - s_I)}\\[0.6em] &= \frac{s_L^f \eta_L/s_N }{1+s_L^f \eta_L(s_K - s_I)/s_N } \end{aligned}\]
Equation (17), (18)#
这一部分引入比例税率，假设 lump-sum tax 恒定，假设\(N\)恒定，我们可以通过 Public Finance Rules 以及生产函数和稳态情况下的 Euler equation 得到两组\(\Delta Y, \Delta \tau, \Delta G\)的式子，从而可以解出\(\Delta G\)分别与\(\Delta Y, \Delta \tau\)的关系，即增加政府支出如何影响经济增长和比例税率。因为我们主要关注稳态情况下的关系，我们将省略下标\(t\)。
由于假设 lump-sum tax 恒定，\(\Delta \text{TR} = 0\).对 Public Finance Rules 取全微分得到\[\begin{align}\tau \Delta Y + Y \Delta \tau = \Delta G\end{align}\]
接下来使用稳态情况下的 Euler Equation \(\beta\bigl(1+(1-\tau)\frac{\partial F(\cdot)}{\partial K}-\delta_K \bigr) = 1\)。
首先我们需要通过生产函数\(Y = A K^{\theta_K}N^{\theta_N}\)得到\(\frac{\partial F(\cdot)}{\partial K}\)：
\[\frac{\partial F(\cdot)}{\partial K} = A \theta_K K^{\theta_K-1}N^{\theta_N} = \frac{A}{K} \theta_K K^{\theta_K}N^{\theta_N} = \frac{\theta_K Y}{K}\]
代入 Euler Equation:
\[\begin{aligned} &\beta\bigl(1+(1-\tau)\frac{\theta_K Y}{K}-\delta_K \bigr) = 1 \\[0.6em] \Leftrightarrow &K = \frac{\theta_K(1-\tau)}{\frac{1}{\beta} -1 + \delta_K}Y \end{aligned}\]
\(Z:= \theta_K/[\frac{1}{\beta} -1 + \delta_K]\)，我们可以将以上式子写成\(K = Z(1-\tau)Y\)。
将这个式子代入到生产函数中
\[\begin{aligned} Y &= A K^{\theta_K}N^{\theta_N} = A (Z(1-\tau)Y)^{\theta_K}N^{\theta_N}\\[0.6em] \Rightarrow Y^{1-\theta_K} &= (1-\tau)^{\theta_K} AZ^{\theta_K}N^{\theta_N}\\[0.6em] \Rightarrow (1-\theta_K)\log Y &= \theta_K \log (1-\tau)+ \log (AZ^{\theta_K}N^{\theta_N})\\[0.6em] \Rightarrow (1-\theta_K)\log Y &= \theta_K \log (1-\tau)+ \log (AZ^{\theta_K}N^{\theta_N})\\[0.6em] \Rightarrow \frac{\Delta Y}{Y} &= \Delta \log Y = \frac{\theta_K}{1-\theta_K} \frac{1}{1-\tau}(-\Delta \tau)\\[0.6em] \Rightarrow \frac{\Delta Y}{\Delta \tau} &= -\frac{\theta_K}{1-\theta_K} \frac{1}{1-\tau}Y = -\frac{\theta_K}{\theta_N} \frac{1}{1-\tau}Y \end{aligned}\]
将这个式子代回(1)替换\(\Delta \tau\)将得到 Eq (17)，替换\(\Delta Y\)将得到 Eq (18)。
Eq (17): \(\Delta Y = -\frac{\theta_K}{\theta_N-\tau}\Delta G\)
\[\begin{aligned} \Delta \tau &= -\frac{\theta_N(1-\tau)}{\theta_K} \frac{\Delta Y}{Y}\\[0.6em] \Rightarrow \Delta G &= \tau \Delta Y - Y \frac{\theta_N(1-\tau)}{\theta_K} \frac{\Delta Y}{Y} \\[0.6em] \Rightarrow \Delta G &= (\tau - \frac{\theta_N(1-\tau)}{\theta_K}) \Delta Y \\[0.6em] \Rightarrow \Delta G &= (\frac{\tau(\theta_K)-\theta_N(1-\tau)}{\theta_K}) \Delta Y \\[0.6em] \Rightarrow \Delta G &= \frac{\tau(1-\theta_N)-\theta_N(1-\tau)}{\theta_K} \Delta Y \\[0.6em] \Rightarrow \Delta Y &= -\frac{\theta_K}{\theta_N-\tau} \Delta Y \\[0.6em] \end{aligned}\]
Eq (18): \(\Delta \tau = (1-\tau)\frac{\theta_N}{\theta_N-\tau}\frac{\Delta G}{Y}\)
\[\begin{aligned} \Delta Y &= -\frac{\theta_K}{\theta_N(1-\tau)}Y \Delta \tau\\[0.6em] \Rightarrow \Delta G &= Y \Delta \tau -\tau \frac{\theta_K}{\theta_N(1-\tau)}Y \Delta \tau \\[0.6em] \Rightarrow \frac{\Delta G}{Y} &= (1 -\frac{\tau \theta_K}{\theta_N(1-\tau)})\Delta \tau\\[0.6em] \Rightarrow \frac{\Delta G}{Y} &= \frac{\theta_N(1-\tau)-\tau(1-\theta_N)}{\theta_N(1-\tau)} \Delta \tau\\[0.6em] \Rightarrow \frac{\Delta G}{Y} &= \frac{\theta_N-\tau}{\theta_N(1-\tau)}\Delta \tau\\[0.6em] \Rightarrow \Delta \tau&= \frac{\theta_N(1-\tau)}{\theta_N-\tau}\frac{\Delta G}{Y} \end{aligned}\]