18. Barro's Model (1979)
18. Barro 模型（1979，Journal of Political Economy）
18.1 设定
给定（外生）：
- 确定的政府支出序列：\(\{G_t\}_{t=0}^{\infty}\)
- 总产出（GDP）序列：\(\{Y_t\}_{t=0}^{\infty}\)
- 我们假设 GDP 序列外生，再从 GDP 中扣除因征税造成的损失。
选择（内生，由政府决定）：
- 政府税收序列：\(\{T_t\}_{t=0}^{\infty}\)
- 每期到期债务：\(\{b_t\}_{t=0}^{\infty}\)
- 唯一可用的政府债务工具是单期短期零息债券。
- \(b_t\) 是政府需在日期 \(t\) 偿付、在日期 \(t-1\) 发行的债务本金。
- 不允许对债务违约。
其他设定：
- 时间上利率恒定：\(r\)
- 无谓损失：\(L_t=Y_t\cdot f\!\left(\dfrac{T_t}{Y_t}\right)\)，其中
- \(f(\cdot)\) 严格递增且严格凸（此处的 \(f(\cdot)\) 类比于图 12 中的三角形 DWL）。
18.2 政府的最小化问题
18.2.1 问题
政府希望最小化超额负担（DWL）：
$$ \min_{\{T_t\}_{t=0}^{\infty}}\ \sum_{t=0}^{\infty}\left(\frac{1}{1+r}\right)^t Y_t\, f\!\left(\frac{T_t}{Y_t}\right) \tag{18.0a} $$
$$ \text{s.t.}\quad \sum_{t=0}^{\infty}\left(\frac{1}{1+r}\right)^t (G_t-T_t)+b_0=0 \tag{18.0b} $$
此处我们把政府的预算约束设在日期 \(0\)。
18.2.2 各日期的预算约束不同
在较晚的日期（例如日期 \(1\)），政府的预算约束不同：
$$ \sum_{t=1}^{\infty}\left(\frac{1}{1+r}\right)^{t-1}(G_t-T_t)+b_1=0 $$
一般地，在日期 \(j\)，政府的预算约束为：
$$ \sum_{t=j}^{\infty}\left(\frac{1}{1+r}\right)^{t-j}(G_t-T_t)+b_j=0 $$
18.2.3 求解
建立拉格朗日函数：
$$ \mathcal{L}=\sum_{t=0}^{\infty}\left(\frac{1}{1+r}\right)^t Y_t\, f\!\left(\frac{T_t}{Y_t}\right)+\lambda\left(\sum_{t=0}^{\infty}\left(\frac{1}{1+r}\right)^t (G_t-T_t)+b_0\right) $$
对 \(T_t\) 求一阶条件：
$$ \left(\frac{1}{1+r}\right)^t \frac{1}{Y_t}\,Y_t\, f'\!\left(\frac{T_t}{Y_t}\right)=\lambda\left(\frac{1}{1+r}\right)^t \ \Rightarrow\ f'\!\left(\frac{T_t}{Y_t}\right)=\lambda\quad \forall t $$
因此可得
$$ \frac{T_t}{Y_t}=\theta $$
其中 \(\theta\) 为常数，由政府预算约束确定。
结论：税收平滑 最优税收在时间上具有平滑（实际上恒定）的路径：税收应作为总产出的一个固定比例来设定与比较。这就是 税收平滑（tax smoothing）。
18.3 几个例子
18.3.1 例 1：\(\{G_t\}\) 与 \(\{Y_t\}\) 均为常数
设政府支出恒定 \(G_t=\overline{G}\) \(\forall t\)，GDP 恒定 \(Y_t=\overline{Y}\) \(\forall t\)。由 §18.2.3 的结论：
$$ T_t=\theta\overline{Y}\equiv \overline{T}\quad \forall t $$
也是常数。考虑日期 \(0\) 与日期 \(1\) 的预算约束：
$$ \sum_{t=0}^{\infty}\left(\frac{1}{1+r}\right)^t (G_t-T_t)+b_0=0 \tag{18.1} $$
$$ \sum_{t=1}^{\infty}\left(\frac{1}{1+r}\right)^{t-1}(G_t-T_t)+b_1=0 \tag{18.2} $$
由 \(G_t=\overline{G}\)、\(Y_t=\overline{Y}\)，(18.1) 与 (18.2) 蕴含 \(b_0=b_1=b_2=\cdots\equiv\bar b\)，亦为常数。
将 (18.2) 两边乘 \(\frac{1}{1+r}\)：
$$ \sum_{t=1}^{\infty}\left(\frac{1}{1+r}\right)^{t}(G_t-T_t)+\frac{1}{1+r}b_1=0 \tag{18.3} $$
用 (18.1) 减去 (18.3)：
$$ G_0-T_0+b_0-\frac{1}{1+r}b_1=0\ \Rightarrow\ \overline{G}-\overline{T}+\bar b-\frac{1}{1+r}\bar b=0\ \Rightarrow\ \overline{T}=\overline{G}+\frac{r}{1+r}\bar b $$
解 政府应在每一期最优地征收恒定的税 \(\overline{T}=\overline{G}+\dfrac{r}{1+r}\bar b\)。
18.3.2 例 2：\(\{G_t\}\) 与 \(\{Y_t\}\) 均以同一常数率增长
设政府支出与 GDP 以同一常数率 \(\sigma\) 增长：
$$ G_t=(1+\sigma)^t G_0\quad \forall t,\qquad Y_t=(1+\sigma)^t Y_0\quad \forall t $$
由 §18.2.3：
$$ T_t=\theta Y_t=(1+\sigma)^t T_0\quad \forall t $$
也以同一常数率 \(\sigma\) 增长。再考虑日期 \(0\) 与日期 \(1\) 的预算约束：
$$ \sum_{t=0}^{\infty}\left(\frac{1}{1+r}\right)^t (G_t-T_t)+b_0=0 \tag{18.4} $$
$$ \sum_{t=1}^{\infty}\left(\frac{1}{1+r}\right)^{t-1}(G_t-T_t)+b_1=0 \tag{18.5} $$
由 \(G_t=(1+\sigma)G_{t-1}\)、\(Y_t=(1+\sigma)Y_{t-1}\)，(18.4) 与 (18.5) 蕴含
$$ b_1=(1+\sigma)b_0\ \Rightarrow\ b_t=(1+\sigma)^t b_0 $$
也以同一常数率 \(\sigma\) 增长。将 (18.5) 两边乘 \(\frac{1}{1+r}\)：
$$ \sum_{t=1}^{\infty}\left(\frac{1}{1+r}\right)^{t}(G_t-T_t)+\frac{1}{1+r}b_1=0 \tag{18.6} $$
用 (18.4) 减去 (18.6)：
$$ G_0-T_0+b_0-\frac{1}{1+r}b_1=0\ \Rightarrow\ G_0-T_0+b_0-\frac{1+\sigma}{1+r}b_0=0\ \Rightarrow\ T_0=G_0+\frac{r-\sigma}{1+r}b_0 $$
$$ \Rightarrow\ T_t=(1+\sigma)^t\left(G_0+\frac{r-\sigma}{1+r}b_0\right) $$
解 在日期 \(t\)，政府应最优地征收 \(T_t=(1+\sigma)^t\!\left(G_0+\dfrac{r-\sigma}{1+r}b_0\right)\)，其增长率与 GDP、政府支出相同。
注记 18.1 在该模型中，我们得到的结论是：经济的最优债务为 GDP 的一个固定比例，即 \(\dfrac{b_t}{Y_t}\) 恒定。若该比例非常高，经济会陷入困境，因为高债务比会使市场（债务投资者）紧张，从而导致更高的风险溢价、更高的利率 \(r\)，进而抬升政府的债务成本，使政府面临更糟的局面。
注记 18.2 在 Barro 模型的设定中，利率 \(r\) 固定、GDP 序列 \(\{Y_t\}\) 外生，故二者都无法随政府对债务与税率的选择而调整。因此，这不是一个一般均衡模型，我们无法据此对政府计划的时间一致性（time consistency）下结论——因为随着时间推移没有信息更新，政府在日期 \(0\) 即可作出决策且永不调整，这使得时间一致性问题变得平凡。
（时间一致性：在每一期，政府重新优化其税收政策；若每期所作的计划彼此一致，则称政府的计划具有时间一致性。）
18. Barro's Model (1979, Journal of Political Economy)
18.1 Set-up
Given (exogenous):
- The deterministic government spending sequence: \(\{G_t\}_{t=0}^{\infty}\)
- The total output (GDP) sequence: \(\{Y_t\}_{t=0}^{\infty}\)
- we suppose the GDP sequence is exogenous, and then subtract the loss due to tax collection from the GDP.
Choice (endogenous, decision made by government):
- The government tax revenue sequence: \(\{T_t\}_{t=0}^{\infty}\)
- Debt due in each period: \(\{b_t\}_{t=0}^{\infty}\)
- The only government bond instrument available is one-period short-term zero coupon bond.
- \(b_t\) is the debt principal that the government needs to repay at date \(t\), which is issued at date \(t-1\).
- No default is allowed for the debt obligation.
Other details:
- constant interest rate over time: \(r\)
- dead weight loss: \(L_t=Y_t\cdot f\!\left(\dfrac{T_t}{Y_t}\right)\) where
- \(f(\cdot)\) is strictly increasing and strictly convex (here \(f(\cdot)\) is analogous to the triangle in Figure 12).
18.2 The government's minimization problem
18.2.1 The problem
The government wants to minimize the excess burden (DWL), i.e.
$$ \min_{\{T_t\}_{t=0}^{\infty}}\ \sum_{t=0}^{\infty}\left(\frac{1}{1+r}\right)^t Y_t\, f\!\left(\frac{T_t}{Y_t}\right) \tag{18.0a} $$
$$ \text{s.t.}\quad \sum_{t=0}^{\infty}\left(\frac{1}{1+r}\right)^t (G_t-T_t)+b_0=0 \tag{18.0b} $$
where we are imposing the government's budget constraint at date \(0\).
18.2.2 Budget constraints are different at each date
At a later date, for example at date \(1\), the government's budget constraint is different:
$$ \sum_{t=1}^{\infty}\left(\frac{1}{1+r}\right)^{t-1}(G_t-T_t)+b_1=0 $$
In general, at date \(j\), the government's budget constraint is
$$ \sum_{t=j}^{\infty}\left(\frac{1}{1+r}\right)^{t-j}(G_t-T_t)+b_j=0 $$
18.2.3 Solving the problem
Set up the Lagrangian as
$$ \mathcal{L}=\sum_{t=0}^{\infty}\left(\frac{1}{1+r}\right)^t Y_t\, f\!\left(\frac{T_t}{Y_t}\right)+\lambda\left(\sum_{t=0}^{\infty}\left(\frac{1}{1+r}\right)^t (G_t-T_t)+b_0\right) $$
whose f.o.c. for \(T_t\) is
$$ \left(\frac{1}{1+r}\right)^t \frac{1}{Y_t}\,Y_t\, f'\!\left(\frac{T_t}{Y_t}\right)=\lambda\left(\frac{1}{1+r}\right)^t \ \Rightarrow\ f'\!\left(\frac{T_t}{Y_t}\right)=\lambda\quad \forall t $$
Therefore, we can conclude that
$$ \frac{T_t}{Y_t}=\theta $$
is constant with \(\theta\) determined by the government's budget constraint.
Conclusion: tax smoothing The optimal taxation has a smooth (actually constant) path over time, and the taxes should be considered and compared as a fraction of total outcome. This is tax smoothing.
18.3 Some examples
18.3.1 Example 1: constant \(\{G_t\}\) and constant \(\{Y_t\}\)
Suppose constant government spending \(G_t=\overline{G}\) \(\forall t\) and constant GDP \(Y_t=\overline{Y}\) \(\forall t\). By the result in §18.2.3:
$$ T_t=\theta\overline{Y}\equiv \overline{T}\quad \forall t $$
which is also a constant. Consider the government's budget constraint at date \(0\) and date \(1\):
$$ \sum_{t=0}^{\infty}\left(\frac{1}{1+r}\right)^t (G_t-T_t)+b_0=0 \tag{18.1} $$
$$ \sum_{t=1}^{\infty}\left(\frac{1}{1+r}\right)^{t-1}(G_t-T_t)+b_1=0 \tag{18.2} $$
Since \(G_t=\overline{G}\) and \(Y_t=\overline{Y}\), (18.1) and (18.2) imply \(b_0=b_1=b_2=\cdots\equiv\bar b\), which is also a constant.
Multiply both sides of (18.2) by \(\frac{1}{1+r}\):
$$ \sum_{t=1}^{\infty}\left(\frac{1}{1+r}\right)^{t}(G_t-T_t)+\frac{1}{1+r}b_1=0 \tag{18.3} $$
Subtracting (18.3) from (18.1):
$$ G_0-T_0+b_0-\frac{1}{1+r}b_1=0\ \Rightarrow\ \overline{G}-\overline{T}+\bar b-\frac{1}{1+r}\bar b=0\ \Rightarrow\ \overline{T}=\overline{G}+\frac{r}{1+r}\bar b $$
Solution The government should optimally collect a constant tax \(\overline{T}=\overline{G}+\dfrac{r}{1+r}\bar b\) in each period.
18.3.2 Example 2: constant growth rate for both \(\{G_t\}\) and \(\{Y_t\}\)
Suppose government spending and GDP grow at the same constant rate \(\sigma\):
$$ G_t=(1+\sigma)^t G_0\quad \forall t,\qquad Y_t=(1+\sigma)^t Y_0\quad \forall t $$
By the result in §18.2.3:
$$ T_t=\theta Y_t=(1+\sigma)^t T_0\quad \forall t $$
which is also growing at the same constant rate \(\sigma\). Again, consider the government's budget constraint at date \(0\) and date \(1\):
$$ \sum_{t=0}^{\infty}\left(\frac{1}{1+r}\right)^t (G_t-T_t)+b_0=0 \tag{18.4} $$
$$ \sum_{t=1}^{\infty}\left(\frac{1}{1+r}\right)^{t-1}(G_t-T_t)+b_1=0 \tag{18.5} $$
Since \(G_t=(1+\sigma)G_{t-1}\) and \(Y_t=(1+\sigma)Y_{t-1}\), (18.4) and (18.5) imply
$$ b_1=(1+\sigma)b_0\ \Rightarrow\ b_t=(1+\sigma)^t b_0 $$
which is also growing at the same constant rate \(\sigma\). Multiply both sides of (18.5) by \(\frac{1}{1+r}\):
$$ \sum_{t=1}^{\infty}\left(\frac{1}{1+r}\right)^{t}(G_t-T_t)+\frac{1}{1+r}b_1=0 \tag{18.6} $$
Subtracting (18.6) from (18.4):
$$ G_0-T_0+b_0-\frac{1}{1+r}b_1=0\ \Rightarrow\ G_0-T_0+b_0-\frac{1+\sigma}{1+r}b_0=0\ \Rightarrow\ T_0=G_0+\frac{r-\sigma}{1+r}b_0 $$
$$ \Rightarrow\ T_t=(1+\sigma)^t\left(G_0+\frac{r-\sigma}{1+r}b_0\right) $$
Solution At date \(t\), the government should optimally collect a tax \(T_t=(1+\sigma)^t\!\left(G_0+\dfrac{r-\sigma}{1+r}b_0\right)\), which grows at the same rate as GDP and government spending.
Remark 18.1 In this model, we have obtained the result that the optimal debt for the economy is a constant fraction of GDP, i.e. \(\dfrac{b_t}{Y_t}\) is constant. If this number is very high, then the economy is in trouble because a high debt ratio makes the market (debt investors) nervous, so it leads to a higher risk premium and thus higher interest rate \(r\), which will increase the cost of debt of government and make the government face worse situations.
Remark 18.2 In the set-up of Barro's Model, interest rate \(r\) is fixed and the GDP sequence \(\{Y_t\}_{t=0}^{\infty}\) is exogenous, so they both cannot adjust with the government's choice of debt and tax rate. Therefore, it is not a general equilibrium model, and we cannot draw conclusion about time consistency in government's plan from it since there is no update in information as time passes, so the government can make a decision at date \(0\) and never adjust it, which makes time consistency trivial.
(Time consistency: in each period, the government re-optimizes its tax policy. If the plans made in each period are consistent with each other, then we say the government's plan has time consistency.)