17. General Modeling Set-up

Jun He June 01, 2026
中文English 最优税收Optimal Taxation 李嘉图等价Ricardian Equivalence 政府预算约束Government Budget Constraint 瓦尔拉斯定律Walras' Law
Note

本组导读：最优税收（Optimal Taxation） 当经济中存在政府时，政府问题有两面：支出面与收入面。支出面研究最优的支出水平与时机；本组聚焦收入面，即税收。

自然的问题是：该对什么征税？ 可能的答案有财富、国际贸易、资本收入等等。例如，国际贸易较易监管，故关税在很多国家是主要税收来源；劳动收入税在发展中国家则难以征收，因为其执行依赖完善的雇佣结构，而发展中国家常以自雇农户为主。一言以蔽之，不同发展阶段的国家对各税源赋予不同权重。

几篇有影响力的税收文献： - Diamond & Mirrlees (1971)：最优税应使所有中间品税率为零；应对初级品或最终产出征税，扭曲较小。 - Lucas & Stokey (1983)：对消费与劳动收入征税是类似的，因为二者都会扭曲家庭决策，使其偏离消费、转向闲暇（闲暇不可征税）。 - Chamley (1986)：资本收入不应被征税（此结论基于诸多假设，脱离这些假设则不能断言资本税应为零）。 - Jaimovich & Rebelo (2016)：收入税对增长的影响是非线性的——企业家精神驱动增长，财富最高一档的人是创新者；若对这些创新者征税，会严重影响经济增长，因为对他们的激励削减会降低创业与创新；但对其他主体征税并不扭曲。 - Aiyagari (1994)：在未来市场不完备的经济中，主体无法对抗特质性劳动供给冲击，故会出于预防动机储蓄以自保，导致资本（储蓄）的过度累积。

本组从（扭曲性）线性劳动税入手。我们关心的是：在给定固定且外生（确定或随机）的支出路径下，最优的税收路径（水平与时机）。在深入不同模型之前，先明确：政府的目标是在满足其支出与债务义务的前提下，最小化无谓损失（DWL）。

对不同商品/不同日期征税的效果是类似的（同日期不同商品、同商品不同日期）：蓝色阴影三角形即为两种情形下的 DWL（图 12，已转述）。税收平滑（tax smoothing）在多数情形下是政府 DWL 最小化问题的解。

17. 一般建模框架

在接下来的各模型中，我们考虑无限期、单一商品（为简化）的经济。

17.1 模型设定

给定（外生）：

  • 政府支出序列：\(\{G_t\}_{t=0}^{\infty}\)
    • 该序列可以是确定的，也可以是随机的。

选择（内生，由政府决定）：

  • 政府税收序列：\(\{T_t\}_{t=0}^{\infty}\)
  • 政府债券的期限结构 \(i\)：\(\{{}_i b_t\}_{t=0}^{\infty}\)，其中 \(i=0,1,2,\ldots\)
    • 这里 \({}_i b_t\) 是在日期 \(i\) 发行（\(t\ge i\)）、在日期 \(t\) 到期偿付的零息债券本金。

可被政府影响：

  • 相对价格 \(p_t\)
    • \(p_t\) 是日期 \(t\) 的商品以日期 \(0\) 的商品计价的价格。
    • 归一化：\(p_0=1\)。
    • 相对价格 \(p_t\) 内嵌了利率：

$$ p_t=\prod_{s=1}^{t}\frac{1}{1+r_s},\quad t=1,2,\ldots \tag{17.1} $$

政府在日期 \(0\) 的预算约束：

$$ \sum_{t=0}^{\infty} p_t\left(G_t+{}_0 b_t-T_t\right)\le 0 \tag{17.2} $$

Tip

注记 注意以下两点：

  1. 瓦尔拉斯定律（Walras' Law）：若家庭的跨期预算约束成立，且每一期商品市场出清，则政府的预算约束自动成立（由瓦尔拉斯定律保证）。

  2. 李嘉图等价（Ricardian Equivalence）：家庭只关心税收的总量 \(\sum_{t=0}^{\infty} p_t T_t\)，而不关心其路径（水平与时机）。

Note

Group overview: Optimal Taxation When the economy includes a government, the government problem has two sides: the spending side and the revenue side. The spending side concerns the optimal level and timing of expenditure; this group focuses on the revenue side, i.e. taxation.

The natural question is: what should be taxed? Possible answers include wealth, international trade, capital income, and so on. For example, international trade is easier to monitor, so tariffs are a major tax revenue source for many countries; labor income tax, on the other hand, is hard to collect for developing countries since its enforcement requires a well-developed employment structure, which is rarely the case in developing countries with self-employed farmers dominating the economy. In a word, different stages of development typically mean different weights on tax income sources for a country.

Some influential papers on taxation: - Diamond & Mirrlees (1971): optimal taxes are zero on all intermediate goods; instead, we can tax on primary goods or final outputs, which tends to be less distorting. - Lucas & Stokey (1983): taxes on consumption and labor income are similar since they both distort the household away from consumption towards leisure as leisure is not taxed. - Chamley (1986): capital income should not be taxed (this rests on many assumptions; without them we cannot say capital tax should be zero). - Jaimovich & Rebelo (2016): the effect of income taxes on growth is nonlinear — entrepreneurship drives growth, which means the upper percentage of wealthiest individuals are the innovative people; if we impose taxes on these innovative people, then it will seriously affect the growth of our economy, because the disincentive for them will reduce entrepreneurship and innovation; but imposing taxes on other agents are not that distorting. - Aiyagari (1994): agents in an economy with incomplete future markets cannot insure themselves against idiosyncratic labor supply shocks, so they will do the precautionary saving to insure themselves, which results in over-accumulation of capital (savings).

This group starts with (distorting) linear tax on labor. We are interested in the optimal path (level and timing) of tax collection given a fixed and exogenous (deterministic or stochastic) expenditure path. Before delving into the different models, make it clear that the goal of the government is to minimize the dead weight loss (DWL) given its expenditure and debt obligations are met.

The effects of taxes levied on different goods of the same date and on the same good of different dates are analogous: the blue shaded triangle is the DWL in both cases (Figure 12, paraphrased). Tax smoothing is the solution to the government DWL minimization problem in most cases.

17. General Modeling Set-up

In the following models, we consider an infinite horizon economy with a single good (for simplicity).

17.1 Model set-up

Given (exogenous):

  • The government spending sequence: \(\{G_t\}_{t=0}^{\infty}\)
    • this sequence can be deterministic or stochastic.

Choice (endogenous, decision made by government):

  • The government tax revenue sequence: \(\{T_t\}_{t=0}^{\infty}\)
  • The government bond maturity structure \(i\): \(\{{}_i b_t\}_{t=0}^{\infty}\) for \(i=0,1,2,\ldots\)
    • where \({}_i b_t\) is the outstanding zero coupon bond principal due (to be repaid) at date \(t\) which is issued at date \(i\) (\(t\ge i\)).

Can be affected by government:

  • relative price \(p_t\)
    • \(p_t\) is the price of the good at date \(t\) in terms of good at date \(0\).
    • normalize: \(p_0=1\).
    • the relative price \(p_t\) has the interest rates embedded in it:

$$ p_t=\prod_{s=1}^{t}\frac{1}{1+r_s},\quad t=1,2,\ldots \tag{17.1} $$

Government's budget constraint at date \(0\):

$$ \sum_{t=0}^{\infty} p_t\left(G_t+{}_0 b_t-T_t\right)\le 0 \tag{17.2} $$

Tip

Remark Note the following two things:

  1. Walras' Law: if the household's inter-temporal budget constraint holds, and the good market clears for each period, then the government's budget constraint holds (by Walras' Law).

  2. Ricardian Equivalence: households only care about the total amount \(\sum_{t=0}^{\infty} p_t T_t\), not the path (level and timing) of taxes.