An Exploration in the Theory of Optimum Income Taxation ^(1,2){ }^{1,2} 最优所得税理论探讨
J. A. MIRRLEES J. A. 米尔利斯Nuffield College, Oxford 牛津大学纳菲尔德学院
1. INTRODUCTION 1. 引言
One would suppose that in any economic system where equality is valued, progressive income taxation would be an important instrument of policy. Even in a highly socialist economy, where all who work are employed by the State, the shadow price of highly skilled labour should surely be considerably greater than the disposable income actually available to the labourer. In Western Europe and America, tax rates on both high and low incomes are widely and lengthily discussed ^(3){ }^{3} : but there is virtually no relevant economic theory to appeal to, despite the importance of the tax. 人们可能会认为,在任何重视平等的经济体系中,累进所得税将是一个重要的政策工具。即使在一个高度社会主义的经济中,所有工作者都由国家雇佣,高技能劳动的影子价格也应该远远高于劳动者实际可支配的收入。在西欧和美国,高收入和低收入的税率都被广泛且深入地讨论 ^(3){ }^{3} :但几乎没有相关的经济理论可供参考,尽管税收的重要性不言而喻。
Redistributive progressive taxation is usually related to a man’s income (or, rather, his estimated income). One might obtain information about a man’s income-earning potential from his apparent I.Q., the number of his degrees, his address, age or colour: but the natural, and one would suppose the most reliable, indicator of his income-earning potential is his income. As a result of using men’s economic performance as evidence of their economic potentialities, complete equality of social marginal utilities of income ceases to be desirable, for the tax system that would bring about that result would completely discourage unpleasant work. The questions therefore arise what principles should govern an optimum income tax; what such a tax schedule would look like; and what degree of inequality would remain once it was established. 再分配的累进税制通常与一个人的收入(或者说,他的预估收入)相关。人们可能通过一个人的显性智商、学位数量、住址、年龄或肤色来获取关于其收入潜力的信息;但自然的、也可以说是最可靠的收入潜力指标是他的实际收入。因此,使用人们的经济表现作为其经济潜力的证据,社会边际效用的完全平等不再是可取的,因为实现这一结果的税制将完全抑制不愉快的工作。因此,问题随之而来:应当遵循什么原则来制定最佳所得税;这样的税率表将是什么样子;以及一旦建立后,将剩下什么程度的不平等。
The problem seems to be a rather difficult one even in the simplest cases. In this paper, I make the following simplifying assumptions: 这个问题似乎即使在最简单的情况下也相当困难。在本文中,我做出以下简化假设:
(1) Intertemporal problems are ignored. It is usual to levy income tax upon each year’s income, with only limited possibilities of transferring one year’s income to another for tax purposes. In an optimum system, one would no doubt wish to relate tax payments to the whole life pattern of income, ^(4){ }^{4} and to initial wealth; and in scheduling payments one would wish to pay attention to imperfect personal capital markets and imperfect foresight. The economy discussed below is timeless. Thus the effects of taxation on saving are ignored. One might perhaps regard the theory presented as a theory of “earned income” taxation (i.e. non-property income). (1)跨期问题被忽视。通常对每年的收入征收所得税,只有有限的可能性将一年的收入转移到另一年以用于税务目的。在一个最优系统中,无疑希望将税款与整个生命周期的收入 ^(4){ }^{4} 以及初始财富相关联;在安排支付时,也希望关注不完善的个人资本市场和不完善的预见性。下面讨论的经济是无时间的。因此,税收对储蓄的影响被忽视。或许可以将所提出的理论视为“劳动收入”税收理论(即非财产收入)。
(2) Differences in tastes, in family size and composition, and in voluntary transfers, are ignored. These raise rather different kinds of problems, and it is natural to assume them away. (2)口味、家庭规模和构成以及自愿转移的差异被忽略。这些会引发相当不同类型的问题,因此假设它们不存在是合乎自然的。
(3) Individuals are supposed to determine the quantity and kind of labour they provide by rational calculation, corresponding to the maximization of a utility function, and social welfare is supposed to be a function of individual utility levels. It is also supposed that the quantity of labour a man offers may be varied within wide limits without affecting the price paid for it. The first assumption may well be seriously unrealistic, especially at higher income levels, where it does sometimes appear that there is consumption satiation and that work is done for reasons barely connected with the income it provides to the " labourer ". (3)个人应通过理性计算来确定他们提供的劳动数量和种类,这与效用函数的最大化相对应,社会福利应被视为个体效用水平的函数。还假设一个人提供的劳动数量可以在广泛的范围内变化,而不影响为其支付的价格。第一个假设在较高收入水平下可能显得非常不现实,因为在这种情况下,确实有时会出现消费饱和现象,工作往往是出于与其为“劳动者”提供的收入几乎无关的原因。
(4) Migration is supposed to be impossible. Since the threat of migration is a major influence on the degree of progression in actual tax systems, at any rate outside the United States, this is another assumption one would rather not make. ^(1){ }^{1} (4) 移民被认为是不可能的。由于移民的威胁是影响实际税制进展程度的主要因素,至少在美国以外,这又是一个不愿意做出的假设。
(5) The State is supposed to have perfect information about the individuals in the economy, their utilities and, consequently, their actions. In practice, this is certainly not the case for certain kinds of income from self-employment, in particular work done for the worker himself and his family; and in some countries, the extent of uncertainty about incomes is very great. Yet it seems doubtful whether the neglect of this uncertainty is a simplification of much significance. (5)国家应该对经济中的个人、他们的效用以及因此而产生的行为拥有完美的信息。实际上,对于某些类型的自雇收入,尤其是为自己和家庭所做的工作,这显然并非如此;在一些国家,关于收入的不确定性程度非常大。然而,忽视这种不确定性是否是一种具有重要意义的简化似乎值得怀疑。
(6) Various formal simplifications are made to render the mathematics more manageable: there is supposed to be one kind of labour (in a special sense to be explained below); there is one consumer good; welfare is separable in terms of the different individuals of the economy, and symmetric-i.e. it can be expressed as the sum of the utilities of individuals when the individual utility function (the same for all) is suitably chosen). (6)为了使数学更易于处理,进行了各种形式上的简化:假设存在一种劳动(在下面将进行特别说明);存在一种消费品;福利在经济中不同个体之间是可分的,并且是对称的——即当个体效用函数(对所有人相同)被适当选择时,可以表示为个体效用的总和。
(7) The costs of administering the optimum tax schedule are assumed to be negligible. (7)管理最佳税率表的成本被假定为微不足道。
In sections 2-5, the more general properties of the optimum income-tax schedule, and the rules governing it, are discussed. The treatment is not rigorous. Nevertheless a reader who wants to avoid mathematical details can omit the last page or two of section 3, and will probably want to glance through section 4 rather rapidly. In section 6, I begin the discussion of special cases. The mathematical arguments in sections 6-86-8 are frequently complicated. If the reader goes straight to section 9 , where numerical results are presented and discussed, he should not find the omission of the previous sections any handicap. He may, nevertheless, find it interesting to look at the results and conjectures presented at the beginning of section 7, and at the diagrams for the two cases discussed in section 8. 在第 2-5 节中,讨论了最优所得税制度的更一般特性及其规则。处理并不严格。然而,想要避免数学细节的读者可以省略第 3 节的最后一两页,并且可能希望快速浏览第 4 节。在第 6 节中,我开始讨论特殊情况。第 6-86-8 节中的数学论证通常比较复杂。如果读者直接跳到第 9 节,在那里呈现和讨论了数值结果,他应该不会觉得省略前面的章节有任何障碍。然而,他可能会发现查看第 7 节开头呈现的结果和猜想,以及第 8 节讨论的两个案例的图表是有趣的。
Rigorous proofs of the main theorems will be given in a subsequent paper, [4]. 主要定理的严格证明将在后续论文中给出,[4]。
2. MODEL AND PROBLEM 2. 模型与问题
Individuals have identical preferences. We shall suppose that consumption and working time enter the individual’s utility function. When consumption is xx and the time worked yy, utility is 个体具有相同的偏好。我们假设消费和工作时间进入个体的效用函数。当消费为 xx 且工作时间为 yy 时,效用为
u(x,y)". "u(x, y) \text {. }
xx and yy both have to be non-negative, and there is an upper limit to yy, which is taken to be 1 . In fact, it is assumed that: uu is a strictly concave, continuously differentiable, function (strictly) increasing in xx, (strictly) decreasing in yy, defined for x > 0x>0 and 0 <= y < 1.u0 \leqq y<1 . u tends to -oo-\infty as xx tends to 0 from above or yy tends to 1 from below. xx 和 yy 都必须是非负的,并且 yy 有一个上限,取为 1。实际上,假设 uu 是一个严格凹的、连续可微的函数,在 xx 上是(严格)递增的,在 yy 上是(严格)递减的,定义在 x > 0x>0 上,并且当 xx 从上方趋近于 0 或 yy 从下方趋近于 1 时, 0 <= y < 1.u0 \leqq y<1 . u 趋近于 -oo-\infty 。
The usefulness of a man’s time, from the point of view of production, is assumed to vary from person to person. To each individual corresponds a number nn such that the quantity of labour provided, per unit of his time, is nn. If he works for time yy, he provides a quantity of labour nyn y. There is a known distribution of skills, measured by the parameter nn, in the population. The number of persons with labour parameter nn or less is F(n)F(n). It 一个人的时间在生产方面的有用性被假定因人而异。每个个体对应一个数字 nn ,使得他每单位时间提供的劳动量为 nn 。如果他工作时间为 yy ,则他提供的劳动量为 nyn y 。在总体中,技能的分布是已知的,以参数 nn 来衡量。劳动参数为 nn 或更少的人数为 F(n)F(n) 。
will be assumed that FF is differentiable, so that there is a density function for ability, f(n)=F^(')(n)f(n)=F^{\prime}(n). Call an individual whose ability-parameter is nn an nn-man. 将假设 FF 是可微的,因此存在一个能力的密度函数 f(n)=F^(')(n)f(n)=F^{\prime}(n) 。称能力参数为 nn 的个体为 nn -人。
The consumption choice of an nn-man is denoted by (x_(n),y_(n))\left(x_{n}, y_{n}\right). Write z_(n)=ny_(n)z_{n}=n y_{n} for the labour he provides. Then the total labour available for use in production in the economy is 一个 nn -人的消费选择用 (x_(n),y_(n))\left(x_{n}, y_{n}\right) 表示。设他提供的劳动为 z_(n)=ny_(n)z_{n}=n y_{n} 。那么经济中可用于生产的总劳动量为
Z=int_(0)^(oo)z_(n)f(n)dnZ=\int_{0}^{\infty} z_{n} f(n) d n
and the aggregate demand for consumer goods is 和消费品的总需求是
X=int_(0)^(oo)x_(n)f(n)dnX=\int_{0}^{\infty} x_{n} f(n) d n
In order to avoid the possibility of infinite labour supply, I assume that 为了避免无限劳动供给的可能性,我假设
int_(0)^(oo)nf(n)dn < oo\int_{0}^{\infty} n f(n) d n<\infty
Each individual makes his choice of (x_(n),y_(n))\left(x_{n}, y_{n}\right) in the light of his budget constraint. Using an income tax, the government can arrange that a man who supplies a quantity of labour zz can consume no more than c(z)c(z) after tax: the government can choose the function cc arbitrarily. It makes sense to impose the restriction on the government’s choice of cc, that cc be upper semi-continuous, for then all individuals have available to them consumption choices that maximize their utility, subject to the budget constraint ^(1){ }^{1} : 每个个体在其预算约束的情况下做出对 (x_(n),y_(n))\left(x_{n}, y_{n}\right) 的选择。通过征收所得税,政府可以安排一个提供数量为 zz 的劳动者在税后消费不超过 c(z)c(z) :政府可以任意选择函数 cc 。对政府选择 cc 施加限制是有意义的,即 cc 应为上半连续的,这样所有个体在预算约束 ^(1){ }^{1} 下都有可用的消费选择,以最大化他们的效用。
(x_(n),y_(n))" maximizes "u(x,y)" subject to "x <= c(ny)". "\left(x_{n}, y_{n}\right) \text { maximizes } u(x, y) \text { subject to } x \leqq c(n y) \text {. }
Notice that (x_(n),y_(n))\left(x_{n}, y_{n}\right) may not be uniquely determined for every n^(2)n^{2} I write: 注意到 (x_(n),y_(n))\left(x_{n}, y_{n}\right) 可能并不是对每个 n^(2)n^{2} 唯一确定的
Proof. If m < nm<n, and y_(m) > 0,u[c(my_(m)),y_(m)] < u[c(n*(m)/(n)y_(m)),(m)/(n)y_(m)] <= u_(n)y_{m}>0, u\left[c\left(m y_{m}\right), y_{m}\right]<u\left[c\left(n \cdot \frac{m}{n} y_{m}\right), \frac{m}{n} y_{m}\right] \leqq u_{n}. Consequently, y_(m)=0y_{m}=0 if y_(n)=0y_{n}=0, since then y_(m)=0y_{m}=0 gives the utility u_(n)u_{n} to nn-man. Thus 证明。如果 m < nm<n ,并且 y_(m) > 0,u[c(my_(m)),y_(m)] < u[c(n*(m)/(n)y_(m)),(m)/(n)y_(m)] <= u_(n)y_{m}>0, u\left[c\left(m y_{m}\right), y_{m}\right]<u\left[c\left(n \cdot \frac{m}{n} y_{m}\right), \frac{m}{n} y_{m}\right] \leqq u_{n} 。因此,如果 y_(n)=0y_{n}=0 ,则 y_(m)=0y_{m}=0 ,因为那时 y_(m)=0y_{m}=0 给 nn -人带来了效用 u_(n)u_{n} 。因此
has the desired properties. ||\| 具有所需的特性。 ||\|
n_(0)=i n f[n∣y_(n) > 0]n_{0}=\inf \left[n \mid y_{n}>0\right]
Proposition 2. Any function ^(3){ }^{3} of n,(x_(n),y_(n))n,\left(x_{n}, y_{n}\right), that satisfies (4) for some upper semicontinuous function c also satisfies (4) for some non-decreasing, right-continuous function c^(')c^{\prime}. 命题 2. 任何满足 (4) 的 n,(x_(n),y_(n))n,\left(x_{n}, y_{n}\right) 的函数 ^(3){ }^{3} ,如果对于某个上半连续函数 c 也满足 (4),则对于某个非递减、右连续的函数 c^(')c^{\prime} 也满足 (4)。
Proof. Define c^(')(z)=s u p_(z^(') <= z)c(z^('))c^{\prime}(z)=\sup _{z^{\prime} \leqq z} c\left(z^{\prime}\right). If x_(n)^(') <= c^(')(ny_(n)^('))x_{n}^{\prime} \leqq c^{\prime}\left(n y_{n}^{\prime}\right), then, for any epsi > 0\varepsilon>0, there exists y_(n)^('') <= y_(n)^(')y_{n}^{\prime \prime} \leqq y_{n}^{\prime} such that x_(n)^(')-epsi <= c(ny_(n)^(''))x_{n}^{\prime}-\varepsilon \leqq c\left(n y_{n}^{\prime \prime}\right). Thus u(x_(n)^(')-epsi,y_(n)^('')) <= u_(n)u\left(x_{n}^{\prime}-\varepsilon, y_{n}^{\prime \prime}\right) \leqq u_{n}, which implies, since uu is a decreasing function in yy, that u(x_(n)^(')-epsi,y_(n)^(')) <= u_(n)u\left(x_{n}^{\prime}-\varepsilon, y_{n}^{\prime}\right) \leqq u_{n}. Letting epsi rarr0,u(x_(n)^('),y_(n)^(')) <= u_(n)\varepsilon \rightarrow 0, u\left(x_{n}^{\prime}, y_{n}^{\prime}\right) \leqq u_{n}. It follows that (x_(n),y_(n):}\left(x_{n}, y_{n}\right. ) maximizes uu subject to x <= c^(')(ny)x \leqq c^{\prime}(n y). 证明。定义 c^(')(z)=s u p_(z^(') <= z)c(z^('))c^{\prime}(z)=\sup _{z^{\prime} \leqq z} c\left(z^{\prime}\right) 。如果 x_(n)^(') <= c^(')(ny_(n)^('))x_{n}^{\prime} \leqq c^{\prime}\left(n y_{n}^{\prime}\right) ,那么,对于任何 epsi > 0\varepsilon>0 ,存在 y_(n)^('') <= y_(n)^(')y_{n}^{\prime \prime} \leqq y_{n}^{\prime} 使得 x_(n)^(')-epsi <= c(ny_(n)^(''))x_{n}^{\prime}-\varepsilon \leqq c\left(n y_{n}^{\prime \prime}\right) 。因此 u(x_(n)^(')-epsi,y_(n)^('')) <= u_(n)u\left(x_{n}^{\prime}-\varepsilon, y_{n}^{\prime \prime}\right) \leqq u_{n} ,这意味着,由于 uu 在 yy 中是一个递减函数, u(x_(n)^(')-epsi,y_(n)^(')) <= u_(n)u\left(x_{n}^{\prime}-\varepsilon, y_{n}^{\prime}\right) \leqq u_{n} 。令 epsi rarr0,u(x_(n)^('),y_(n)^(')) <= u_(n)\varepsilon \rightarrow 0, u\left(x_{n}^{\prime}, y_{n}^{\prime}\right) \leqq u_{n} 。因此, (x_(n),y_(n):}\left(x_{n}, y_{n}\right. )在约束条件 x <= c^(')(ny)x \leqq c^{\prime}(n y) 下最大化 uu 。 c^(')c^{\prime} is clearly a non-decreasing function of zz. To prove that it is right-continuous, take a decreasing sequence z^(i)rarr z.quadc^(')(z^(i))z^{i} \rightarrow z . \quad c^{\prime}\left(z^{i}\right) is a non-increasing sequence, and therefore tends to a limit, which is not less than c^(')(z)c^{\prime}(z). If it is equal to c^(')(z)c^{\prime}(z), there is no more to prove. Suppose it is greater. Then for some epsi > 0\varepsilon>0 each c^(')(z^(i)) > c^(')(z)+epsic^{\prime}\left(z^{i}\right)>c^{\prime}(z)+\varepsilon. Therefore, there exists a sequence ( {: bar(z)^(i))\left.\bar{z}^{i}\right) such that bar(z)^(i) <= z^(i)\bar{z}^{i} \leqq z^{i} and c^(')(z^(i)) >= c( bar(z)^(i)) > c^(')(z)+epsic^{\prime}\left(z^{i}\right) \geqq c\left(\bar{z}^{i}\right)>c^{\prime}(z)+\varepsilon