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An Exploration in the Theory of Optimum Income Taxation 1 , 2 1 , 2 ^(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 ^(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 ^(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 ^(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 ^(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 ^(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 − 8 6 − 8 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 − 8 6 − 8 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 x x xx and the time worked y y 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 x x xx 且工作时间为 y y yy 时，效用为
u ( x , y ) . u ( x , y ) .  u(x,y)". "u(x, y) \text {. }
x x xx and y y yy both have to be non-negative, and there is an upper limit to y y yy, which is taken to be 1 . In fact, it is assumed that: u u uu is a strictly concave, continuously differentiable, function (strictly) increasing in x x xx, (strictly) decreasing in y y yy, defined for x > 0 x > 0 x > 0x>0 and 0 ≦ y < 1 . u 0 ≦ y < 1 . u 0 <= y < 1.u0 \leqq y<1 . u tends to − ∞ − ∞ -oo-\infty as x x xx tends to 0 from above or y y yy tends to 1 from below.
x x xx 和 y y yy 都必须是非负的，并且 y y yy 有一个上限，取为 1。实际上，假设 u u uu 是一个严格凹的、连续可微的函数，在 x x xx 上是（严格）递增的，在 y y yy 上是（严格）递减的，定义在 x > 0 x > 0 x > 0x>0 上，并且当 x x xx 从上方趋近于 0 或 y y yy 从下方趋近于 1 时， 0 ≦ y < 1 . u 0 ≦ y < 1 . u 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 n n nn such that the quantity of labour provided, per unit of his time, is n n nn. If he works for time y y yy, he provides a quantity of labour n y n y nyn y. There is a known distribution of skills, measured by the parameter n n nn, in the population. The number of persons with labour parameter n n nn or less is F ( n ) F ( n ) 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 n n nn ，使得他每单位时间提供的劳动量为 n n nn 。如果他工作时间为 y y yy ，则他提供的劳动量为 n y n y nyn y 。在总体中，技能的分布是已知的，以参数 n n nn 来衡量。劳动参数为 n n nn 或更少的人数为 F ( n ) F ( n ) F(n)F(n) 。
will be assumed that F F FF is differentiable, so that there is a density function for ability, f ( n ) = F ′ ( n ) f ( n ) = F ′ ( n ) f(n)=F^(')(n)f(n)=F^{\prime}(n). Call an individual whose ability-parameter is n n nn an n n nn-man.
将假设 F F 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 ′ ( n ) f(n)=F^(')(n)f(n)=F^{\prime}(n) 。称能力参数为 n n nn 的个体为 n n nn -人。
The consumption choice of an n n nn-man is denoted by ( x n , y n ) x n , y n (x_(n),y_(n))\left(x_{n}, y_{n}\right). Write z n = n y n z n = n y n 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 n n nn -人的消费选择用 ( x n , y n ) x n , y n (x_(n),y_(n))\left(x_{n}, y_{n}\right) 表示。设他提供的劳动为 z n = n y n z n = n y n 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 = ∫ 0 ∞ z n f ( n ) d n Z = ∫ 0 ∞   z n f ( n ) d n 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 = ∫ 0 ∞ x n f ( n ) d n X = ∫ 0 ∞   x n f ( n ) d n 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
∫ 0 ∞ n f ( n ) d n < ∞ ∫ 0 ∞   n f ( n ) d n < ∞ 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 ) x n , y n (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 z z zz can consume no more than c ( z ) c ( z ) c(z)c(z) after tax: the government can choose the function c c cc arbitrarily. It makes sense to impose the restriction on the government’s choice of c c cc, that c c 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 ^(1){ }^{1} :
每个个体在其预算约束的情况下做出对 ( x n , y n ) x n , y n (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 z z zz 的劳动者在税后消费不超过 c ( z ) c ( z ) c(z)c(z) ：政府可以任意选择函数 c c cc 。对政府选择 c c cc 施加限制是有意义的，即 c c cc 应为上半连续的，这样所有个体在预算约束 1 1 ^(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 ( n y ) . x n , y n  maximizes  u ( x , y )  subject to  x ≦ c ( n y ) .  (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 ) x n , y n (x_(n),y_(n))\left(x_{n}, y_{n}\right) may not be uniquely determined for every n 2 n 2 n^(2)n^{2} I write:
注意到 ( x n , y n ) x n , y n (x_(n),y_(n))\left(x_{n}, y_{n}\right) 可能并不是对每个 n 2 n 2 n^(2)n^{2} 唯一确定的
u n = u ( x n , y n ) u n = u x n , y n u_(n)=u(x_(n),y_(n))u_{n}=u\left(x_{n}, y_{n}\right)
Proposition 1. There exists a number n 0 ≧ 0 n 0 ≧ 0 n_(0) >= 0n_{0} \geqq 0 such that
命题 1. 存在一个数字 n 0 ≧ 0 n 0 ≧ 0 n_(0) >= 0n_{0} \geqq 0 使得
y n = 0 ( n ≦ n 0 ) , y n > 0 ( n > n 0 ) . y n = 0      n ≦ n 0 , y n > 0      n > n 0 . {:[y_(n)=0,(n <= n_(0))","],[y_(n) > 0,(n > n_(0)).]:}\begin{array}{ll} y_{n}=0 & \left(n \leqq n_{0}\right), \\ y_{n}>0 & \left(n>n_{0}\right) . \end{array}
Proof. If m < n m < n m < nm<n, and y m > 0 , u [ c ( m y m ) , y m ] < u [ c ( n ⋅ m n y m ) , m n y m ] ≦ u n y m > 0 , u c m y m , y m < u c n ⋅ m n y m , m n y m ≦ u n 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 = 0 y m = 0 y_(m)=0y_{m}=0 if y n = 0 y n = 0 y_(n)=0y_{n}=0, since then y m = 0 y m = 0 y_(m)=0y_{m}=0 gives the utility u n u n u_(n)u_{n} to n n nn-man. Thus
证明。如果 m < n m < n m < nm<n ，并且 y m > 0 , u [ c ( m y m ) , y m ] < u [ c ( n ⋅ m n y m ) , m n y m ] ≦ u n y m > 0 , u c m y m , y m < u c n ⋅ m n y m , m n y m ≦ u n 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 = 0 y n = 0 y_(n)=0y_{n}=0 ，则 y m = 0 y m = 0 y_(m)=0y_{m}=0 ，因为那时 y m = 0 y m = 0 y_(m)=0y_{m}=0 给 n n nn -人带来了效用 u n u n u_(n)u_{n} 。因此
has the desired properties. ‖ ‖ ||\|
具有所需的特性。 ‖ ‖ ||\|
n 0 = inf [ n ∣ y n > 0 ] n 0 = inf n ∣ y n > 0 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 ^(3){ }^{3} of n , ( x n , y n ) n , x n , y n 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 ′ c^(')c^{\prime}.
命题 2. 任何满足 (4) 的 n , ( x n , y n ) n , x n , y n n,(x_(n),y_(n))n,\left(x_{n}, y_{n}\right) 的函数 3 3 ^(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 ′ c^(')c^{\prime} 也满足 (4)。
Proof. Define c ′ ( z ) = sup z ′ ≦ z c ( z ′ ) c ′ ( z ) = sup z ′ ≦ z   c z ′ 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 ′ ( n y n ′ ) x n ′ ≦ c ′ n y n ′ x_(n)^(') <= c^(')(ny_(n)^('))x_{n}^{\prime} \leqq c^{\prime}\left(n y_{n}^{\prime}\right), then, for any ε > 0 ε > 0 epsi > 0\varepsilon>0, there exists y n ′ ′ ≦ y n ′ y n ′ ′ ≦ y n ′ y_(n)^('') <= y_(n)^(')y_{n}^{\prime \prime} \leqq y_{n}^{\prime} such that x n ′ − ε ≦ c ( n y n ′ ′ ) x n ′ − ε ≦ c n y n ′ ′ x_(n)^(')-epsi <= c(ny_(n)^(''))x_{n}^{\prime}-\varepsilon \leqq c\left(n y_{n}^{\prime \prime}\right). Thus u ( x n ′ − ε , y n ′ ′ ) ≦ u n u x n ′ − ε , y n ′ ′ ≦ u n 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 u u uu is a decreasing function in y y yy, that u ( x n ′ − ε , y n ′ ) ≦ u n u x n ′ − ε , y n ′ ≦ u n u(x_(n)^(')-epsi,y_(n)^(')) <= u_(n)u\left(x_{n}^{\prime}-\varepsilon, y_{n}^{\prime}\right) \leqq u_{n}. Letting ε → 0 , u ( x n ′ , y n ′ ) ≦ u n ε → 0 , u x n ′ , y n ′ ≦ u n 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 x n , y n (x_(n),y_(n):}\left(x_{n}, y_{n}\right. ) maximizes u u uu subject to x ≦ c ′ ( n y ) x ≦ c ′ ( n y ) x <= c^(')(ny)x \leqq c^{\prime}(n y).
证明。定义 c ′ ( z ) = sup z ′ ≦ z c ( z ′ ) c ′ ( z ) = sup z ′ ≦ z   c z ′ 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 ′ ( n y n ′ ) x n ′ ≦ c ′ n y n ′ x_(n)^(') <= c^(')(ny_(n)^('))x_{n}^{\prime} \leqq c^{\prime}\left(n y_{n}^{\prime}\right) ，那么，对于任何 ε > 0 ε > 0 epsi > 0\varepsilon>0 ，存在 y n ′ ′ ≦ y n ′ y n ′ ′ ≦ y n ′ y_(n)^('') <= y_(n)^(')y_{n}^{\prime \prime} \leqq y_{n}^{\prime} 使得 x n ′ − ε ≦ c ( n y n ′ ′ ) x n ′ − ε ≦ c n y n ′ ′ x_(n)^(')-epsi <= c(ny_(n)^(''))x_{n}^{\prime}-\varepsilon \leqq c\left(n y_{n}^{\prime \prime}\right) 。因此 u ( x n ′ − ε , y n ′ ′ ) ≦ u n u x n ′ − ε , y n ′ ′ ≦ u n 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 u u uu 在 y y yy 中是一个递减函数， u ( x n ′ − ε , y n ′ ) ≦ u n u x n ′ − ε , y n ′ ≦ u n u(x_(n)^(')-epsi,y_(n)^(')) <= u_(n)u\left(x_{n}^{\prime}-\varepsilon, y_{n}^{\prime}\right) \leqq u_{n} 。令 ε → 0 , u ( x n ′ , y n ′ ) ≦ u n ε → 0 , u x n ′ , y n ′ ≦ u n 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 x n , y n (x_(n),y_(n):}\left(x_{n}, y_{n}\right. ）在约束条件 x ≦ c ′ ( n y ) x ≦ c ′ ( n y ) x <= c^(')(ny)x \leqq c^{\prime}(n y) 下最大化 u u uu 。
c ′ c ′ c^(')c^{\prime} is clearly a non-decreasing function of z z zz. To prove that it is right-continuous, take a decreasing sequence z i → z . c ′ ( z i ) z i → z . c ′ z i 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 ′ ( z ) c^(')(z)c^{\prime}(z). If it is equal to c ′ ( z ) c ′ ( z ) c^(')(z)c^{\prime}(z), there is no more to prove. Suppose it is greater. Then for some ε > 0 ε > 0 epsi > 0\varepsilon>0 each c ′ ( z i ) > c ′ ( z ) + ε c ′ z i > c ′ ( z ) + ε c^(')(z^(i)) > c^(')(z)+epsic^{\prime}\left(z^{i}\right)>c^{\prime}(z)+\varepsilon. Therefore, there exists a sequence ( z ¯ i ) z ¯ i {: bar(z)^(i))\left.\bar{z}^{i}\right) such that z ¯ i ≦ z i z ¯ i ≦ z i bar(z)^(i) <= z^(i)\bar{z}^{i} \leqq z^{i} and c ′ ( z i ) ≧ c ( z ¯ i ) > c ′ ( z ) + ε c ′ z i ≧ c z ¯ i > c ′ ( z ) + ε 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. The second inequality implies that z ¯ i > z z ¯ i > z bar(z)^(i) > z\bar{z}^{i}>z. Thus z ¯ i → z z ¯ i → z bar(z)^(i)rarr z\bar{z}^{i} \rightarrow z. Yet lim sup c ( z ¯ i ) > c ( z ) lim sup c z ¯ i > c ( z ) lim s u p c( bar(z)^(i)) > c(z)\lim \sup c\left(\bar{z}^{i}\right)>c(z), which contradicts upper semi-continuity. Thus in fact, c c cc is right-continuous. |
c ′ c ′ c^(')c^{\prime} 显然是 z z zz 的一个非递减函数。为了证明它是右连续的，取一个递减序列 z i → z . c ′ ( z i ) z i → z . c ′ z i z^(i)rarr z.quadc^(')(z^(i))z^{i} \rightarrow z . \quad c^{\prime}\left(z^{i}\right) 是一个非递增序列，因此趋向于一个不小于 c ′ ( z ) c ′ ( z ) c^(')(z)c^{\prime}(z) 的极限。如果它等于 c ′ ( z ) c ′ ( z ) c^(')(z)c^{\prime}(z) ，则无需再证明。假设它大于。则对于某些 ε > 0 ε > 0 epsi > 0\varepsilon>0 ，每个 c ′ ( z i ) > c ′ ( z ) + ε c ′ z i > c ′ ( z ) + ε c^(')(z^(i)) > c^(')(z)+epsic^{\prime}\left(z^{i}\right)>c^{\prime}(z)+\varepsilon 。因此，存在一个序列 ( z ¯ i ) z ¯ i {: bar(z)^(i))\left.\bar{z}^{i}\right) ，使得 z ¯ i ≦ z i z ¯ i ≦ z i bar(z)^(i) <= z^(i)\bar{z}^{i} \leqq z^{i} 和 c ′ ( z i ) ≧ c ( z ¯ i ) > c ′ ( z ) + ε c ′ z i ≧ c z ¯ i > c ′ ( z ) + ε 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 。第二个不等式意味着 z ¯ i > z z ¯ i > z bar(z)^(i) > z\bar{z}^{i}>z 。因此 z ¯ i → z z ¯ i → z bar(z)^(i)rarr z\bar{z}^{i} \rightarrow z 。然而 lim sup c ( z ¯ i ) > c ( z ) lim sup c z ¯ i > c ( z ) lim s u p c( bar(z)^(i)) > c(z)\lim \sup c\left(\bar{z}^{i}\right)>c(z) ，这与上半连续性相矛盾。因此实际上， c c cc 是右连续的。
This proposition says that the marginal tax rate may as well be not greater than 100 per cent. We shall consider later whether it should be positive.
该命题表明，边际税率也可以不超过 100%。我们稍后将考虑它是否应该为正。
The government chooses the function c c cc so as to maximize a welfare function
政府选择函数 c c cc 以最大化福利函数
W = ∫ 0 ∞ G ( u n ) f ( n ) d n W = ∫ 0 ∞   G u n f ( n ) d n W=int_(0)^(oo)G(u_(n))f(n)dnW=\int_{0}^{\infty} G\left(u_{n}\right) f(n) d n
I use the function G G GG here, rather than writing u n u n u_(n)u_{n} alone, because I shall later want to devote special attention to the case u x y = 0 u x y = 0 u_(xy)=0u_{x y}=0 (when u u uu can be written as the sum of a function depending only on x x xx and a function depending only on y y yy ). In maximizing welfare, the government is constrained by production possibilities: it must be possible to produce the consumption demands, X X XX, arising from its choice of c c cc, with labour input no greater than Z Z ZZ. The production constraint is written
我在这里使用函数 G G GG ，而不是单独写 u n u n u_(n)u_{n} ，因为我稍后将特别关注案例 u x y = 0 u x y = 0 u_(xy)=0u_{x y}=0 （当 u u uu 可以写成仅依赖于 x x xx 的函数和仅依赖于 y y yy 的函数的和时）。在最大化福利时，政府受到生产可能性的限制：必须能够以不超过 Z Z ZZ 的劳动投入来满足其选择的 c c cc 所产生的消费需求 X X XX 。生产约束写为
X ≦ H ( Z ) . X ≦ H ( Z ) .  X <= H(Z)". "X \leqq H(Z) \text {. }
We have not yet fully specified the possibilities available to the government, since, if ( x n , y n ) x n , y n (x_(n),y_(n))\left(x_{n}, y_{n}\right) is not uniquely defined, it is not clear whether the government or the consumer is allowed to choose the particular utility-maximizing point. Perhaps it is reasonable to suppose that the government can choose, and that the necessity for market-clearing will make its choices actual. But it will turn out that the issue is of no significance when we make the following assumption, as we shall:
我们尚未完全明确政府可用的可能性，因为如果 ( x n , y n ) x n , y n (x_(n),y_(n))\left(x_{n}, y_{n}\right) 没有唯一定义，政府或消费者是否被允许选择特定的效用最大化点就不清楚。也许可以合理地假设政府可以选择，并且市场清算的必要性将使其选择变为现实。但当我们做出以下假设时，这个问题将显得无关紧要。
(A) y n y n y_(n)y_{n} is uniquely defined for all n n nn except for a set of measure 0 .
(A) y n y n y_(n)y_{n} 对于所有 n n nn 唯一确定，除了一个测度为 0 的集合。
Thus the class of functions c c cc from which the government chooses is further restricted by the requirement that the function lead to choices satisfying (A). It will appear in due course that ( A ) ( A ) (A)(\mathrm{A}) is satisfied for all functions c c cc in the particular cases we shall be most concerned with.
因此，政府选择的函数类 c c cc 进一步受到限制，要求该函数导致满足(A)的选择。随后将会出现， ( A ) ( A ) (A)(\mathrm{A}) 在我们最关心的特定情况下对所有函数 c c cc 都是满足的。

3. NECESSARY CONDITIONS FOR THE OPTIMUM
3. 最优的必要条件

On the assumption that an optimum for our problem exists, we shall now obtain conditions that it must satisfy. The mathematical argument will not be rigorous. To do the analysis properly, one must attend to a number of rather tricky points. Since these technical details tend to obscure the main lines of the argument, rigorous proofs will be presented separately, in the continuation of this paper. The nature of these neglected difficulties will be discussed briefly in the next section.
在假设我们的研究问题存在最优解的前提下，我们将获得必须满足的条件。数学论证不会是严格的。要正确进行分析，必须关注一些相当棘手的细节。由于这些技术细节往往会掩盖论证的主要思路，因此严格的证明将在本文的后续部分单独呈现。下一节将简要讨论这些被忽视的困难的性质。
The key to a reasonably neat solution of the problem is to find a convenient expression of the condition that each man maximizes his utility subject to the imposed " consumption function " c c cc. If we suppose that c c cc is differentiable, the derivative of u [ c ( n y ) , y ] u [ c ( n y ) , y ] u[c(ny),y]u[c(n y), y] with respect to y y yy must be zero. Denoting the derivative of u u uu with respect to its first and second arguments by u 1 u 1 u_(1)u_{1} and u 2 u 2 u_(2)u_{2}, respectively, we have
解决该问题的合理整洁方案的关键在于找到一个方便的表达式，表明每个人在施加的“消费函数” c c cc 的约束下最大化其效用。如果我们假设 c c cc 是可微的，则 u [ c ( n y ) , y ] u [ c ( n y ) , y ] u[c(ny),y]u[c(n y), y] 对 y y yy 的导数必须为零。将 u u uu 对其第一个和第二个参数的导数分别表示为 u 1 u 1 u_(1)u_{1} 和 u 2 u 2 u_(2)u_{2} ，我们有
u 1 n c ′ ( n y ) + u 2 = 0 u 1 n c ′ ( n y ) + u 2 = 0 u_(1)nc^(')(ny)+u_(2)=0u_{1} n c^{\prime}(n y)+u_{2}=0
Recollect that u n u n u_(n)u_{n} is the utility of n n nn-man. Then a straightforward calculation, using the first-order condition (9), yields
回忆一下 u n u n u_(n)u_{n} 是 n n nn -人所带来的效用。然后，使用一阶条件（9）进行简单计算，得出：
d u n d n = u 1 y c ′ = − y u 2 n d u n d n = u 1 y c ′ = − y u 2 n (du_(n))/(dn)=u_(1)yc^(')=-(yu_(2))/(n)\frac{d u_{n}}{d n}=u_{1} y c^{\prime}=-\frac{y u_{2}}{n}
(The expressions on the right are, of course, alternative expressions for the partial derivative of u u uu with respect to n n nn, evaluated at the maximum. The case where n n nn enters u u uu in a more general manner can be analyzed by using this more general equation. We shall return to this point later.)
（右侧的表达式当然是对 u u uu 关于 n n nn 的偏导数在最大值处的替代表达式。 n n nn 以更一般的方式进入 u u uu 的情况可以通过使用这个更一般的方程进行分析。我们稍后将回到这一点。）
Our problem is to maximize w w ww subject to the constraint of the production function, X ≦ H ( Z ) X ≦ H ( Z ) X <= H(Z)X \leqq H(Z), the differential equation (10), and the definition u n = u ( x n , y n ) u n = u x n , y n u_(n)=u(x_(n),y_(n))u_{n}=u\left(x_{n}, y_{n}\right). Those who are familiar with the Pontriyagin Maximum Principle will see that this is a form of problem fairly suitable for treatment by it. Shadow prices p p pp and w w ww have to be introduced for X X XX and Z Z ZZ. Then we would like to maximize
我们的问题是最大化 w w ww ，同时满足生产函数的约束 X ≦ H ( Z ) X ≦ H ( Z ) X <= H(Z)X \leqq H(Z) 、微分方程 (10) 和定义 u n = u ( x n , y n ) u n = u x n , y n u_(n)=u(x_(n),y_(n))u_{n}=u\left(x_{n}, y_{n}\right) 。熟悉庞特里亚金极大原理的人会发现，这是一种相当适合用该原理处理的问题。需要为 X X XX 和 Z Z ZZ 引入影子价格 p p pp 和 w w ww 。然后我们希望最大化
W − p X + w Z = ∫ [ G ( u n ) − p x n + w y n n ] f ( n ) d n W − p X + w Z = ∫ G u n − p x n + w y n n f ( n ) d n W-pX+wZ=int[G(u_(n))-px_(n)+wy_(n)n]f(n)dnW-p X+w Z=\int\left[G\left(u_{n}\right)-p x_{n}+w y_{n} n\right] f(n) d n
subject to (10). u n u n quadu_(n)\quad u_{n} is to be regarded as the state variable, y n y n y_(n)y_{n} (say) as the control variable, while x n x n x_(n)x_{n} is determined as a function of u n u n u_(n)u_{n} and y n y n y_(n)y_{n} from the equation u n = u ( x n , y n ) u n = u x n , y n u_(n)=u(x_(n),y_(n))u_{n}=u\left(x_{n}, y_{n}\right). The Hamiltonian is
受限于 (10)。 u n u n quadu_(n)\quad u_{n} 被视为状态变量， y n y n y_(n)y_{n} （例如）作为控制变量，而 x n x n x_(n)x_{n} 则根据方程 u n = u ( x n , y n ) u n = u x n , y n u_(n)=u(x_(n),y_(n))u_{n}=u\left(x_{n}, y_{n}\right) 中的 u n u n u_(n)u_{n} 和 y n y n y_(n)y_{n} 的函数来确定。哈密顿量是
M = G [ ( u n ) − p x n + w y n n ] f ( n ) − ϕ n y n u 2 n , M = G u n − p x n + w y n n f ( n ) − ϕ n y n u 2 n , M=G[(u_(n))-px_(n)+wy_(n)n]f(n)-phi_(n)(y_(n)u_(2))/(n),M=G\left[\left(u_{n}\right)-p x_{n}+w y_{n} n\right] f(n)-\phi_{n} \frac{y_{n} u_{2}}{n},
where ϕ n ϕ n phi_(n)\phi_{n} is a function of n n nn satisfying the differential equation
其中 ϕ n ϕ n phi_(n)\phi_{n} 是满足微分方程的 n n nn 的函数
d ϕ d n = − ∂ M ∂ u = − [ G ′ ( u n ) − p u 1 ] f ( n ) + ϕ y n u 12 n u 1 d ϕ d n = − ∂ M ∂ u = − G ′ u n − p u 1 f ( n ) + ϕ y n u 12 n u 1 {:[(d phi)/(dn)=-(del M)/(del u)],[=-[G^(')(u_(n))-(p)/(u_(1))]f(n)+phi(y_(n)u_(12))/(nu_(1))]:}\begin{aligned} \frac{d \phi}{d n} & =-\frac{\partial M}{\partial u} \\ & =-\left[G^{\prime}\left(u_{n}\right)-\frac{p}{u_{1}}\right] f(n)+\phi \frac{y_{n} u_{12}}{n u_{1}} \end{aligned}
y n y n y_(n)y_{n} should then be chosen so as to maximize M M MM :
y n y n y_(n)y_{n} 应该被选择以最大化 M M MM ：
[ w n + p u 2 u 1 ] f ( n ) + ϕ n ψ y n = 0 w n + p u 2 u 1 f ( n ) + ϕ n ψ y n = 0 [wn+(pu_(2))/(u_(1))]f(n)+phi_(n)(psi_(y))/(n)=0\left[w n+\frac{p u_{2}}{u_{1}}\right] f(n)+\phi_{n} \frac{\psi_{y}}{n}=0
where the function ψ ( u , y ) ψ ( u , y ) psi(u,y)\psi(u, y) is defined by
函数 ψ ( u , y ) ψ ( u , y ) psi(u,y)\psi(u, y) 的定义为
ψ ( u , y ) = − y u 2 ( x , y ) , u = u ( x , y ) ψ ( u , y ) = − y u 2 ( x , y ) , u = u ( x , y ) psi(u,y)=-yu_(2)(x,y),quad u=u(x,y)\psi(u, y)=-y u_{2}(x, y), \quad u=u(x, y)
and ψ y ψ y psi_(y)\psi_{y} is its partial derivative with respect to y y yy. (Notice, at the same time, that
和 ψ y ψ y psi_(y)\psi_{y} 是关于 y y yy 的偏导数。（同时注意，
ψ u = − y u 12 / u 1 . ) ψ u = − y u 12 / u 1 . {:psi_(u)=-yu_(12)//u_(1).)\left.\psi_{u}=-y u_{12} / u_{1} .\right)
Equation (12) can now be integrated to obtain an expression for ϕ n ϕ n phi_(n)\phi_{n}; which, when substituted in (13), provides us with an equation to be satisfied by the optimum we seek. Before going on to use this equation, however, we shall derive it in a different way, by a more explicit use of the methods of the calculus of variations. The use of the Maximum Principle has a number of serious disadvantages. It does not show us how to obtain certain important supplementary conditions on the optimum. The analysis provides no hint as to how it could be made rigorous. It does not provide any insight into the kind of maximization that is going on. When we have done a more explicit variational analysis, we shall be better able to see where the logical holes are, and to understand why things come out the way they do.
方程（12）现在可以被积分以获得 ϕ n ϕ n phi_(n)\phi_{n} 的表达式；当将其代入（13）时，给我们提供了一个需要满足的方程，以达到我们所寻求的最优解。然而，在使用这个方程之前，我们将通过更明确地运用变分法的方法以不同的方式推导它。使用极大值原理有许多严重的缺点。它没有告诉我们如何获得关于最优解的一些重要补充条件。分析没有提供任何关于如何使其严谨的提示。它也没有提供对正在进行的最大化过程的任何洞察。当我们进行更明确的变分分析时，我们将更好地看到逻辑上的漏洞，并理解事物为何会以这种方式出现。
For this purpose, I prefer to write (10) in integrated form:
为此，我更喜欢将（10）以综合形式写出：
u n = − ∫ 0 n y m u 2 ( x m , y m ) d m m + u ( c ( 0 ) , 0 ) = ∫ 0 n ψ ( u m , y m ) d m m + u 0 u n = − ∫ 0 n   y m u 2 x m , y m d m m + u ( c ( 0 ) , 0 ) = ∫ 0 n   ψ u m , y m d m m + u 0 {:[u_(n)=-int_(0)^(n)y_(m)u_(2)(x_(m),y_(m))(dm)/(m)+u(c(0)","0)],[=int_(0)^(n)psi(u_(m),y_(m))(dm)/(m)+u_(0)]:}\begin{aligned} u_{n} & =-\int_{0}^{n} y_{m} u_{2}\left(x_{m}, y_{m}\right) \frac{d m}{m}+u(c(0), 0) \\ & =\int_{0}^{n} \psi\left(u_{m}, y_{m}\right) \frac{d m}{m}+u_{0} \end{aligned}
using the notation ψ ψ psi\psi introduced above, and denoting the utility allowed to a man who does no work by u 0 u 0 u_(0)u_{0}. Suppose first that ψ ψ psi\psi is independent of u u uu (corresponding to the sepcial case u 12 = 0 u 12 = 0 u_(12)=0u_{12}=0 ). If we consider a variation from the optimum which changes the functions u n u n u_(n)u_{n} and y n y n y_(n)y_{n} by “small” variations δ u n δ u n deltau_(n)\delta u_{n} and δ y n δ y n deltay_(n)\delta y_{n}, we deduce from (15) that these variations must be related by
使用上述引入的符号 ψ ψ psi\psi ，并用 u 0 u 0 u_(0)u_{0} 表示一个不工作的人的效用。首先假设 ψ ψ psi\psi 与 u u uu 是独立的（对应于特殊情况 u 12 = 0 u 12 = 0 u_(12)=0u_{12}=0 ）。如果我们考虑从最优解的变化，这种变化通过“微小”的变化 δ u n δ u n deltau_(n)\delta u_{n} 和 δ y n δ y n deltay_(n)\delta y_{n} 改变函数 u n u n u_(n)u_{n} 和 y n y n y_(n)y_{n} ，我们从（15）中推导出这些变化必须相关。
δ u n = ∫ 0 n ψ ν δ y m d m m + δ u 0 δ u n = ∫ 0 n   ψ ν δ y m d m m + δ u 0 deltau_(n)=int_(0)^(n)psi_(nu)deltay_(m)(dm)/(m)+deltau_(0)\delta u_{n}=\int_{0}^{n} \psi_{\nu} \delta y_{m} \frac{d m}{m}+\delta u_{0}
This variation will bring about changes in W , X W , X W,XW, X, and Z Z ZZ. As before, introduce shadow prices (in terms of welfare) for X X XX and Z Z ZZ. Then the variation must leave (11) stationary:
这种变化将导致 W , X W , X W,XW, X 和 Z Z ZZ 的变化。与之前一样，为 X X XX 和 Z Z ZZ 引入影子价格（以福利为单位）。然后，该变化必须使（11）保持不变：
0 = δ ∫ [ G ( u n ) − p x n + w y n n ] f ( n ) d n = ∫ [ G ′ ( u n ) δ u n − p ( 1 u 1 δ u n − u 2 u 1 δ y n ) + w δ y n n ] f ( n ) d n , 0 = δ ∫ G u n − p x n + w y n n f ( n ) d n = ∫ G ′ u n δ u n − p 1 u 1 δ u n − u 2 u 1 δ y n + w δ y n n f ( n ) d n , {:[0=delta int[G(u_(n))-px_(n)+wy_(n)n]f(n)dn],[=int[G^(')(u_(n))deltau_(n)-p((1)/(u_(1))deltau_(n)-(u_(2))/(u_(1))deltay_(n))+w deltay_(n)n]f(n)dn","]:}\begin{aligned} 0 & =\delta \int\left[G\left(u_{n}\right)-p x_{n}+w y_{n} n\right] f(n) d n \\ & =\int\left[G^{\prime}\left(u_{n}\right) \delta u_{n}-p\left(\frac{1}{u_{1}} \delta u_{n}-\frac{u_{2}}{u_{1}} \delta y_{n}\right)+w \delta y_{n} n\right] f(n) d n, \end{aligned}
where the variation in x x xx is calculated as follows:
在这里， x x xx 的变化计算如下：
δ u n = δ u ( x n , y n ) = u 1 δ x n + u 2 δ y n δ u n = δ u x n , y n = u 1 δ x n + u 2 δ y n deltau_(n)=delta u(x_(n),y_(n))=u_(1)deltax_(n)+u_(2)deltay_(n)\delta u_{n}=\delta u\left(x_{n}, y_{n}\right)=u_{1} \delta x_{n}+u_{2} \delta y_{n}
It remains to substitute (16) in (17), yielding,
仍需将（16）代入（17），得到：
0 = ∫ 0 ∞ { [ G ′ ( u n ) − p u 1 ] [ ∫ 0 n ψ y δ y m d m m + δ u 0 ] + [ w n + p u 2 u 1 ] δ y n } f ( n ) d n = ∫ 0 ∞ { ∫ n ∞ [ G ′ ( u m ) − p u 1 ] f ( m ) d m ⋅ ψ y n + ( w n + p u 2 u 1 ) f ( n ) } δ y n d n + ∫ 0 ∞ [ G ′ ( u n ) − p u 1 ] f ( n ) d n . δ u 0 . 0 = ∫ 0 ∞   G ′ u n − p u 1 ∫ 0 n   ψ y δ y m d m m + δ u 0 + w n + p u 2 u 1 δ y n f ( n ) d n = ∫ 0 ∞   ∫ n ∞   G ′ u m − p u 1 f ( m ) d m ⋅ ψ y n + w n + p u 2 u 1 f ( n ) δ y n d n + ∫ 0 ∞   G ′ u n − p u 1 f ( n ) d n . δ u 0 . {:[0=int_(0)^(oo){[G^(')(u_(n))-(p)/(u_(1))][int_(0)^(n)psi_(y)deltay_(m)(dm)/(m)+deltau_(0)]+[wn+p(u_(2))/(u_(1))]deltay_(n)}f(n)dn],[=int_(0)^(oo){int_(n)^(oo)[G^(')(u_(m))-(p)/(u_(1))]f(m)dm*(psi_(y))/(n)+(wn+p(u_(2))/(u_(1)))f(n)}deltay_(n)dn],[+int_(0)^(oo)[G^(')(u_(n))-(p)/(u_(1))]f(n)dn.deltau_(0).]:}\begin{aligned} & 0=\int_{0}^{\infty}\left\{\left[G^{\prime}\left(u_{n}\right)-\frac{p}{u_{1}}\right]\left[\int_{0}^{n} \psi_{y} \delta y_{m} \frac{d m}{m}+\delta u_{0}\right]+\left[w n+p \frac{u_{2}}{u_{1}}\right] \delta y_{n}\right\} f(n) d n \\ & =\int_{0}^{\infty}\left\{\int_{n}^{\infty}\left[G^{\prime}\left(u_{m}\right)-\frac{p}{u_{1}}\right] f(m) d m \cdot \frac{\psi_{y}}{n}+\left(w n+p \frac{u_{2}}{u_{1}}\right) f(n)\right\} \delta y_{n} d n \\ & +\int_{0}^{\infty}\left[G^{\prime}\left(u_{n}\right)-\frac{p}{u_{1}}\right] f(n) d n . \delta u_{0} . \end{aligned}
The second equation is obtained by inverting the order of integration in the double integral. 1 1 ^(1){ }^{1} (19) is to be satisfied for all possible variations of the function y n y n y_(n)y_{n}, and the number u 0 u 0 u_(0)u_{0}. Since u 0 u 0 u_(0)u_{0} can be either increased or decreased at the optimum (if, as is to be expected in general, some people will do no work at the optimum),
第二个方程是通过反转双重积分中的积分顺序得到的。 1 1 ^(1){ }^{1} (19) 必须对函数 y n y n y_(n)y_{n} 和数字 u 0 u 0 u_(0)u_{0} 的所有可能变化成立。由于 u 0 u 0 u_(0)u_{0} 在最优情况下可以增加或减少（如果，正如一般所预期的那样，有些人在最优情况下将不进行工作），
∫ 0 ∞ [ G ′ ( u n ) − p u 1 ] f ( n ) d n = 0 ∫ 0 ∞   G ′ u n − p u 1 f ( n ) d n = 0 int_(0)^(oo)[G^(')(u_(n))-(p)/(u_(1))]f(n)dn=0\int_{0}^{\infty}\left[G^{\prime}\left(u_{n}\right)-\frac{p}{u_{1}}\right] f(n) d n=0
at the optimum. 在最佳状态下。
1 1 ^(1){ }^{1} The double integral is
双重积分是
∫ 0 ∞ [ G ′ ( u n ) − p u 1 ] f ( n ) ∫ 0 n ψ ν δ y m d m m ⋅ d n . ∫ 0 ∞   G ′ u n − p u 1 f ( n ) ∫ 0 n   ψ ν δ y m d m m ⋅ d n . int_(0)^(oo)[G^(')(u_(n))-(p)/(u_(1))]f(n)int_(0)^(n)psi_(nu)deltay_(m)(dm)/(m)*dn.\int_{0}^{\infty}\left[G^{\prime}\left(u_{n}\right)-\frac{p}{u_{1}}\right] f(n) \int_{0}^{n} \psi_{\nu} \delta y_{m} \frac{d m}{m} \cdot d n .
The region over which the integration takes place is defined by 0 ≦ m ≦ n 0 ≦ m ≦ n 0 <= m <= n0 \leqq m \leqq n. Thus, when the order of integration is inverted, n n nn ranges between m m mm and ∞ ∞ oo\infty for given m m mm. The integral can therefore be written
积分所进行的区域由 0 ≦ m ≦ n 0 ≦ m ≦ n 0 <= m <= n0 \leqq m \leqq n 定义。因此，当积分的顺序被颠倒时， n n nn 在给定的 m m mm 下范围在 m m mm 和 ∞ ∞ oo\infty 之间。因此，积分可以写为
∫ 0 ∞ ∫ m ∞ [ G ′ − p u 1 ] f ( n ) d n ⋅ ψ v δ y m ⋅ d m m , ∫ 0 ∞   ∫ m ∞   G ′ − p u 1 f ( n ) d n ⋅ ψ v δ y m ⋅ d m m , int_(0)^(oo)int_(m)^(oo)[G^(')-(p)/(u_(1))]f(n)dn*psi_(v)deltay_(m)*(dm)/(m),\int_{0}^{\infty} \int_{m}^{\infty}\left[G^{\prime}-\frac{p}{u_{1}}\right] f(n) d n \cdot \psi_{v} \delta y_{m} \cdot \frac{d m}{m},
which is seen to justify (19) on permuting the symbols m m mm and n n nn.
这被视为在对符号 m m mm 和 n n nn 进行置换时证明(19)的合理性。
If all variations in y n y n y_(n)y_{n} were possible-and this is a question we shall take up shortlywe could also claim that the expression within curly brackets ought to be zero:
如果 y n y n y_(n)y_{n} 中的所有变体都是可能的——这是我们稍后将讨论的问题——我们也可以声称大括号内的表达式应该为零：
( w n + p u 2 u 1 ) f ( n ) = ψ y n ∫ n ∞ [ p u 1 − G ′ ( u m ) ] f ( m ) d m w n + p u 2 u 1 f ( n ) = ψ y n ∫ n ∞   p u 1 − G ′ u m f ( m ) d m (wn+p(u_(2))/(u_(1)))f(n)=(psi_(y))/(n)int_(n)^(oo)[(p)/(u_(1))-G^(')(u_(m))]f(m)dm\left(w n+p \frac{u_{2}}{u_{1}}\right) f(n)=\frac{\psi_{y}}{n} \int_{n}^{\infty}\left[\frac{p}{u_{1}}-G^{\prime}\left(u_{m}\right)\right] f(m) d m
It should be noticed that this equation will only be valid for n ≧ n 0 n ≧ n 0 n >= n_(0)n \geqq n_{0} : it does not apply to n n nn for which y n = 0 y n = 0 y_(n)=0y_{n}=0 (except n 0 n 0 n_(0)n_{0} ) because, there, not all variations of the function y n y n y_(n)y_{n} are possible, since y n y n y_(n)y_{n} cannot be negative.
需要注意的是，这个方程仅在 n ≧ n 0 n ≧ n 0 n >= n_(0)n \geqq n_{0} 有效：它不适用于 n n nn ，因为 y n = 0 y n = 0 y_(n)=0y_{n}=0 （除了 n 0 n 0 n_(0)n_{0} ），因为在那里，函数 y n y n y_(n)y_{n} 的所有变体都不可能，因为 y n y n y_(n)y_{n} 不能为负。
Finally we know that the marginal product of labour should be equal to the shadow wage:
最后我们知道，劳动的边际产出应该等于影子工资：
p H ′ ( Z ) = w . p H ′ ( Z ) = w . pH^(')(Z)=w.p H^{\prime}(Z)=w .
These equations, (20) and (21), have been worked out under the special assumption that ψ ψ psi\psi is independent of u u uu. In the more general case, we have to replace (16) by
这些方程（20）和（21）是在假设 ψ ψ psi\psi 与 u u uu 独立的特殊情况下推导出来的。在更一般的情况下，我们必须用(16)替换为
δ u n = ∫ 0 n T m n ψ y δ y m d m m + δ u 0 δ u n = ∫ 0 n   T m n ψ y δ y m d m m + δ u 0 deltau_(n)=int_(0)^(n)T_(mn)psi_(y)deltay_(m)(dm)/(m)+deltau_(0)\delta u_{n}=\int_{0}^{n} T_{m n} \psi_{y} \delta y_{m} \frac{d m}{m}+\delta u_{0}
where 哪里
T m n = exp ⁡ ∫ m n ψ u d m ′ m ′ T m n = exp ⁡ ∫ m n   ψ u d m ′ m ′ T_(mn)=exp int_(m)^(n)psi_(u)(dm^('))/(m^('))T_{m n}=\exp \int_{m}^{n} \psi_{u} \frac{d m^{\prime}}{m^{\prime}}
To show this, we can go back to the differential equation (10). Applying the variation, we obtain from it,
为了证明这一点，我们可以回到微分方程（10）。通过应用变分，我们从中得到，
d d n δ u n = 1 n ψ u δ u n + 1 n ψ y δ y n . d d n δ u n = 1 n ψ u δ u n + 1 n ψ y δ y n . (d)/(dn)deltau_(n)=(1)/(n)psi_(u)deltau_(n)+(1)/(n)psi_(y)deltay_(n).\frac{d}{d n} \delta u_{n}=\frac{1}{n} \psi_{u} \delta u_{n}+\frac{1}{n} \psi_{y} \delta y_{n} .
This is a first order linear equation, and can therefore be solved by the standard method to give the solution (23).
这是一个一阶线性方程，因此可以通过标准方法求解，得到解（23）。
Having replaced (16) by (23), we can now go through the rest of the calculation as before. We find that (20) is generalized into
将（16）替换为（23）后，我们可以像之前一样继续进行其余的计算。我们发现（20）被推广为
∫ 0 ∞ [ G ′ ( u n ) − p / u 1 ] T 0 n f ( n ) d n = 0 ∫ 0 ∞   G ′ u n − p / u 1 T 0 n f ( n ) d n = 0 int_(0)^(oo)[G^(')(u_(n))-p//u_(1)]T_(0n)f(n)dn=0\int_{0}^{\infty}\left[G^{\prime}\left(u_{n}\right)-p / u_{1}\right] T_{0 n} f(n) d n=0
while (21) becomes 当 (21) 变为
( w n + p u 2 / u 1 ) f ( n ) = ψ y n ∫ n ∞ [ p / u 1 − G ′ ( u m ) ] T n m f ( m ) d m w n + p u 2 / u 1 f ( n ) = ψ y n ∫ n ∞   p / u 1 − G ′ u m T n m f ( m ) d m (wn+pu_(2)//u_(1))f(n)=(psi_(y))/(n)int_(n)^(oo)[p//u_(1)-G^(')(u_(m))]T_(nm)f(m)dm\left(w n+p u_{2} / u_{1}\right) f(n)=\frac{\psi_{y}}{n} \int_{n}^{\infty}\left[p / u_{1}-G^{\prime}\left(u_{m}\right)\right] T_{n m} f(m) d m
Notice that we have T n m T n m T_(nm)T_{n m} here, although it was T m n T m n T_(mn)T_{m n} that appeared in (23).
注意到这里有 T n m T n m T_(nm)T_{n m} ，尽管在(23)中出现的是 T m n T m n T_(mn)T_{m n} 。
If these equations are correct, the two integral equations, (15) and (27) may be thought of as determining the two functions u n u n u_(n)u_{n} and y n y n y_(n)y_{n}, given the three parameters u 0 , w u 0 , w u_(0),wu_{0}, w, and p p pp. The values of these parameters are fixed by the three equations (26), (22), and (8). We have enough relations to determine the optimum tax schedule, since the function c c cc can be determined once we know u n u n u_(n)u_{n} and y n y n y_(n)y_{n}.
如果这些方程是正确的，那么两个积分方程（15）和（27）可以被视为在给定三个参数 u 0 , w u 0 , w u_(0),wu_{0}, w 和 p p pp 的情况下确定两个函数 u n u n u_(n)u_{n} 和 y n y n y_(n)y_{n} 。这些参数的值由三个方程（26）、（22）和（8）固定。我们有足够的关系来确定最佳税收计划，因为一旦我们知道 u n u n u_(n)u_{n} 和 y n y n y_(n)y_{n} ，就可以确定函数 c c cc 。

4. NECESSARY CONDITIONS: A COMPLETE STATEMENT
4. 必要条件：完整陈述

The argument used to derive these conditions for the optimum tax schedule had a number of weak points. It is indeed unlikely that the relationships derived above hold in general. Among the weak points of the argument, notice that
用于推导最佳税收计划条件的论点存在一些弱点。确实，上述推导的关系在一般情况下不太可能成立。在论点的弱点中，请注意：
(i) the existence of the shadow prices p p pp and w w ww was assumed without proof;
（i）假设存在影子价格 p p pp 和 w w ww ，但未提供证明；
(ii) the optimum tax schedule, and the resulting functions x n , y n x n , y n x_(n),y_(n)x_{n}, y_{n}, and u n u n u_(n)u_{n} were assumed to be differentiable;
（ii）最优税率表，以及由此产生的函数 x n , y n x n , y n x_(n),y_(n)x_{n}, y_{n} 和 u n u n u_(n)u_{n} 被假定为可微分的；
(iii) the application of the variation was quite heuristic; and
(iii) 该变异的应用相当启发性；并且
(iv) no justification was provided for assuming that the function y n y n y_(n)y_{n} could be varied arbitrarily (for n > n 0 n > n 0 n > n_(0)n>n_{0} ).
（iv）没有提供任何理由来假设函数 y n y n y_(n)y_{n} 可以任意变化（对于 n > n 0 n > n 0 n > n_(0)n>n_{0} ）。
I shall not comment on (i) and (iii), which, though important, are technical matters: they can be justified. (ii) is not satisfied in general: there was no reason to suppose that it would be. When (ii) is not satisfied, the first-order condition, (9), for maximization of utility ceases to be meaningful. Finally, (iv) is never justified. The function y n y n y_(n)y_{n} is derived from the imposition of the consumption function c c cc, and we have no a priori information about it. We must expect that some conceivable functions y n y n y_(n)y_{n} can never arise from the imposition of a consumption function. The class of possible y y yy-functions is no doubt quite complicated in certain cases. Fortunately it is possible to specify that class quite simply in the realistic cases, and it is then possible to use the variational argument rigorously.
我将不对（i）和（iii）进行评论，尽管它们很重要，但属于技术性问题：它们是可以被证明的。（ii）一般情况下不满足：没有理由假设它会满足。当（ii）不满足时，效用最大化的一阶条件（9）就失去了意义。最后，（iv）从未被证明。函数 y n y n y_(n)y_{n} 是由消费函数 c c cc 的强加得出的，我们对其没有先验信息。我们必须预期某些可想象的函数 y n y n y_(n)y_{n} 永远无法从消费函数的强加中产生。可能的 y y yy -函数的类别在某些情况下无疑是相当复杂的。幸运的是，在现实情况下，可以相对简单地指定该类别，然后可以严格地使用变分论证。
Problem (ii) is dealt with in the rigorous analysis by depending on equation (15) instead of the differential first-order condition (9). It is a remarkable fact that this condition holds if and only if the various functions arise from utility-maximization under an imposed consumption function, even when that function is not differentiable. For proof, the reader is referred to [4].
问题（ii）通过依赖方程（15）而不是微分一阶条件（9）进行严格分析。一个显著的事实是，当且仅当各种函数源于在施加的消费函数下的效用最大化时，该条件成立，即使该函数不可微分。有关证明，请参阅[4]。
To deal with problem (iv), we have to restrict the class of utility functions considered. We assume that
为了处理问题（iv），我们必须限制考虑的效用函数类别。我们假设
(B) V ( x , y ) = − y u 2 / u 1 V ( x , y ) = − y u 2 / u 1 V(x,y)=-yu_(2)//u_(1)V(x, y)=-y u_{2} / u_{1} is an increasing function of y y yy for each x > 0 x > 0 x > 0x>0 (and bounded in 0 ≦ x ≦ x ¯ , 0 ≦ y ≦ y ¯ 0 ≦ x ≦ x ¯ , 0 ≦ y ≦ y ¯ 0 <= x <= bar(x),0 <= y <= bar(y)0 \leqq x \leqq \bar{x}, 0 \leqq y \leqq \bar{y} for any x ¯ < ∞ x ¯ < ∞ bar(x) < oo\bar{x}<\infty and y ¯ < 1 y ¯ < 1 bar(y) < 1\bar{y}<1 ).
(B) V ( x , y ) = − y u 2 / u 1 V ( x , y ) = − y u 2 / u 1 V(x,y)=-yu_(2)//u_(1)V(x, y)=-y u_{2} / u_{1} 是 y y yy 的一个递增函数，对于每个 x > 0 x > 0 x > 0x>0 （并且在任何 x ¯ < ∞ x ¯ < ∞ bar(x) < oo\bar{x}<\infty 和 y ¯ < 1 y ¯ < 1 bar(y) < 1\bar{y}<1 的 0 ≦ x ≦ x ¯ , 0 ≦ y ≦ y ¯ 0 ≦ x ≦ x ¯ , 0 ≦ y ≦ y ¯ 0 <= x <= bar(x),0 <= y <= bar(y)0 \leqq x \leqq \bar{x}, 0 \leqq y \leqq \bar{y} 中是有界的）。
It will be noticed that this is an assumption about preferences, not just about the form of the utility function used to represent preferences. The second part of the assumption is readily acceptable. The first, and main part of the assumption holds if and only if, for a given level of consumption x x xx, a one per cent increase in the amount of work done requires a larger increase in consumption to maintain the same utility level, the greater is the amount of work being done. It is equivalent to assuming that (in the absence of taxation) the consumer’s demand for goods is an increasing function of the real wage rate (at any given non-wage income. 1 1 ^(1){ }^{1} Few individuals appear to have preferences violating (B), and intuitively it is rather plausible. We shall later use the fact that (B) holds if preferences can be represented by an additive utility function. (It will be noticed that, as y → 1 , V → + ∞ y → 1 , V → + ∞ y rarr1,V rarr+ooy \rightarrow 1, V \rightarrow+\infty, so that the assumption must hold for some ranges of y y yy.) If the assumption does not hold, the theory of optimum taxation is more complicated.
需要注意的是，这是假设关于偏好的，而不仅仅是关于用于表示偏好的效用函数的形式。假设的第二部分是可以接受的。假设的第一部分和主要部分成立的条件是，对于给定的消费水平 x x xx ，工作量增加 1%所需的消费增加量必须更大，以维持相同的效用水平，且工作量越大，这种需求越明显。这等同于假设（在没有税收的情况下）消费者对商品的需求是实际工资率的一个递增函数（在任何给定的非工资收入 1 1 ^(1){ }^{1} 下）。似乎很少有人具有违反（B）的偏好，直观上这也是相当合理的。我们稍后将利用（B）成立的事实，即如果偏好可以用加性效用函数表示，则（B）成立。（需要注意的是，随着 y → 1 , V → + ∞ y → 1 , V → + ∞ y rarr1,V rarr+ooy \rightarrow 1, V \rightarrow+\infty ，假设必须在某些范围内成立 y y yy 。）如果假设不成立，最优税收理论将更加复杂。
The point of the assumption is indicated in
假设的要点在于
Theorem 1. Under Assumption ( B B BB ), z n = n y n z n = n y n z_(n)=ny_(n)z_{n}=n y_{n} maximizes utility for every n n nn under some consumption function c c cc if and only if
定理 1。在假设 ( B B BB ) 下， z n = n y n z n = n y n z_(n)=ny_(n)z_{n}=n y_{n} 在某个消费函数 c c cc 下对每个 n n nn 最大化效用当且仅当
(i) z n z n z_(n)z_{n} is a non-decreasing function defined for n > 0 n > 0 n > 0n>0;
(i) z n z n z_(n)z_{n} 是一个在 n > 0 n > 0 n > 0n>0 上定义的非递减函数；
(ii) 0 ≦ z n < n 0 ≦ z n < n 0 <= z_(n) < n0 \leqq z_{n}<n for all n > 0 n > 0 n > 0n>0.
（ii） 0 ≦ z n < n 0 ≦ z n < n 0 <= z_(n) < n0 \leqq z_{n}<n 对于所有 n > 0 n > 0 n > 0n>0 。
1 1 ^(1){ }^{1} This equivalence is fairly obvious from an indifference curve diagram. For a formal proof that (B) implies that consumption is an increasing function of the wage rate, let w w ww be the wage rate, and m m mm non-labour income (both measured in terms of goods). (B) states that wy, regarded as a function of x x xx and y y yy, is an increasing function of y y yy. Write x x xx and y y yy as functions of w w ww and m m mm, putting x = x ( w , m ) , y = y ( w , m ) x = x ( w , m ) , y = y ( w , m ) x=x(w,m),y=y(w,m)x=x(w, m), y=y(w, m) and x ′ = x ( w ′ , m ) , y ′ = y ( w ′ , m ) x ′ = x w ′ , m , y ′ = y w ′ , m x^(')=x(w^('),m),y^(')=y(w^('),m)x^{\prime}=x\left(w^{\prime}, m\right), y^{\prime}=y\left(w^{\prime}, m\right) where w ′ > w w ′ > w w^(') > ww^{\prime}>w. I shall show that x ′ > x x ′ > x x^(') > xx^{\prime}>x. To do this, choose w ′ ′ w ′ ′ w^('')w^{\prime \prime} and m ′ ′ m ′ ′ m^('')m^{\prime \prime} such that x ′ ′ = x ( w ′ ′ , m ′ ′ ) = x x ′ ′ = x w ′ ′ , m ′ ′ = x x^('')=x(w^(''),m^(''))=xx^{\prime \prime}=x\left(w^{\prime \prime}, m^{\prime \prime}\right)=x, and
这个等式从无差异曲线图中是相当明显的。为了正式证明（B）意味着消费是工资率的一个递增函数，设 w w ww 为工资率， m m mm 为非劳动收入（均以商品计）。（B）指出 wy 被视为 x x xx 和 y y yy 的函数，是 y y yy 的一个递增函数。将 x x xx 和 y y yy 写成 w w ww 和 m m mm 的函数，放置 x = x ( w , m ) , y = y ( w , m ) x = x ( w , m ) , y = y ( w , m ) x=x(w,m),y=y(w,m)x=x(w, m), y=y(w, m) 和 x ′ = x ( w ′ , m ) , y ′ = y ( w ′ , m ) x ′ = x w ′ , m , y ′ = y w ′ , m x^(')=x(w^('),m),y^(')=y(w^('),m)x^{\prime}=x\left(w^{\prime}, m\right), y^{\prime}=y\left(w^{\prime}, m\right) 在 w ′ > w w ′ > w w^(') > ww^{\prime}>w 的位置。我将证明 x ′ > x x ′ > x x^(') > xx^{\prime}>x 。为此，选择 w ′ ′ w ′ ′ w^('')w^{\prime \prime} 和 m ′ ′ m ′ ′ m^('')m^{\prime \prime} 使得 x ′ ′ = x ( w ′ ′ , m ′ ′ ) = x x ′ ′ = x w ′ ′ , m ′ ′ = x x^('')=x(w^(''),m^(''))=xx^{\prime \prime}=x\left(w^{\prime \prime}, m^{\prime \prime}\right)=x ，并且
y ′ ′ = y ( w ′ ′ , m ′ ′ ) = w w ′ y y ′ ′ = y w ′ ′ , m ′ ′ = w w ′ y y^('')=y(w^(''),m^(''))=(w)/(w^('))yy^{\prime \prime}=y\left(w^{\prime \prime}, m^{\prime \prime}\right)=\frac{w}{w^{\prime}} y
Since x ′ ′ − w ′ y ′ ′ = m , ( x ′ , y ′ ) x ′ ′ − w ′ y ′ ′ = m , x ′ , y ′ x^('')-w^(')y^('')=m,(x^('),y^('))x^{\prime \prime}-w^{\prime} y^{\prime \prime}=m,\left(x^{\prime}, y^{\prime}\right) is preferred to ( x ′ ′ , y ′ ′ ) x ′ ′ , y ′ ′ (x^(''),y^(''))\left(x^{\prime \prime}, y^{\prime \prime}\right); and therefore
由于 x ′ ′ − w ′ y ′ ′ = m , ( x ′ , y ′ ) x ′ ′ − w ′ y ′ ′ = m , x ′ , y ′ x^('')-w^(')y^('')=m,(x^('),y^('))x^{\prime \prime}-w^{\prime} y^{\prime \prime}=m,\left(x^{\prime}, y^{\prime}\right) 优于 ( x ′ ′ , y ′ ′ ) x ′ ′ , y ′ ′ (x^(''),y^(''))\left(x^{\prime \prime}, y^{\prime \prime}\right) ；因此
x ′ − x > w ′ ′ ( y ′ − y ′ ′ ) = w ′ ′ w ′ ( w ′ y ′ − w ′ y ′ ′ ) = w ′ ′ w ′ ( w ′ y ′ − w y ) = w ′ ′ w ′ ( x ′ − x ) x ′ − x > w ′ ′ y ′ − y ′ ′ = w ′ ′ w ′ w ′ y ′ − w ′ y ′ ′ = w ′ ′ w ′ w ′ y ′ − w y = w ′ ′ w ′ x ′ − x {:[x^(')-x > w^('')(y^(')-y^(''))],[=(w^(''))/(w^('))(w^(')y^(')-w^(')y^(''))=(w^(''))/(w^('))(w^(')y^(')-wy)],[=(w^(''))/(w^('))(x^(')-x)]:}\begin{aligned} x^{\prime}-x & >w^{\prime \prime}\left(y^{\prime}-y^{\prime \prime}\right) \\ & =\frac{w^{\prime \prime}}{w^{\prime}}\left(w^{\prime} y^{\prime}-w^{\prime} y^{\prime \prime}\right)=\frac{w^{\prime \prime}}{w^{\prime}}\left(w^{\prime} y^{\prime}-w y\right) \\ & =\frac{w^{\prime \prime}}{w^{\prime}}\left(x^{\prime}-x\right) \end{aligned}
since x ′ − w ′ y ′ = m = x − w y x ′ − w ′ y ′ = m = x − w y x^(')-w^(')y^(')=m=x-wyx^{\prime}-w^{\prime} y^{\prime}=m=x-w y. This implies, with our assumption w ′ ′ < w ′ w ′ ′ < w ′ w^('') < w^(')w^{\prime \prime}<w^{\prime}, that x ′ > x x ′ > x x^(') > xx^{\prime}>x.
自 x ′ − w ′ y ′ = m = x − w y x ′ − w ′ y ′ = m = x − w y x^(')-w^(')y^(')=m=x-wyx^{\prime}-w^{\prime} y^{\prime}=m=x-w y 以来。这意味着，根据我们的假设 w ′ ′ < w ′ w ′ ′ < w ′ w^('') < w^(')w^{\prime \prime}<w^{\prime} ， x ′ > x x ′ > x x^(') > xx^{\prime}>x 。
The converse proposition can be proved by reversing the steps.
逆命题可以通过反转步骤来证明。
For a rigorous proof of this theorem, the reader is referred to [4]. For a heuristic justification, suppose that z n z n z_(n)z_{n} is differentiable, and that c c cc is twice differentiable. The first order condition, (9), can be written
对于该定理的严格证明，读者请参阅[4]。为了进行启发式的论证，假设 z n z n z_(n)z_{n} 是可微的，并且 c c cc 是二次可微的。第一阶条件(9)可以写成
∂ ∂ z u ( c ( z ) , z / n ) = u 1 z [ z c ′ ( z ) − V ( c ( z ) , z / n ) ] = 0 ∂ ∂ z u ( c ( z ) , z / n ) = u 1 z z c ′ ( z ) − V ( c ( z ) , z / n ) = 0 (del)/(del z)u(c(z),z//n)=(u_(1))/(z)[zc^(')(z)-V(c(z),z//n)]=0\frac{\partial}{\partial z} u(c(z), z / n)=\frac{u_{1}}{z}\left[z c^{\prime}(z)-V(c(z), z / n)\right]=0
Furthermore, we have the second-order condition, that the derivative is non-increasing at z n z n z_(n)z_{n}. Since it is zero there, this is also true when we drop the positive factor u 1 / z u 1 / z u_(1)//zu_{1} / z. In other words,
此外，我们有第二阶条件，即在 z n z n z_(n)z_{n} 处导数是非递增的。由于在该点导数为零，因此当我们去掉正因子 u 1 / z u 1 / z u_(1)//zu_{1} / z 时，这一条件仍然成立。换句话说，
∂ ∂ z [ z c ′ ( z ) − V ( c ( z ) , z / n ) ] ≦ 0 , at z = z n ∂ ∂ z z c ′ ( z ) − V ( c ( z ) , z / n ) ≦ 0 ,  at  z = z n (del)/(del z)[zc^(')(z)-V(c(z),z//n)] <= 0," at "z=z_(n)\frac{\partial}{\partial z}\left[z c^{\prime}(z)-V(c(z), z / n)\right] \leqq 0, \text { at } z=z_{n}
Now differentiate the equation z n c ′ ( z n ) − V ( c ( z n ) z n / n ) = 0 z n c ′ z n − V c z n z n / n = 0 z_(n)c^(')(z_(n))-V(c(z_(n))z_(n)//n)=0z_{n} c^{\prime}\left(z_{n}\right)-V\left(c\left(z_{n}\right) z_{n} / n\right)=0 with respect to n n nn :
现在对方程 z n c ′ ( z n ) − V ( c ( z n ) z n / n ) = 0 z n c ′ z n − V c z n z n / n = 0 z_(n)c^(')(z_(n))-V(c(z_(n))z_(n)//n)=0z_{n} c^{\prime}\left(z_{n}\right)-V\left(c\left(z_{n}\right) z_{n} / n\right)=0 关于 n n nn 进行求导：
∂ ∂ z [ z c ′ − V ] | z = z n ⋅ d z n d n = − V y ( c ( z n ) , z n / n ) z / n 2 ∂ ∂ z z c ′ − V z = z n ⋅ d z n d n = − V y c z n , z n / n z / n 2 (del)/(del z)[zc^(')-V]|_(z=z_(n))*(dz_(n))/(dn)=-V_(y)(c(z_(n)),z_(n)//n)z//n^(2)\left.\frac{\partial}{\partial z}\left[z c^{\prime}-V\right]\right|_{z=z_{n}} \cdot \frac{d z_{n}}{d n}=-V_{y}\left(c\left(z_{n}\right), z_{n} / n\right) z / n^{2}
It follows from (29) and assumption (B) that
根据（29）和假设（B），可以得出以下结论：
d z n d n > 0 d z n d n > 0 (dz_(n))/(dn) > 0\frac{d z_{n}}{d n}>0
unless z n = 0 z n = 0 z_(n)=0z_{n}=0. In fact z n z n z_(n)z_{n} is strictly increasing when n > n 0 n > n 0 n > n_(0)n>n_{0} and c c cc is differentiable; a corner in c c cc causes z n z n z_(n)z_{n} to be constant for a range of values of n n nn. (An indifference curve diagram makes this clear.) Condition (ii) of the theorem clearly has to be satisfied by the utility maximizing choice.
除非 z n = 0 z n = 0 z_(n)=0z_{n}=0 。事实上，当 n > n 0 n > n 0 n > n_(0)n>n_{0} 时， z n z n z_(n)z_{n} 是严格递增的，并且 c c cc 是可微的；在 c c cc 中的一个拐角导致 z n z n z_(n)z_{n} 在 n n nn 的一系列值上保持不变。（无差异曲线图使这一点变得清晰。）定理的条件（ii）显然必须由效用最大化选择满足。
To prove that a suitable consumption function exists for a given z z zz-function satisfying the two conditions, one defines c c cc by the first-order condition (28). (30) then shows (nearly) that the second-order condition for a maximum is satisfied. This does not yet prove global maximization of utility, but that also is true.
为了证明对于满足两个条件的给定 z z zz -函数存在一个合适的消费函数，首先通过一阶条件（28）定义 c c cc 。然后（30）几乎表明最大值的二阶条件得到了满足。这尚未证明效用的全局最大化，但这也是正确的。
It should be noticed that, as a corollary of Theorem 1, condition (A) holds when condition (B) holds, for z n z n z_(n)z_{n} is shown to be non-decreasing even if it is a correspondence. It therefore takes a single value for all but a countable set of values of n n nn. A fortiori, condition ( A ) ( A ) (A)(\mathrm{A}) is satisfied in this case.
应注意的是，作为定理 1 的推论，当条件(B)成立时，条件(A)也成立，因为 z n z n z_(n)z_{n} 即使是一个对应关系也被证明是非递减的。因此，对于除可数个 n n nn 值之外的所有值，它取一个单一值。因此，在这种情况下，条件 ( A ) ( A ) (A)(\mathrm{A}) 也得到了满足。
Theorem 1 at once implies that z n z n z_(n)z_{n} and therefore also x n x n x_(n)x_{n} are non-decreasing functions when the optimum tax schedule is imposed. Furthermore, it shows us quite straightforwardly what changes in the function y n y n y_(n)y_{n} we are allowed to contemplate when applying the variational argument that allowable small changes should make only a second-order difference to the maximand. The rigorous argument is still complicated, in part because one has to allow for the possibility that z n z n z_(n)z_{n} is constant over some intervals, and discontinuous at some values of n n nn. The full statement of the result, which is proved in [4], is as follows:
定理 1 立即意味着 z n z n z_(n)z_{n} 以及因此也意味着 x n x n x_(n)x_{n} 在施加最优税率表时是非递减函数。此外，它相当直接地向我们展示了在应用变分论证时，我们可以考虑对函数 y n y n y_(n)y_{n} 进行哪些变化，因为允许的小变化只应对最大值产生二阶差异。严格的论证仍然复杂，部分原因是必须考虑 z n z n z_(n)z_{n} 在某些区间内是常数的可能性，并且在某些 n n nn 值处是不连续的。结果的完整陈述在[4]中得到了证明，内容如下：
Theorem 2. If preferences satisfy assumption (B) and ( u n , x n , y n ) u n , x n , y n (u_(n),x_(n),y_(n))\left(u_{n}, x_{n}, y_{n}\right) arise from optimum income taxation, then
定理 2。如果偏好满足假设(B)，并且 ( u n , x n , y n ) u n , x n , y n (u_(n),x_(n),y_(n))\left(u_{n}, x_{n}, y_{n}\right) 源于最优收入税收，那么
(i) z n = n y n z n = n y n z_(n)=ny_(n)z_{n}=n y_{n} is a non-decreasing function of n n nn;
(i) z n = n y n z n = n y n z_(n)=ny_(n)z_{n}=n y_{n} 是 n n nn 的非递减函数；
(ii) u n = u 0 − ∫ 0 n [ y m u 2 ( x m , y m ) / m ] d m ( n ≧ 0 ) u n = u 0 − ∫ 0 n   y m u 2 x m , y m / m d m ( n ≧ 0 ) u_(n)=u_(0)-int_(0)^(n)[y_(m)u_(2)(x_(m),y_(m))//m]dm quad(n >= 0)u_{n}=u_{0}-\int_{0}^{n}\left[y_{m} u_{2}\left(x_{m}, y_{m}\right) / m\right] d m \quad(n \geqq 0);
(iii) at all points of increase of z n z n z_(n)z_{n} (i.e., where z n > z n z n > z n z_(n) > z_(n)z_{n}>z_{n}, for all n ′ < n n ′ < n n^(') < nn^{\prime}<n, or z n < z n z n < z n z_(n) < z_(n)z_{n}<z_{n}, for all n ′ > n n ′ > n n^(') > nn^{\prime}>n )
在 z n z n z_(n)z_{n} 的所有增值点（即，在 z n > z n z n > z n z_(n) > z_(n)z_{n}>z_{n} ，对于所有 n ′ < n n ′ < n n^(') < nn^{\prime}<n ，或 z n < z n z n < z n z_(n) < z_(n)z_{n}<z_{n} ，对于所有 n ′ > n n ′ > n n^(') > nn^{\prime}>n ）
A n ≡ [ w + u 2 ( n ) / n u 1 ( n ) ] f ( n ) − ψ y n 2 ∫ n ∞ [ 1 u 1 ( m ) − λ G ′ ( u m ) ] T n m f ( m ) d m = 0 A n ≡ w + u 2 ( n ) / n u 1 ( n ) f ( n ) − ψ y n 2 ∫ n ∞   1 u 1 ( m ) − λ G ′ u m T n m f ( m ) d m = 0 A_(n)-=[w+u_(2)^((n))//nu_(1)^((n))]f(n)-(psi_(y))/(n^(2))int_(n)^(oo)[(1)/(u_(1)^((m)))-lambdaG^(')(u_(m))]T_(nm)f(m)dm=0A_{n} \equiv\left[w+u_{2}^{(n)} / n u_{1}^{(n)}\right] f(n)-\frac{\psi_{y}}{n^{2}} \int_{n}^{\infty}\left[\frac{1}{u_{1}^{(m)}}-\lambda G^{\prime}\left(u_{m}\right)\right] T_{n m} f(m) d m=0
where superscripts " ( n ) ( n ) (n)(n) ", etc. indicate that the function is evaluated at n-man (etc.)'s utility-maximizing choice, and
其中上标“ ( n ) ( n ) (n)(n) ”等表示该函数在 n-man（等）的效用最大化选择下进行评估。
ψ y = − u 2 ( n ) − y n u 22 ( n ) + y n u 2 ( n ) u 12 ( n ) / u 1 ( n ) T n m ′ = exp ⁡ [ − ∫ n m y m ′ u 12 ( x m ′ , y m ′ ) / u 1 ( x m ′ , y m ′ ) ⋅ d m ′ ] ψ y = − u 2 ( n ) − y n u 22 ( n ) + y n u 2 ( n ) u 12 ( n ) / u 1 ( n ) T n m ′ = exp ⁡ − ∫ n m   y m ′ u 12 x m ′ , y m ′ / u 1 x m ′ , y m ′ ⋅ d m ′ {:[psi_(y)=-u_(2)^((n))-y_(n)u_(22)^((n))+y_(n)u_(2)^((n))u_(12)^((n))//u_(1)^((n))],[T_(nm)^(')=exp[-int_(n)^(m)y_(m^('))u_(12)(x_(m^(')),y_(m^(')))//u_(1)(x_(m^(')),y_(m^(')))*dm^(')]]:}\begin{gathered} \psi_{y}=-u_{2}^{(n)}-y_{n} u_{22}^{(n)}+y_{n} u_{2}^{(n)} u_{12}^{(n)} / u_{1}^{(n)} \\ T_{n m}^{\prime}=\exp \left[-\int_{n}^{m} y_{m^{\prime}} u_{12}\left(x_{m^{\prime}}, y_{m^{\prime}}\right) / u_{1}\left(x_{m^{\prime}}, y_{m^{\prime}}\right) \cdot d m^{\prime}\right] \end{gathered}
(iv) If n ∈ [ n 1 , n 2 ] n ∈ n 1 , n 2 n in[n_(1),n_(2)]n \in\left[n_{1}, n_{2}\right], where z z zz is constant on [ n 1 , n 2 ] n 1 , n 2 [n_(1),n_(2)]\left[n_{1}, n_{2}\right], and [ n 1 , n 2 ] n 1 , n 2 [n_(1),n_(2)]\left[n_{1}, n_{2}\right] is a maximal interval of constancy for z z zz,
（iv）如果 n ∈ [ n 1 , n 2 ] n ∈ n 1 , n 2 n in[n_(1),n_(2)]n \in\left[n_{1}, n_{2}\right] ，其中 z z zz 在 [ n 1 , n 2 ] n 1 , n 2 [n_(1),n_(2)]\left[n_{1}, n_{2}\right] 上是常数，并且 [ n 1 , n 2 ] n 1 , n 2 [n_(1),n_(2)]\left[n_{1}, n_{2}\right] 是 z z zz 的最大常数区间，
∫ n 1 n A m d m ≧ 0 , ∫ n n 2 A m d m ≦ 0 ∫ n 1 n   A m d m ≧ 0 , ∫ n n 2   A m d m ≦ 0 int_(n_(1))^(n)A_(m)dm >= 0,quadint_(n)^(n_(2))A_(m)dm <= 0\int_{n_{1}}^{n} A_{m} d m \geqq 0, \quad \int_{n}^{n_{2}} A_{m} d m \leqq 0
(v) If z z zz is discontinuous at n , y ¯ n n , y ¯ n n, bar(y)_(n)n, \bar{y}_{n} is defined to be lim m → n − y m , x ¯ n lim m → n −   y m , x ¯ n lim_(m rarr n-)y_(m), bar(x)_(n)\lim _{m \rightarrow n-} y_{m}, \bar{x}_{n} is defined by
如果 z z zz 在 n , y ¯ n n , y ¯ n n, bar(y)_(n)n, \bar{y}_{n} 处不连续，则定义为 lim m → n − y m , x ¯ n lim m → n −   y m , x ¯ n lim_(m rarr n-)y_(m), bar(x)_(n)\lim _{m \rightarrow n-} y_{m}, \bar{x}_{n}
u ( x ¯ n , y ¯ n ) = u n = u ( x n , y n ) u x ¯ n , y ¯ n = u n = u x n , y n u( bar(x)_(n), bar(y)_(n))=u_(n)=u(x_(n),y_(n))u\left(\bar{x}_{n}, \bar{y}_{n}\right)=u_{n}=u\left(x_{n}, y_{n}\right)
and u ¯ 1 u ¯ 1 bar(u)_(1)\bar{u}_{1}, etc., denote u 1 u 1 u_(1)u_{1} evaluated at x ¯ n , y ¯ n x ¯ n , y ¯ n bar(x)_(n), bar(y)_(n)\bar{x}_{n}, \bar{y}_{n}, while u 1 u 1 u_(1)u_{1}, etc., denote evaluation at x n , y n x n , y n x_(n),y_(n)x_{n}, y_{n},
和 u ¯ 1 u ¯ 1 bar(u)_(1)\bar{u}_{1} 等表示在 x ¯ n , y ¯ n x ¯ n , y ¯ n bar(x)_(n), bar(y)_(n)\bar{x}_{n}, \bar{y}_{n} 处评估的 u 1 u 1 u_(1)u_{1} ，而 u 1 u 1 u_(1)u_{1} 等表示在 x n , y n x n , y n x_(n),y_(n)x_{n}, y_{n} 处的评估，
( w y n − x n / n ) − ( w y ¯ n − x ¯ n / n ) y ¯ n u ¯ 2 − y n u 2 = w + u 2 / n u 1 ψ y = w + u ¯ 2 / n u ¯ 1 ψ ¯ y . w y n − x n / n − w y ¯ n − x ¯ n / n y ¯ n u ¯ 2 − y n u 2 = w + u 2 / n u 1 ψ y = w + u ¯ 2 / n u ¯ 1 ψ ¯ y . ((wy_(n)-x_(n)//n)-(w bar(y)_(n)- bar(x)_(n)//n))/( bar(y)_(n) bar(u)_(2)-y_(n)u_(2))=(w+u_(2)//nu_(1))/(psi_(y))=(w+ bar(u)_(2)//n bar(u)_(1))/( bar(psi)_(y)).\frac{\left(w y_{n}-x_{n} / n\right)-\left(w \bar{y}_{n}-\bar{x}_{n} / n\right)}{\bar{y}_{n} \bar{u}_{2}-y_{n} u_{2}}=\frac{w+u_{2} / n u_{1}}{\psi_{y}}=\frac{w+\bar{u}_{2} / n \bar{u}_{1}}{\bar{\psi}_{y}} .
If ψ y ψ y psi_(y)\psi_{y} is a non-decreasing function of y y yy for constant u , z n u , z n u,z_(n)u, z_{n} is continuous for all n n nn.
如果 ψ y ψ y psi_(y)\psi_{y} 是关于常数 u , z n u , z n u,z_(n)u, z_{n} 的非递减函数，并且在所有 n n nn 上是连续的。
(vi) ∫ 0 ∞ [ 1 u 1 − λ G ′ ( u m ) ] T 0 m f ( m ) d m = 0 ∫ 0 ∞   1 u 1 − λ G ′ u m T 0 m f ( m ) d m = 0 int_(0)^(oo)[(1)/(u_(1))-lambdaG^(')(u_(m))]T_(0m)f(m)dm=0\int_{0}^{\infty}\left[\frac{1}{u_{1}}-\lambda G^{\prime}\left(u_{m}\right)\right] T_{0 m} f(m) d m=0,
(vii) X = H ( Z ) X = H ( Z ) X=H(Z)X=H(Z),
w = H ′ ( Z ) w = H ′ ( Z ) w=H^(')(Z)w=H^{\prime}(Z).
It will be noticed that in this statement w w ww is the commodity shadow wage rate ( w / p w / p w//pw / p in the earlier notation), while λ λ lambda\lambda ( 1 / p 1 / p 1//p1 / p in the previous notation) is the inverse of the marginal social utility of commodities (national income). The second part of ( v ) should be particularly noted, since we are quite likely to be willing to assume that ψ y ψ y psi_(y)\psi_{y} is a non-decreasing function of y y yy, and it is a great advantage not to have to worry about possible discontinuities in z n z n z_(n)z_{n}. It does not seem possible, unfortunately, to delimit a class of cases in which one can be sure that [ 0 , n 0 ] 0 , n 0 [0,n_(0)]\left[0, n_{0}\right] will be the only interval of constancy for z z zz. It should be mentioned that, when ψ y ψ y psi_(y)\psi_{y} is not non-decreasing, and the equations (37) may possibly apply, the conditions of Theorem 1 may define more than one candidate for optimality, and then only direct comparison of the welfare generated by the alternative paths so defined will solve the problem.
需要注意的是，在这个陈述中， w w ww 是商品影子工资率（在之前的符号中为 w / p w / p w//pw / p ），而 λ λ lambda\lambda （在之前的符号中为 1 / p 1 / p 1//p1 / p ）是商品的边际社会效用（国民收入）的倒数。特别需要注意 (v) 的第二部分，因为我们很可能愿意假设 ψ y ψ y psi_(y)\psi_{y} 是 y y yy 的非递减函数，并且不必担心 z n z n z_(n)z_{n} 中可能存在的不连续性是一个很大的优势。不幸的是，似乎无法界定一个案例类别，在这个类别中可以确定 [ 0 , n 0 ] 0 , n 0 [0,n_(0)]\left[0, n_{0}\right] 将是 z z zz 的唯一恒定区间。需要提到的是，当 ψ y ψ y psi_(y)\psi_{y} 不是非递减时，方程 (37) 可能适用，定理 1 的条件可能定义多个最优候选，然后只有对所定义的替代路径所产生的福利进行直接比较才能解决问题。

5. INTERPRETATION 5. 解释

If n n nn is not in an interval of constancy for z z zz, and c ( c ( c(c(. ) i s t h e r e f o r e a d i f f e r e n t i a b l e f u n c t i o n ) i s t h e r e f o r e a d i f f e r e n t i a b l e f u n c t i o n )isthereforeadifferentiablefunction) is therefore a differentiable function at z n z n z_(n)z_{n}, the first-order condition (9) applies. It can be written
如果 n n nn 不在 z z zz 和 c ( c ( c(c( 的常数区间内，且在 z n z n z_(n)z_{n} 处的 ) i s t h e r e f o r e a d i f f e r e n t i a b l e f u n c t i o n ) i s t h e r e f o r e a d i f f e r e n t i a b l e f u n c t i o n )isthereforeadifferentiablefunction) is therefore a differentiable function ，则适用一阶条件（9）。可以写成
− u 2 / n u 1 = c ′ ( z ) − u 2 / n u 1 = c ′ ( z ) -u_(2)//nu_(1)=c^(')(z)-u_{2} / n u_{1}=c^{\prime}(z)
If we denote the marginal tax rate, d d ( w z ) [ w z − c ( z ) ] d d ( w z ) [ w z − c ( z ) ] (d)/(d(wz))[wz-c(z)]\frac{d}{d(w z)}[w z-c(z)], by θ θ theta\theta, we have
如果我们用 θ θ theta\theta 表示边际税率 d d ( w z ) [ w z − c ( z ) ] d d ( w z ) [ w z − c ( z ) ] (d)/(d(wz))[wz-c(z)]\frac{d}{d(w z)}[w z-c(z)] ，那么我们有
w θ = d d z [ w z − c ( z ) ] = w + u 2 / n u 1 = ψ y n 2 f ( n ) ∫ n ∞ 1 − λ G ′ u 1 u 1 T n m f ( m ) d m w θ = d d z [ w z − c ( z ) ] = w + u 2 / n u 1 = ψ y n 2 f ( n ) ∫ n ∞   1 − λ G ′ u 1 u 1 T n m f ( m ) d m {:[w theta=(d)/(dz)[wz-c(z)]=w+u_(2)//nu_(1)],[=(psi_(y))/(n^(2)f(n))int_(n)^(oo)(1-lambdaG^(')u_(1))/(u_(1))T_(nm)f(m)dm]:}\begin{aligned} w \theta & =\frac{d}{d z}[w z-c(z)]=w+u_{2} / n u_{1} \\ & =\frac{\psi_{y}}{n^{2} f(n)} \int_{n}^{\infty} \frac{1-\lambda G^{\prime} u_{1}}{u_{1}} T_{n m} f(m) d m \end{aligned}
by (33). (42) suggests the considerations that should influence the magnitude of the marginal tax rate. First, it can tell us something about the sign of θ θ theta\theta : we already know that θ θ theta\theta will not be greater than 1 , but we were not previously able to say anything about its sign. Of course, we expect that it will not usually be negative. Using (42) and the conditions in Theorem 1, we can establish this rigorously.
通过 (33)。(42) 提出了应影响边际税率大小的考虑因素。首先，它可以告诉我们关于 θ θ theta\theta 的一些信息：我们已经知道 θ θ theta\theta 不会大于 1，但之前我们无法对其符号做出任何判断。当然，我们期望它通常不会是负数。利用 (42) 和定理 1 中的条件，我们可以严格地建立这一点。
Note first that 1 − λ G ′ u 1 1 − λ G ′ u 1 1-lambdaG^(')u_(1)1-\lambda G^{\prime} u_{1} is a non-decreasing function of n n nn, since x n x n x_(n)x_{n} is a non-decreasing function of n n nn, and ∂ ∂ x G = G ′ u 1 ∂ ∂ x G = G ′ u 1 (del)/(del x)G=G^(')u_(1)\frac{\partial}{\partial x} G=G^{\prime} u_{1} a decreasing function of x x xx. If 1 − λ G ′ u 1 1 − λ G ′ u 1 1-lambdaG^(')u_(1)1-\lambda G^{\prime} u_{1} were always positive or always negative, Equation (38) could not be satisfied. Therefore
首先注意到 1 − λ G ′ u 1 1 − λ G ′ u 1 1-lambdaG^(')u_(1)1-\lambda G^{\prime} u_{1} 是 n n nn 的非递减函数，因为 x n x n x_(n)x_{n} 是 n n nn 的非递减函数，而 ∂ ∂ x G = G ′ u 1 ∂ ∂ x G = G ′ u 1 (del)/(del x)G=G^(')u_(1)\frac{\partial}{\partial x} G=G^{\prime} u_{1} 是 x x xx 的递减函数。如果 1 − λ G ′ u 1 1 − λ G ′ u 1 1-lambdaG^(')u_(1)1-\lambda G^{\prime} u_{1} 始终为正或始终为负，则方程（38）无法满足。因此
∫ n ∞ 1 u 1 ( 1 − λ G ′ u 1 ) T n m f ( m ) d m ∫ n ∞   1 u 1 1 − λ G ′ u 1 T n m f ( m ) d m int_(n)^(oo)(1)/(u_(1))(1-lambdaG^(')u_(1))T_(nm)f(m)dm\int_{n}^{\infty} \frac{1}{u_{1}}\left(1-\lambda G^{\prime} u_{1}\right) T_{n m} f(m) d m
is increasing in n n nn for n n nn less than some n ¯ n ¯ bar(n)\bar{n}, and decreasing for n > n ¯ n > n ¯ n > bar(n)n>\bar{n}; but in any case positive for n > n ¯ n > n ¯ n > bar(n)n>\bar{n}. (Here we use the properties u 1 > 0 , T m n > 0 u 1 > 0 , T m n > 0 u_(1) > 0,T_(mn) > 0u_{1}>0, T_{m n}>0.) Since the integral is zero when n = 0 n = 0 n=0n=0, it is non-negative for all n n nn. Consequently the marginal tax rate is non-negative at all points of increase of z z zz. If n n nn is not a point of increase of z , c z , c z,cz, c is not differentiable at z n z n z_(n)z_{n}. It is easily seen that, if [ n 1 , n 2 n 1 , n 2 n_(1),n_(2)n_{1}, n_{2} ] is a maximal interval of constancy of z , − u 2 / n u 1 z , − u 2 / n u 1 z,-u_(2)//nu_(1)z,-u_{2} / n u_{1} is equal to the left derivative of c c cc at n 1 n 1 n_(1)n_{1}, and the right derivative at n 2 n 2 n_(2)n_{2}. Thus both the " right " and " left" marginal tax rates are non-negative in this case. Summarizing:
在 n n nn 中对于 n n nn 小于某些 n ¯ n ¯ bar(n)\bar{n} 时是增加的，而对于 n > n ¯ n > n ¯ n > bar(n)n>\bar{n} 则是减少的；但在任何情况下对于 n > n ¯ n > n ¯ n > bar(n)n>\bar{n} 都是正的。（在这里我们使用属性 u 1 > 0 , T m n > 0 u 1 > 0 , T m n > 0 u_(1) > 0,T_(mn) > 0u_{1}>0, T_{m n}>0 。）由于当 n = 0 n = 0 n=0n=0 时积分为零，因此对于所有 n n nn 都是非负的。因此，边际税率在 z z zz 的所有增加点都是非负的。如果 n n nn 不是 z , c z , c z,cz, c 的增加点，则在 z n z n z_(n)z_{n} 处不可微分。很容易看出，如果[ n 1 , n 2 n 1 , n 2 n_(1),n_(2)n_{1}, n_{2} ]是 z , − u 2 / n u 1 z , − u 2 / n u 1 z,-u_(2)//nu_(1)z,-u_{2} / n u_{1} 的最大常数区间，则在 n 1 n 1 n_(1)n_{1} 处等于 c c cc 的左导数，在 n 2 n 2 n_(2)n_{2} 处等于右导数。因此，在这种情况下，“右”边际税率和“左”边际税率都是非负的。总结：
Proposition 3. 1 1 ^(1){ }^{1} If assumption (B) is satisfied, wz-c(z) (the " tax function") is a nondecreasing function for all z z zz that actually occur (and may therefore be taken to be a nondecreasing function for all z z zz ).
命题 3. 1 1 ^(1){ }^{1} 如果假设 (B) 得到满足，则 wz-c(z)（“税收函数”）对于所有实际发生的 z z zz 是一个非递减函数（因此可以认为对于所有 z z zz 是一个非递减函数）。
Having established that the integral in Equation (42) is non-negative for all n n nn, we can see that the marginal tax rate will be greater if there are relatively few n n nn-men than otherwise; or if the utility-value of work, − y u y − y u y -yu_(y)-y u_{y}, is more sensitive to work done (utility being held constant); or if n n nn is closer to n ¯ n ¯ bar(n)\bar{n}, the value of n n nn at which 1 = λ G ′ u 1 1 = λ G ′ u 1 1=lambdaG^(')u_(1)1=\lambda G^{\prime} u_{1} (and the integral is therefore a maximum). If f f ff is a single-peaked distribution, the first consideration suggests that marginal tax rates should be greatest for the richest and the poorest; but the last consideration tells the other way.
已确定方程（42）中的积分对于所有 n n nn 都是非负的，我们可以看到，如果相对较少的 n n nn -人存在，边际税率将会更高；或者如果工作的效用值 − y u y − y u y -yu_(y)-y u_{y} 对所做工作的敏感性更强（效用保持不变）；或者如果 n n nn 更接近 n ¯ n ¯ bar(n)\bar{n} ，即 1 = λ G ′ u 1 1 = λ G ′ u 1 1=lambdaG^(')u_(1)1=\lambda G^{\prime} u_{1} 时 n n nn 的值（因此积分达到最大值）。如果 f f ff 是单峰分布，第一个考虑因素表明边际税率在最富有和最贫穷的人群中应该是最高的；但最后一个考虑因素则表明相反的情况。
In any case, it is important to note than n 0 n 0 n_(0)n_{0}, the largest n n nn for which y n = 0 y n = 0 y_(n)=0y_{n}=0, may be quite large: if the number who do not work in the optimum regime is large, the marginal tax rate may not be high at zero income. Explicitly, we can rewrite Equation (38) in the form
无论如何，重要的是要注意到 n 0 n 0 n_(0)n_{0} ，这是最大的 n n nn ，对于 y n = 0 y n = 0 y_(n)=0y_{n}=0 ，可能相当大：如果在最佳状态下不工作的数量很大，边际税率在零收入时可能并不高。我们可以明确地将方程（38）重写为以下形式：
[ 1 u 1 ( x 0 , 0 ) − i G ′ ( u 0 ) ] F ( n 0 ) + ∫ n 0 ∞ [ 1 u 1 − λ G ′ ] T n 0 m f ( m ) d m = 0 1 u 1 x 0 , 0 − i G ′ u 0 F n 0 + ∫ n 0 ∞   1 u 1 − λ G ′ T n 0 m f ( m ) d m = 0 [(1)/(u_(1)(x_(0),0))-iG^(')(u_(0))]F(n_(0))+int_(n_(0))^(oo)[(1)/(u_(1))-lambdaG^(')]T_(n_(0)m)f(m)dm=0\left[\frac{1}{u_{1}\left(x_{0}, 0\right)}-i G^{\prime}\left(u_{0}\right)\right] F\left(n_{0}\right)+\int_{n_{0}}^{\infty}\left[\frac{1}{u_{1}}-\lambda G^{\prime}\right] T_{n_{0} m} f(m) d m=0
which, when combined with Equation (33) (for n = n 0 n = n 0 n=n_(0)n=n_{0} ) gives
当与方程（33）（对于 n = n 0 n = n 0 n=n_(0)n=n_{0} ）结合时，得到
w + u 2 ( x 0 , 0 ) n 0 u 1 ( x 0 , 0 ) = ψ y ( u 0 , 0 ) F ( n 0 ) n 0 2 f ( n 0 ) [ λ G ′ ( u 0 ) − 1 u 1 ( x 0 , 0 ) ] . w + u 2 x 0 , 0 n 0 u 1 x 0 , 0 = ψ y u 0 , 0 F n 0 n 0 2 f n 0 λ G ′ u 0 − 1 u 1 x 0 , 0 . w+(u_(2)(x_(0),0))/(n_(0)u_(1)(x_(0),0))=psi_(y)(u_(0),0)(F(n_(0)))/(n_(0)^(2)f(n_(0)))[lambdaG^(')(u_(0))-(1)/(u_(1)(x_(0),0))].w+\frac{u_{2}\left(x_{0}, 0\right)}{n_{0} u_{1}\left(x_{0}, 0\right)}=\psi_{y}\left(u_{0}, 0\right) \frac{F\left(n_{0}\right)}{n_{0}^{2} f\left(n_{0}\right)}\left[\lambda G^{\prime}\left(u_{0}\right)-\frac{1}{u_{1}\left(x_{0}, 0\right)}\right] .
Unfortunately, one cannot get much information from these " local" conditions, at least for small n n nn. For any detail, and in particular for numerical results, one must examine the whole system of equations. It is easier to do that for particular examples of the general problem, and that is what we shall do in succeeding sections. It may be noted, however, that Equation (44) does provide us with some information about n 0 n 0 n_(0)n_{0} and x 0 x 0 x_(0)x_{0}. For example, it is clear that n 0 n 0 n_(0)n_{0} can be zero only if F / n f F / n f F//nfF / n f tends to 0 as n n nn tends to 0 ; indeed, since the left hand side of Equation (44) is bounded, n 0 = 0 n 0 = 0 n_(0)=0n_{0}=0 only if x 0 = 0 x 0 = 0 x_(0)=0x_{0}=0, and therefore 1 / u 1 = 0 1 / u 1 = 0 1//u_(1)=01 / u_{1}=0. It follows that n 0 = 0 n 0 = 0 n_(0)=0n_{0}=0 only if F / ( n 2 f ) F / n 2 f F//(n^(2)f)F /\left(n^{2} f\right) is bounded as n → 0 n → 0 n rarr0n \rightarrow 0, which means that F F FF tends to zero faster than exp ⁡ ( − 1 / n ) exp ⁡ ( − 1 / n ) exp(-1//n)\exp (-1 / n). This excludes the cases usually considered by economists. We
不幸的是，从这些“局部”条件中无法获得太多信息，至少对于小的 n n nn 来说。对于任何细节，特别是数值结果，必须检查整个方程组。对于一般问题的特定例子，这样做更容易，这也是我们将在接下来的章节中所做的。然而，需要注意的是，方程 (44) 确实为我们提供了一些关于 n 0 n 0 n_(0)n_{0} 和 x 0 x 0 x_(0)x_{0} 的信息。例如，很明显，只有当 F / n f F / n f F//nfF / n f 趋近于 0 时， n 0 n 0 n_(0)n_{0} 才能为零；实际上，由于方程 (44) 的左侧是有界的， n 0 = 0 n 0 = 0 n_(0)=0n_{0}=0 仅在 x 0 = 0 x 0 = 0 x_(0)=0x_{0}=0 的情况下成立，因此 1 / u 1 = 0 1 / u 1 = 0 1//u_(1)=01 / u_{1}=0 。因此，只有当 F / ( n 2 f ) F / n 2 f F//(n^(2)f)F /\left(n^{2} f\right) 像 n → 0 n → 0 n rarr0n \rightarrow 0 一样有界时， n 0 = 0 n 0 = 0 n_(0)=0n_{0}=0 才成立，这意味着 F F FF 的趋近速度快于 exp ⁡ ( − 1 / n ) exp ⁡ ( − 1 / n ) exp(-1//n)\exp (-1 / n) 。这排除了经济学家通常考虑的情况。我们
may conclude at this stage that it will be optimal, in the most interesting cases, to encourage some of the population to be idle.
在这个阶段可以得出结论，在最有趣的案例中，鼓励部分人口处于闲置状态将是最优的。
A number of conclusions have been obtained, but they are fairly weak: the marginal tax rate lies between zero and one; in a large class of cases, consumption and labour supply vary continuously with the skill of the individual; there will usually be a group of people who ought to work only if they enjoy it. The main feature of the results is that the optimum tax schedule depends upon the distribution of skills within the population, and the labourconsumption preferences of the population, in such a complicated way that it is not possible to say in general whether marginal tax rates should be higher for high-income, low-income, or intermediate-income groups. The two integral equations that characterise the optimum tax schedule are, however, of a reasonably manageable form. One expects to be able to calculate the schedule in particular cases without great difficulty. In the next sections of the paper, we shall show how this can be done in certain special cases, and obtain further properties of the optimum tax in these cases.
已经得出了一些结论，但它们相当薄弱：边际税率介于零和一之间；在大多数情况下，消费和劳动供给随着个体技能的变化而连续变化；通常会有一群人只有在享受工作的情况下才应该工作。结果的主要特征是，最优税收方案依赖于人口中技能的分布以及人口的劳动消费偏好，以一种复杂的方式，使得无法一般性地判断高收入、低收入或中等收入群体的边际税率应该更高。 然而，描述最优税收方案的两个积分方程形式相对可管理。人们期望能够在特定情况下毫不费力地计算出该方案。在本文的下一部分，我们将展示如何在某些特殊情况下做到这一点，并获得这些情况下最优税收的进一步特性。

6. ADDITIVE UTILITY 6. 加性效用

An interesting case arises when, for all x x xx and y y yy,
当对于所有 x x xx 和 y y yy 时，出现一个有趣的情况
u 12 = 0 u 12 = 0 u_(12)=0u_{12}=0
Thus u 1 u 1 u_(1)u_{1} depends only on x x xx, and u 2 u 2 u_(2)u_{2} only on y y yy.
因此 u 1 u 1 u_(1)u_{1} 仅依赖于 x x xx ，而 u 2 u 2 u_(2)u_{2} 仅依赖于 y y yy 。
Proposition 4. If assumption (45) is satisfied, V ( x , y ) V ( x , y ) V(x,y)V(x, y) is an increasing function of y y yy, bounded for small x x xx and y y yy.
命题 4。如果假设（45）成立， V ( x , y ) V ( x , y ) V(x,y)V(x, y) 是 y y yy 的一个递增函数，对于小的 x x xx 和 y y yy 是有界的。
Proof. V = − y u 2 ( y ) / u 1 ( x ) V = − y u 2 ( y ) / u 1 ( x ) V=-yu_(2)(y)//u_(1)(x)V=-y u_{2}(y) / u_{1}(x), and V 2 = ( − u 2 − y u 22 ) / u 1 > 0 V 2 = − u 2 − y u 22 / u 1 > 0 V_(2)=(-u_(2)-yu_(22))//u_(1) > 0V_{2}=\left(-u_{2}-y u_{22}\right) / u_{1}>0. Boundedness is obvious. |
证明。 V = − y u 2 ( y ) / u 1 ( x ) V = − y u 2 ( y ) / u 1 ( x ) V=-yu_(2)(y)//u_(1)(x)V=-y u_{2}(y) / u_{1}(x) 和 V 2 = ( − u 2 − y u 22 ) / u 1 > 0 V 2 = − u 2 − y u 22 / u 1 > 0 V_(2)=(-u_(2)-yu_(22))//u_(1) > 0V_{2}=\left(-u_{2}-y u_{22}\right) / u_{1}>0 。有界性显而易见。|
Corollary. Under assumption (45), Theorem 1 applies.
推论。在假设（45）下，定理 1 适用。
In particular we know, from statement ( v v vv ) of that Theorem that y n y n y_(n)y_{n} is continuous provided that ψ y ψ y psi_(y)\psi_{y} is non-decreasing. In the present case, this condition is equivalent to the requirement that
特别地，我们知道，从该定理的陈述（ v v vv ）中，如果 ψ y ψ y psi_(y)\psi_{y} 是非递减的，则 y n y n y_(n)y_{n} 是连续的。在当前情况下，这个条件等价于要求
− y u 2 ( y ) is convex. − y u 2 ( y )  is convex.  -yu_(2)(y)" is convex. "-y u_{2}(y) \text { is convex. }
There is no reason why this assumption should hold in general, but it is easily checked for any particular case. We shall now restrict attention to cases for which (46) holds. 1 1 ^(1){ }^{1}
没有理由认为这一假设在一般情况下成立，但可以很容易地检查任何特定情况。我们现在将注意力限制在满足（46）式的情况。
If we restrict attention also to cases where z z zz is strictly increasing when n > n 0 n > n 0 n > n_(0)n>n_{0}, the optimum situation will be a solution of the equations
如果我们还将注意力限制在当 n > n 0 n > n 0 n > n_(0)n>n_{0} 时 z z zz 严格递增的情况，那么最佳情况将是方程的解
{ ( w + u 2 n u 1 ) n 2 f ( n ) = ψ y ∫ n ∞ ( 1 u 1 − λ G ′ ) f ( m ) d m u n = u 0 − ∫ 0 n y m u 2 d m m w + u 2 n u 1 n 2 f ( n ) = ψ y ∫ n ∞   1 u 1 − λ G ′ f ( m ) d m u n = u 0 − ∫ 0 n   y m u 2 d m m {[(w+(u_(2))/(nu_(1)))n^(2)f(n)=psi_(y)int_(n)^(oo)((1)/(u_(1))-lambdaG^('))f(m)dm],[u_(n)=u_(0)-int_(0)^(n)y_(m)u_(2)(dm)/(m)]:}\left\{\begin{array}{l} \left(w+\frac{u_{2}}{n u_{1}}\right) n^{2} f(n)=\psi_{y} \int_{n}^{\infty}\left(\frac{1}{u_{1}}-\lambda G^{\prime}\right) f(m) d m \\ u_{n}=u_{0}-\int_{0}^{n} y_{m} u_{2} \frac{d m}{m} \end{array}\right.
We shall further assume that f f ff is continuously differentiable. Since x n , y n x n , y n x_(n),y_(n)x_{n}, y_{n} are continuous in this case, it follows that u n u n u_(n)u_{n} and ( w + u 2 n u 1 ) / ψ y w + u 2 n u 1 / ψ y (w+(u_(2))/(nu_(1)))//psi_(y)\left(w+\frac{u_{2}}{n u_{1}}\right) / \psi_{y} are differentiable functions of n n nn. Write
我们进一步假设 f f ff 是连续可微的。由于 x n , y n x n , y n x_(n),y_(n)x_{n}, y_{n} 在这种情况下是连续的，因此 u n u n u_(n)u_{n} 和 ( w + u 2 n u 1 ) / ψ y w + u 2 n u 1 / ψ y (w+(u_(2))/(nu_(1)))//psi_(y)\left(w+\frac{u_{2}}{n u_{1}}\right) / \psi_{y} 是 n n nn 的可微函数。写作
v = w + u 2 n u 1 ψ y v = w + u 2 n u 1 ψ y v=(w+(u_(2))/(nu_(1)))/(psi_(y))v=\frac{w+\frac{u_{2}}{n u_{1}}}{\psi_{y}}
u u uu and v v vv are continuously differentiable functions of x x xx and y y yy. Since ∂ u ∂ x > 0 , ∂ u ∂ y < 0 ∂ u ∂ x > 0 , ∂ u ∂ y < 0 (del u)/(del x) > 0,(del u)/(del y) < 0\frac{\partial u}{\partial x}>0, \frac{\partial u}{\partial y}<0, and, as can easily be seen, ∂ v ∂ x < 0 , ∂ v ∂ y < 0 ∂ v ∂ x < 0 , ∂ v ∂ y < 0 (del v)/(del x) < 0,(del v)/(del y) < 0\frac{\partial v}{\partial x}<0, \frac{\partial v}{\partial y}<0, the Jacobian ∂ ( u , v ) ∂ ( x , y ) ∂ ( u , v ) ∂ ( x , y ) (del(u,v))/(del(x,y))\frac{\partial(u, v)}{\partial(x, y)} is always negative. Consequently x x xx and y y yy can be expressed as continuously differentiable functions of u u uu and v v vv, and are therefore themselves differentiable functions of n n nn.
u u uu 和 v v vv 是 x x xx 和 y y yy 的连续可微函数。由于 ∂ u ∂ x > 0 , ∂ u ∂ y < 0 ∂ u ∂ x > 0 , ∂ u ∂ y < 0 (del u)/(del x) > 0,(del u)/(del y) < 0\frac{\partial u}{\partial x}>0, \frac{\partial u}{\partial y}<0 ，并且可以很容易看出 ∂ v ∂ x < 0 , ∂ v ∂ y < 0 ∂ v ∂ x < 0 , ∂ v ∂ y < 0 (del v)/(del x) < 0,(del v)/(del y) < 0\frac{\partial v}{\partial x}<0, \frac{\partial v}{\partial y}<0 ，雅可比矩阵 ∂ ( u , v ) ∂ ( x , y ) ∂ ( u , v ) ∂ ( x , y ) (del(u,v))/(del(x,y))\frac{\partial(u, v)}{\partial(x, y)} 始终为负。因此 x x xx 和 y y yy 可以表示为 u u uu 和 v v vv 的连续可微函数，因此它们本身也是 n n nn 的可微函数。
We can now write Equations (47) and (48) as differential equations:
我们现在可以将方程（47）和（48）写成微分方程：
d v d n = − v n ( 2 + n f ′ f ) − 1 n 2 u 1 + λ G ′ n 2 d u d n = − y u 2 n d v d n = − v n 2 + n f ′ f − 1 n 2 u 1 + λ G ′ n 2 d u d n = − y u 2 n {:[(dv)/(dn)=-(v)/(n)(2+(nf^('))/(f))-(1)/(n^(2)u_(1))+(lambdaG^('))/(n^(2))],[(du)/(dn)=-(yu_(2))/(n)]:}\begin{aligned} & \frac{d v}{d n}=-\frac{v}{n}\left(2+\frac{n f^{\prime}}{f}\right)-\frac{1}{n^{2} u_{1}}+\frac{\lambda G^{\prime}}{n^{2}} \\ & \frac{d u}{d n}=-\frac{y u_{2}}{n} \end{aligned}
which, as we have just shown, can be thought of as equations in u u uu and v v vv. The particular solution we seek, and the particular value of λ λ lambda\lambda, are defined by the boundary conditions, Equations (39), (40),
我们刚刚展示的，可以被视为 u u uu 和 v v vv 中的方程。我们所寻求的特解，以及 λ λ lambda\lambda 的特定值，由边界条件定义，即方程（39）、（40）。
v n 0 = F ( n 0 ) n 0 2 f ( n 0 ) [ λ G ′ ( u n 0 ) − 1 u 1 ( x n 0 ) ] v n 0 = F n 0 n 0 2 f n 0 λ G ′ u n 0 − 1 u 1 x n 0 v_(n_(0))=(F(n_(0)))/(n_(0)^(2)f(n_(0)))[lambdaG^(')(u_(n_(0)))-(1)/(u_(1)(x_(n_(0))))]v_{n_{0}}=\frac{F\left(n_{0}\right)}{n_{0}^{2} f\left(n_{0}\right)}\left[\lambda G^{\prime}\left(u_{n_{0}}\right)-\frac{1}{u_{1}\left(x_{n_{0}}\right)}\right]
which is the form (38) takes here, and
形式 (38) 在这里所呈现的
v n n 2 f ( n ) → 0 ( n → ∞ ) v n n 2 f ( n ) → 0 ( n → ∞ ) v_(n)n^(2)f(n)rarr0quad(n rarr oo)v_{n} n^{2} f(n) \rightarrow 0 \quad(n \rightarrow \infty)
which is apparent from Equation (47). Provided that z n z n z_(n)z_{n} is strictly increasing for n ≧ n 0 n ≧ n 0 n >= n_(0)n \geqq n_{0}, a solution that satisfies all those conditions will, by Theorem 2 of [4], provide the optimum.
这从方程（47）中显而易见。只要 z n z n z_(n)z_{n} 在 n ≧ n 0 n ≧ n 0 n >= n_(0)n \geqq n_{0} 上是严格递增的，满足所有这些条件的解将根据[4]的定理 2 提供最优解。
Equations (39) and (40), the production function and the marginal productivity equation, may be ignored in the calculations. Corresponding to the particular values of w w ww and λ λ lambda\lambda used in the calculation, one obtains values for X X XX and Z Z ZZ. Thus we know the optimum tax schedule when the marginal product is w w ww and the average product is X / Z X / Z X//ZX / Z. In this way one could obtain a range of tax schedules corresponding to different average products and marginal products-which is what one wants. Of course, it is desirable to choose λ λ lambda\lambda so that the average product will be related to the marginal product, w w ww, in a reasonable way. This should not present any great difficulty.
方程（39）和（40），生产函数和边际生产力方程，可以在计算中忽略。根据计算中使用的特定值 w w ww 和 λ λ lambda\lambda ，可以得到 X X XX 和 Z Z ZZ 的值。因此，当边际产品为 w w ww 且平均产品为 X / Z X / Z X//ZX / Z 时，我们知道最佳税率表。通过这种方式，可以获得一系列对应于不同平均产品和边际产品的税率表——这正是我们所希望的。当然，选择 λ λ lambda\lambda 使得平均产品与边际产品 w w ww 之间有合理的关系是理想的。这应该不会带来太大困难。
To determine the sign of d z n d n = y n + n d y n d n d z n d n = y n + n d y n d n (dz_(n))/(dn)=y_(n)+n(dy_(n))/(dn)\frac{d z_{n}}{d n}=y_{n}+n \frac{d y_{n}}{d n}, we calculate, from Equation (49),
为了确定 d z n d n = y n + n d y n d n d z n d n = y n + n d y n d n (dz_(n))/(dn)=y_(n)+n(dy_(n))/(dn)\frac{d z_{n}}{d n}=y_{n}+n \frac{d y_{n}}{d n} 的符号，我们从方程 (49) 进行计算，
ψ y d v d n = ( u 22 n u 1 − v ψ y y ) d y d n − u 2 n 2 u 1 − u 2 u 11 n u 1 2 d x d n = ( u 22 n u 1 − v ψ y y ) d y d n − u 2 n 2 u 1 + u 2 2 u 11 n u 1 3 d y d n − u 2 u 11 n u 1 3 d u d n = 1 n ( u 22 n u 1 − v ψ y y + u 2 2 u 11 n u 1 3 ) d z d n − y n n ( u 22 n u 1 − v ψ y y ) − u 2 n 2 u 1 ψ y d v d n = u 22 n u 1 − v ψ y y d y d n − u 2 n 2 u 1 − u 2 u 11 n u 1 2 d x d n = u 22 n u 1 − v ψ y y d y d n − u 2 n 2 u 1 + u 2 2 u 11 n u 1 3 d y d n − u 2 u 11 n u 1 3 d u d n = 1 n u 22 n u 1 − v ψ y y + u 2 2 u 11 n u 1 3 d z d n − y n n u 22 n u 1 − v ψ y y − u 2 n 2 u 1 {:[psi_(y)(dv)/(dn)=((u_(22))/(nu_(1))-vpsi_(yy))(dy)/(dn)-(u_(2))/(n^(2)u_(1))-(u_(2)u_(11))/(nu_(1)^(2))(dx)/(dn)],[=((u_(22))/(nu_(1))-vpsi_(yy))(dy)/(dn)-(u_(2))/(n^(2)u_(1))+(u_(2)^(2)u_(11))/(nu_(1)^(3))(dy)/(dn)-(u_(2)u_(11))/(nu_(1)^(3))(du)/(dn)],[=(1)/(n)((u_(22))/(nu_(1))-vpsi_(yy)+(u_(2)^(2)u_(11))/(nu_(1)^(3)))(dz)/(dn)-(y_(n))/(n)((u_(22))/(nu_(1))-vpsi_(yy))-(u_(2))/(n^(2)u_(1))]:}\begin{aligned} \psi_{y} \frac{d v}{d n} & =\left(\frac{u_{22}}{n u_{1}}-v \psi_{y y}\right) \frac{d y}{d n}-\frac{u_{2}}{n^{2} u_{1}}-\frac{u_{2} u_{11}}{n u_{1}^{2}} \frac{d x}{d n} \\ & =\left(\frac{u_{22}}{n u_{1}}-v \psi_{y y}\right) \frac{d y}{d n}-\frac{u_{2}}{n^{2} u_{1}}+\frac{u_{2}^{2} u_{11}}{n u_{1}^{3}} \frac{d y}{d n}-\frac{u_{2} u_{11}}{n u_{1}^{3}} \frac{d u}{d n} \\ & =\frac{1}{n}\left(\frac{u_{22}}{n u_{1}}-v \psi_{y y}+\frac{u_{2}^{2} u_{11}}{n u_{1}^{3}}\right) \frac{d z}{d n}-\frac{y_{n}}{n}\left(\frac{u_{22}}{n u_{1}}-v \psi_{y y}\right)-\frac{u_{2}}{n^{2} u_{1}} \end{aligned}
substituting from (51). Therefore, using (50)
从（51）中替代。因此，使用（50）
[ u 22 n u 1 − v ψ y y + u 2 2 u 11 n u 1 3 ] d z d n = y u 22 n u 1 − y v ψ y y + u 2 n u 1 − ( 2 + n f ′ f ) v ψ y − ψ y n u 1 + λ ψ y G ′ n = − ψ y { ( 2 + n f ′ f + y ψ y y ψ y ) v + 2 n u 1 − λ G ′ n } u 22 n u 1 − v ψ y y + u 2 2 u 11 n u 1 3 d z d n = y u 22 n u 1 − y v ψ y y + u 2 n u 1 − 2 + n f ′ f v ψ y − ψ y n u 1 + λ ψ y G ′ n = − ψ y 2 + n f ′ f + y ψ y y ψ y v + 2 n u 1 − λ G ′ n {:[[(u_(22))/(nu_(1))-vpsi_(yy)+(u_(2)^(2)u_(11))/(nu_(1)^(3))](dz)/(dn)=(yu_(22))/(nu_(1))-yvpsi_(yy)+(u_(2))/(nu_(1))-(2+(nf^('))/(f))vpsi_(y)-(psi_(y))/(nu_(1))+(lambdapsi_(y)G^('))/(n)],[=-psi_(y){(2+(nf^('))/(f)+(ypsi_(yy))/(psi_(y)))v+(2)/(nu_(1))-(lambdaG^('))/(n)}]:}\begin{aligned} {\left[\frac{u_{22}}{n u_{1}}-v \psi_{y y}+\frac{u_{2}^{2} u_{11}}{n u_{1}^{3}}\right] \frac{d z}{d n} } & =\frac{y u_{22}}{n u_{1}}-y v \psi_{y y}+\frac{u_{2}}{n u_{1}}-\left(2+\frac{n f^{\prime}}{f}\right) v \psi_{y}-\frac{\psi_{y}}{n u_{1}}+\frac{\lambda \psi_{y} G^{\prime}}{n} \\ & =-\psi_{y}\left\{\left(2+\frac{n f^{\prime}}{f}+\frac{y \psi_{y y}}{\psi_{y}}\right) v+\frac{2}{n u_{1}}-\frac{\lambda G^{\prime}}{n}\right\} \end{aligned}
We may therefore check the assumption d z d n ≧ 0 d z d n ≧ 0 (dz)/(dn) >= 0\frac{d z}{d n} \geqq 0 by examining the solution to see whether
因此，我们可以通过检查解来验证假设 d z d n ≧ 0 d z d n ≧ 0 (dz)/(dn) >= 0\frac{d z}{d n} \geqq 0 是否成立
( 2 + n f ′ f + y ψ y y ψ y ) v + 2 n u 1 − λ G ′ n ≧ 0 2 + n f ′ f + y ψ y y ψ y v + 2 n u 1 − λ G ′ n ≧ 0 (2+(nf^('))/(f)+(ypsi_(yy))/(psi_(y)))v+(2)/(nu_(1))-(lambdaG^('))/(n) >= 0\left(2+\frac{n f^{\prime}}{f}+\frac{y \psi_{y y}}{\psi_{y}}\right) v+\frac{2}{n u_{1}}-\frac{\lambda G^{\prime}}{n} \geqq 0
Equation (56) is equivalent to d z d n ≧ 0 d z d n ≧ 0 (dz)/(dn) >= 0\frac{d z}{d n} \geqq 0 because the expression in square brackets in Equation (55) is negative, term by term.
方程（56）等价于 d z d n ≧ 0 d z d n ≧ 0 (dz)/(dn) >= 0\frac{d z}{d n} \geqq 0 ，因为方程（55）中方括号内的表达式逐项为负。
In computation, one can proceed as follows:-
在计算中，可以按如下方式进行：
[1] A value of λ λ lambda\lambda is chosen. To get the right order of magnitude, one can calculate ∫ 0 ∞ u 1 − 1 f d n / ∫ 0 ∞ G ′ f d n ∫ 0 ∞   u 1 − 1 f d n / ∫ 0 ∞   G ′ f d n int_(0)^(oo)u_(1)^(-1)fdn//int_(0)^(oo)G^(')fdn\int_{0}^{\infty} u_{1}^{-1} f d n / \int_{0}^{\infty} G^{\prime} f d n (cf. (38)) for some particular feasible, and a priori plausible, allocation of consumption and labour.
选择一个值 λ λ lambda\lambda 。为了获得正确的数量级，可以计算 ∫ 0 ∞ u 1 − 1 f d n / ∫ 0 ∞ G ′ f d n ∫ 0 ∞   u 1 − 1 f d n / ∫ 0 ∞   G ′ f d n int_(0)^(oo)u_(1)^(-1)fdn//int_(0)^(oo)G^(')fdn\int_{0}^{\infty} u_{1}^{-1} f d n / \int_{0}^{\infty} G^{\prime} f d n （参见（38））对于某些特定的可行且事先合理的消费和劳动分配。
[2] A trial value of n 0 > 0 n 0 > 0 n_(0) > 0n_{0}>0 is chosen. (It should be borne in mind that the inequality v n 0 ≧ 0 v n 0 ≧ 0 v_(n_(0)) >= 0v_{n_{0}} \geqq 0 may, with (52), restrict the range of possible n 0 n 0 n_(0)n_{0}.)
选择一个试验值 n 0 > 0 n 0 > 0 n_(0) > 0n_{0}>0 。（应当记住，不等式 v n 0 ≧ 0 v n 0 ≧ 0 v_(n_(0)) >= 0v_{n_{0}} \geqq 0 可能会与（52）一起限制可能的 n 0 n 0 n_(0)n_{0} 范围。）
[3] Bearing in mind that y n 0 = 0 y n 0 = 0 y_(n_(0))=0y_{n_{0}}=0, the values of v n 0 v n 0 v_(n_(0))v_{n_{0}} and u n 0 u n 0 u_(n_(0))u_{n_{0}} are obtained from (49) and (52).
考虑到 y n 0 = 0 y n 0 = 0 y_(n_(0))=0y_{n_{0}}=0 ， v n 0 v n 0 v_(n_(0))v_{n_{0}} 和 u n 0 u n 0 u_(n_(0))u_{n_{0}} 的值是从(49)和(52)中获得的。
[4] The solution of equations (50) and (51) is calculated for increasing n n nn until either (56) fails to be satisfied, or it becomes apparent that (53) will not be satisfied (see [6] below).
方程（50）和（51）的解是通过增加 n n nn 来计算，直到（56）不再满足，或者明显（53）将不被满足（见下面的[6]）。
[5] If (56) fails to be satisfied, z n z n z_(n)z_{n} is kept constant, u n u n u_(n)u_{n} (and v n v n v_(n)v_{n} ) being calculated from (49) until (56) is satisfied again, when z n z n z_(n)z_{n} is allowed to increase and the solution pursued as in [4].
如果（56）未能满足，则保持 z n z n z_(n)z_{n} 不变， u n u n u_(n)u_{n} （和 v n v n v_(n)v_{n} ）根据（49）进行计算，直到（56）再次满足，此时允许 z n z n z_(n)z_{n} 增加，并按照[4]中的方法继续求解。
[6] The attempted solution should be stopped if u n u n u_(n)u_{n} or x n x n x_(n)x_{n} begins to decrease, or v n v n v_(n)v_{n} or y n y n y_(n)y_{n} fall to zero, or x n , y n x n , y n x_(n),y_(n)x_{n}, y_{n} cannot be calculated (e.g. because u n u n u_(n)u_{n} exceeds the upper bound of u u uu, if there is one). Other stopping rules can be given for particular examples, depending on the structure of the solutions of the equations.
如果 u n u n u_(n)u_{n} 或 x n x n x_(n)x_{n} 开始减少，或者 v n v n v_(n)v_{n} 或 y n y n y_(n)y_{n} 降至零，或者 x n , y n x n , y n x_(n),y_(n)x_{n}, y_{n} 无法计算（例如，因为 u n u n u_(n)u_{n} 超过了 u u uu 的上限，如果有的话），则应停止尝试的解决方案。根据方程解的结构，可以为特定示例提供其他停止规则。
[7] A range of trial values of n 0 n 0 n_(0)n_{0} must be used to find the one that most nearly provides a solution satisfying (53). Efficient rules for iteration might be obtained in particular cases.
[7] 必须使用一系列试验值 n 0 n 0 n_(0)n_{0} 来找到最接近满足(53)的解。在特定情况下，可能获得有效的迭代规则。

7. FEATURES OF SOLUTIONS
7. 溶液的特征

Solutions may, for all I know, be very diverse in their characteristics; but examination of the equations suggests a number of comments. First we note that v n v n v_(n)v_{n} will always lie between 0 and 1 ψ y ( 0 ) 1 ψ y ( 0 ) (1)/(psi_(y)(0))\frac{1}{\psi_{y}(0)}, since
解决方案可能在特征上非常多样；但对方程的检查提出了一些评论。首先，我们注意到 v n v n v_(n)v_{n} 将始终位于 0 和 1 ψ y ( 0 ) 1 ψ y ( 0 ) (1)/(psi_(y)(0))\frac{1}{\psi_{y}(0)} 之间，因为
0 ≦ 1 + u 2 n u 1 ψ y ≦ 1 + u 2 n u 1 ψ y ( 0 ) < 1 ψ y ( 0 ) 0 ≦ 1 + u 2 n u 1 ψ y ≦ 1 + u 2 n u 1 ψ y ( 0 ) < 1 ψ y ( 0 ) 0 <= (1+(u_(2))/(nu_(1)))/(psi_(y)) <= (1+(u_(2))/(nu_(1)))/(psi_(y)(0)) < (1)/(psi_(y)(0))0 \leqq \frac{1+\frac{u_{2}}{n u_{1}}}{\psi_{y}} \leqq \frac{1+\frac{u_{2}}{n u_{1}}}{\psi_{y}(0)}<\frac{1}{\psi_{y}(0)}
We are therefore led to expect that v v vv tends to a limit as n → ∞ n → ∞ n rarr oon \rightarrow \infty. (It might cycle for certain forms of f f ff, of a kind one would perhaps be unlikely to use.) y y yy is also bounded, by 0 and 1 , and is therefore likely to tend to a limit. One is then led to certain conjectures about the limits, which ought to hold for sufficiently regular f f ff and u u uu.
因此，我们可以预期 v v vv 在 n → ∞ n → ∞ n rarr oon \rightarrow \infty 时趋向于一个极限。（对于某些形式的 f f ff ，它可能会循环，这种情况可能不太可能使用。） y y yy 也被 0 和 1 所界定，因此很可能趋向于一个极限。由此可以得出关于极限的某些猜想，这些猜想应该适用于足够规则的 f f ff 和 u u uu 。
Let 让
n f ′ f → γ + 2 ≦ ∞ n f ′ f → γ + 2 ≦ ∞ (nf^('))/(f)rarr gamma+2 <= oo\frac{n f^{\prime}}{f} \rightarrow \gamma+2 \leqq \infty
(Since ∫ 0 ∞ n f d n < ∞ , γ ≧ 0 ∫ 0 ∞   n f d n < ∞ , γ ≧ 0 int_(0)^(oo)nfdn < oo,gamma >= 0\int_{0}^{\infty} n f d n<\infty, \gamma \geqq 0 : otherwise n 2 f n 2 f n^(2)fn^{2} f is increasing for large n n nn, therefore bounded below.) Further, suppose
（由于 ∫ 0 ∞ n f d n < ∞ , γ ≧ 0 ∫ 0 ∞   n f d n < ∞ , γ ≧ 0 int_(0)^(oo)nfdn < oo,gamma >= 0\int_{0}^{\infty} n f d n<\infty, \gamma \geqq 0 ：否则 n 2 f n 2 f n^(2)fn^{2} f 在大 n n nn 时是递增的，因此有下界。）进一步假设
u 1 ∼ α x − μ ( μ > 0 ) u 1 ∼ α x − μ ( μ > 0 ) u_(1)∼alphax^(-mu)quad(mu > 0)u_{1} \sim \alpha x^{-\mu} \quad(\mu>0)
as x → ∞ x → ∞ x rarr oox \rightarrow \infty. Then there appear to be three cases; in each of which one expects the following results to hold.
作为 x → ∞ x → ∞ x rarr oox \rightarrow \infty 。然后似乎出现了三种情况；在每种情况下，人们期望以下结果成立。
(i) μ < 1 μ < 1 mu < 1\mu<1. As n → ∞ n → ∞ n rarr oon \rightarrow \infty,
(i) μ < 1 μ < 1 mu < 1\mu<1 。作为 n → ∞ n → ∞ n rarr oon \rightarrow \infty ，
y n → 1 y n → 1 y_(n)rarr1y_{n} \rightarrow 1
and v n → 0 v n → 0 quadv_(n)rarr0\quad v_{n} \rightarrow 0. 和 v n → 0 v n → 0 quadv_(n)rarr0\quad v_{n} \rightarrow 0 。
The marginal tax rate,
边际税率，
θ → 1 θ → 1 theta rarr1\theta \rightarrow 1
(ii) μ = 1 μ = 1 mu=1\mu=1. As n → ∞ n → ∞ n rarr oon \rightarrow \infty,
(ii) μ = 1 μ = 1 mu=1\mu=1 。作为 n → ∞ n → ∞ n rarr oon \rightarrow \infty ，
y n → y ¯ , y n → y ¯ , y_(n)rarr bar(y),y_{n} \rightarrow \bar{y},
where y ¯ y ¯ bar(y)\bar{y} is defined (uniquely) by
其中 y ¯ y ¯ bar(y)\bar{y} 被（唯一）定义为
y ¯ u 2 ( y ¯ ) = − α y ¯ u 2 ( y ¯ ) = − α bar(y)u_(2)( bar(y))=-alpha\bar{y} u_{2}(\bar{y})=-\alpha
and v n → [ − ( 1 + γ ) u 2 ( y ¯ ) − y ¯ u 22 ( y ¯ ) ] − 1 v n → − ( 1 + γ ) u 2 ( y ¯ ) − y ¯ u 22 ( y ¯ ) − 1 quadv_(n)rarr[-(1+gamma)u_(2)(( bar(y)))-( bar(y))u_(22)(( bar(y)))]^(-1)\quad v_{n} \rightarrow\left[-(1+\gamma) u_{2}(\bar{y})-\bar{y} u_{22}(\bar{y})\right]^{-1}. 和 v n → [ − ( 1 + γ ) u 2 ( y ¯ ) − y ¯ u 22 ( y ¯ ) ] − 1 v n → − ( 1 + γ ) u 2 ( y ¯ ) − y ¯ u 22 ( y ¯ ) − 1 quadv_(n)rarr[-(1+gamma)u_(2)(( bar(y)))-( bar(y))u_(22)(( bar(y)))]^(-1)\quad v_{n} \rightarrow\left[-(1+\gamma) u_{2}(\bar{y})-\bar{y} u_{22}(\bar{y})\right]^{-1} 。
Furthermore, 此外，
θ → 1 + v 1 + v + γ θ → 1 + v 1 + v + γ theta rarr(1+v)/(1+v+gamma)\theta \rightarrow \frac{1+v}{1+v+\gamma}
where 哪里
(iii) μ > 1 μ > 1 mu > 1\mu>1. As n → ∞ n → ∞ n rarr oon \rightarrow \infty,
(iii) μ > 1 μ > 1 mu > 1\mu>1 。如 n → ∞ n → ∞ n rarr oon \rightarrow \infty 所示，
v = y ¯ u 22 ( y ¯ ) u 2 ( y ) v = y ¯ u 22 ( y ¯ ) u 2 ( y ) v=(( bar(y))u_(22)(( bar(y))))/(u_(2)(y))v=\frac{\bar{y} u_{22}(\bar{y})}{u_{2}(y)}
y n → 0 , and v n → [ − ( 1 + γ ) u 2 ( 0 ) ] − 1 . θ → 1 1 + γ . y n → 0 ,  and  v n → − ( 1 + γ ) u 2 ( 0 ) − 1 .  θ → 1 1 + γ . {:[y_(n)rarr0","],[" and "quadv_(n)rarr[-(1+gamma)u_(2)(0)]^(-1)". "],[theta rarr(1)/(1+gamma).]:}\begin{aligned} & y_{n} \rightarrow 0, \\ & \text { and } \quad v_{n} \rightarrow\left[-(1+\gamma) u_{2}(0)\right]^{-1} \text {. } \\ & \theta \rightarrow \frac{1}{1+\gamma} . \end{aligned}
(It may be noted that, in a natural sense, (66) holds for all cases.)
（需要注意的是，从自然意义上讲，（66）适用于所有情况。）
Before indicating the reasons for these conjectures, a few words of interpretation may be in place. On the whole, the distribution of income from employment appears to be of Paretian form at the upper tail 1 1 ^(1){ }^{1} : Equation (58) holds with γ γ gamma\gamma between 1 and 2, roughly speaking. It is not improbable, however, that marginal productivity per working year is distributed differently from actual incomes: the lognormal distribution is the most plausible simple distribution. For this, γ = ∞ γ = ∞ gamma=oo\gamma=\infty, and
在指出这些推测的原因之前，可能需要几句解释。总体而言，来自就业的收入分布在上尾部似乎呈现帕累托形式 1 1 ^(1){ }^{1} ：方程（58）在 γ γ gamma\gamma 介于 1 和 2 之间时大致成立。然而，边际生产力每工作年分布与实际收入可能不同并非不可能：对数正态分布是最合理的简单分布。为此， γ = ∞ γ = ∞ gamma=oo\gamma=\infty ，并且
− n f ′ f ∼ log ⁡ n σ 2 − n f ′ f ∼ log ⁡ n σ 2 -(nf^('))/(f)∼(log n)/(sigma^(2))-\frac{n f^{\prime}}{f} \sim \frac{\log n}{\sigma^{2}}
for large n n nn （ σ 2 σ 2 sigma^(2)\sigma^{2} is the variance of the distribution of logarithm of incomes）.
对于大 n n nn （ σ 2 σ 2 sigma^(2)\sigma^{2} 是收入对数分布的方差）。
The realism of alternative assumptions about utility may be assessed by calculating the response of the consumer to a linear budget constraint, x = w y + a x = w y + a x=wy+ax=w y+a. It is easy to see that utility-maximization requires (since u 12 ≡ 0 u 12 ≡ 0 u_(12)-=0u_{12} \equiv 0 )
关于效用的替代假设的现实性可以通过计算消费者对线性预算约束的反应来评估， x = w y + a x = w y + a x=wy+ax=w y+a 。很容易看出，效用最大化要求（因为 u 12 ≡ 0 u 12 ≡ 0 u_(12)-=0u_{12} \equiv 0 ）
− u 1 ( x ) u 2 ( y ) = 1 w , x = w y + a . − u 1 ( x ) u 2 ( y ) = 1 w , x = w y + a . -(u_(1)(x))/(u_(2)(y))=(1)/(w),quad x=wy+a.-\frac{u_{1}(x)}{u_{2}(y)}=\frac{1}{w}, \quad x=w y+a .
If u 1 = α x − μ u 1 = α x − μ u_(1)=alphax^(-mu)u_{1}=\alpha x^{-\mu}, we have to solve
如果 u 1 = α x − μ u 1 = α x − μ u_(1)=alphax^(-mu)u_{1}=\alpha x^{-\mu} ，我们必须解决
α w = − ( a + w y ) μ u 2 ( y ) α w = − ( a + w y ) μ u 2 ( y ) alpha w=-(a+wy)^(mu)u_(2)(y)\alpha w=-(a+w y)^{\mu} u_{2}(y)
(If α w ≦ − a μ u 2 ( 0 ) , y = 0 α w ≦ − a μ u 2 ( 0 ) , y = 0 alpha w <= -a^(mu)u_(2)(0),y=0\alpha w \leqq-a^{\mu} u_{2}(0), y=0.) Clearly the solution has the following properties:
（如果 α w ≦ − a μ u 2 ( 0 ) , y = 0 α w ≦ − a μ u 2 ( 0 ) , y = 0 alpha w <= -a^(mu)u_(2)(0),y=0\alpha w \leqq-a^{\mu} u_{2}(0), y=0 。）显然，解具有以下特性：
y → 1 as w → ∞ if μ < 1 , y → 0 as w → ∞ if μ > 1 . } y → 1  as  w → ∞  if  μ < 1 , y → 0  as  w → ∞  if  μ > 1 . {:[y rarr1" as "w rarr oo" if "mu < 1","],[y rarr0" as "w rarr oo" if "mu > 1.]}\left.\begin{array}{l} y \rightarrow 1 \text { as } w \rightarrow \infty \text { if } \mu<1, \\ y \rightarrow 0 \text { as } w \rightarrow \infty \text { if } \mu>1 . \end{array}\right\}
(Cf. (61) and (68).) Also
（参见（61）和（68）。）此外
x ∼ a + w ( μ < 1 ) x ∼ ( a w u 2 ( 0 ) ) 1 μ ( μ > 1 ) } x ∼ a + w      ( μ < 1 ) x ∼ a w u 2 ( 0 ) 1 μ      ( μ > 1 ) {:[x∼a+w,(mu < 1)],[x∼((aw)/(u_(2)(0)))^((1)/(mu)),(mu > 1)]}\left.\begin{array}{ll} x \sim a+w & (\mu<1) \\ x \sim\left(\frac{a w}{u_{2}(0)}\right)^{\frac{1}{\mu}} & (\mu>1) \end{array}\right\}
These asymptotic properties suggest that the case μ = 1 μ = 1 mu=1\mu=1 is particularly interesting.
这些渐近性质表明情况 μ = 1 μ = 1 mu=1\mu=1 特别有趣。
When μ = 1 μ = 1 mu=1\mu=1, since, by (73)
当 μ = 1 μ = 1 mu=1\mu=1 时，由于（73）
a w = − α u 2 − y − y u 2 → α as w → ∞ a w = − α u 2 − y − y u 2 → α  as  w → ∞ {:[(a)/(w)=-(alpha)/(u_(2))-y],[-yu_(2)rarr alpha" as "w rarr oo]:}\begin{gathered} \frac{a}{w}=-\frac{\alpha}{u_{2}}-y \\ -y u_{2} \rightarrow \alpha \text { as } w \rightarrow \infty \end{gathered}
i.e. 即
y → y ¯ , y → y ¯ , y rarr bar(y),y \rightarrow \bar{y},
where y ¯ y ¯ bar(y)\bar{y} is defined by (64). (Cf. (63).) If in addition,
其中 y ¯ y ¯ bar(y)\bar{y} 由 (64) 定义。（参见 (63)。）如果此外，
we have 我们有
u 2 ( y ) = − ( 1 − y ) − δ ( δ > 0 ) u 2 ( y ) = − ( 1 − y ) − δ ( δ > 0 ) u_(2)(y)=-(1-y)^(-delta)quad(delta > 0)u_{2}(y)=-(1-y)^{-\delta} \quad(\delta>0)
y ¯ ( 1 − y ¯ ) − δ = α y ¯ ( 1 − y ¯ ) − 1 = v y ¯ ( 1 − y ¯ ) − δ = α y ¯ ( 1 − y ¯ ) − 1 = v {:[ bar(y)(1- bar(y))^(-delta)=alpha],[ bar(y)(1- bar(y))^(-1)=v]:}\begin{aligned} & \bar{y}(1-\bar{y})^{-\delta}=\alpha \\ & \bar{y}(1-\bar{y})^{-1}=v \end{aligned}
The choice of α α alpha\alpha may be influenced by considering that y = 0 y = 0 y=0y=0 when w / a ≦ 1 / α w / a ≦ 1 / α w//a <= 1//alphaw / a \leqq 1 / \alpha. It is interesting to note that, if
选择 α α alpha\alpha 可能受到考虑 y = 0 y = 0 y=0y=0 时 w / a ≦ 1 / α w / a ≦ 1 / α w//a <= 1//alphaw / a \leqq 1 / \alpha 的影响。有趣的是，如果
α = 2 , δ = 1 , γ = 2 , y ¯ = 2 / 3 , v = 2 , α = 2 , δ = 1 , γ = 2 , y ¯ = 2 / 3 , v = 2 , {:[alpha=2","quad delta=1","quad gamma=2","],[ bar(y)=2//3","quad v=2","]:}\begin{aligned} & \alpha=2, \quad \delta=1, \quad \gamma=2, \\ & \bar{y}=2 / 3, \quad v=2, \end{aligned}
and, if our conjectures are correct,
如果我们的推测是正确的，
θ → 60 per cent. θ → 60  per cent.  theta rarr60" per cent. "\theta \rightarrow 60 \text { per cent. }
This case is perhaps not completely unrealistic; but it should be remembered that the homogeneous form for u u uu means that the decision not to work depends only on the ratio of earned to unearned income, which is not a very realistic assumption.
这个案例或许并非完全不现实；但应当记住，对于 u u uu 的均匀形式意味着不工作的决定仅依赖于赚取收入与未赚取收入的比例，这并不是一个非常现实的假设。
It will be noticed that, in this case, the asymptotic marginal tax rate is very sensitive to the value of μ μ mu\mu (in the neighbourhood of 1 ).
需要注意的是，在这种情况下，渐近边际税率对 μ μ mu\mu 的值（在 1 附近）非常敏感。
The reasons for the conjectures Equations (60)-(70) (in fact, I can provide a proof of (iii) and will do so below) are as follows. One expects that, as n → ∞ n → ∞ n rarr oon \rightarrow \infty, the relevant solution of the differential equations will tend towards a singularity of the equations: not only will y y yy and v v vv tend to limits, but n d y d n n d y d n n(dy)/(dn)n \frac{d y}{d n} and n d v d n n d v d n n(dv)/(dn)n \frac{d v}{d n} will tend to zero. Denote the postulated limit of y n y n y_(n)y_{n} by y ¯ y ¯ bar(y)\bar{y}. Consider first the case u 1 = α x − μ ( μ < 1 ) u 1 = α x − μ ( μ < 1 ) u_(1)=alphax^(-mu)(mu < 1)u_{1}=\alpha x^{-\mu}(\mu<1).
关于猜想方程（60）-（70）的原因（实际上，我可以提供（iii）的证明，并将在下面进行）如下。人们期望，当 n → ∞ n → ∞ n rarr oon \rightarrow \infty 时，微分方程的相关解将趋向于方程的奇点：不仅 y y yy 和 v v vv 将趋向于极限，而且 n d y d n n d y d n n(dy)/(dn)n \frac{d y}{d n} 和 n d v d n n d v d n n(dv)/(dn)n \frac{d v}{d n} 将趋向于零。将假设的 y n y n y_(n)y_{n} 的极限记为 y ¯ y ¯ bar(y)\bar{y} 。首先考虑情况 u 1 = α x − μ ( μ < 1 ) u 1 = α x − μ ( μ < 1 ) u_(1)=alphax^(-mu)(mu < 1)u_{1}=\alpha x^{-\mu}(\mu<1) 。
  1. 1 1 ^(1){ }^{1} First version received Aug. 1970; final version received October 1970 (Eds.).
    1 1 ^(1){ }^{1} 第一版于 1970 年 8 月收到；最终版于 1970 年 10 月收到（编辑）。
    2 Work on this paper and its continuation was begun during a stimulating and pleasurable visit to the Department of Economics, M.I.T. The influence of Peter Diamond is particularly great, and his comments have been very useful. Earlier versions were presented at the Cowles Foundation, to the Economic Study Society, at the London School of Economics, and to CORE. I am grateful to the members of these seminars and to A. B. Atkinson for valuable comments. I am also greatly indebted to P. G. Hare and J. R. Broome for the computations.
    这篇论文及其后续研究的工作始于对麻省理工学院经济学系一次激动人心且愉快的访问。彼得·戴蒙德的影响尤为显著，他的评论非常有帮助。早期版本在考尔斯基金会、经济研究学会、伦敦经济学院和 CORE 进行了展示。我对这些研讨会的成员以及 A. B. 阿特金森的宝贵评论表示感谢。我还非常感谢 P. G. 哈雷和 J. R. 布鲁姆在计算方面的帮助。
    3 Discussions on (usually) orthodox lines, including many important points neglected in the present paper, can be found in [7], [1], [5, Chapters 5, 7, 8], and [6, Chapters 11 and 12]. [2] is close in spirit to what is attempted here.
    关于（通常）正统的讨论，包括本文中许多重要的被忽视的观点，可以在[7]、[1]、[5，第 5、7、8 章]和[6，第 11 和 12 章]中找到。[2]在精神上与这里所尝试的内容相近。
    4 4 ^(4){ }^{4} Cf. [7, Chapter 6].
    4 4 ^(4){ }^{4} 参见 [7，第 6 章]。
  2. 1 1 ^(1){ }^{1} The relation of optimum tax schedules to propensities to migrate is discussed in another paper under preparation.
    最佳税率与迁移倾向的关系将在另一篇正在准备的论文中讨论。
  3. 1 To say that c c cc is upper semi-continuous means that
    说 c c cc 是上半连续的意味着
    lim sup c ( z i ) = c ( z ) when lim i → ∞ z i = z lim sup c z i = c ( z )  when  lim i → ∞   z i = z lim s u p c(z_(i))=c(z)" when "lim_(i rarr oo)z_(i)=z\lim \sup c\left(z_{i}\right)=c(z) \text { when } \lim _{i \rightarrow \infty} z_{i}=z
    If 如果
    u n = sup { u ( x , y ) ∣ x ≦ c ( n y ) } , and u ( x i , y i ) → u n , x i ≦ c ( n y i ) u n = sup { u ( x , y ) ∣ x ≦ c ( n y ) } ,  and  u x i , y i → u n , x i ≦ c n y i u_(n)=s u p{u(x,y)∣x <= c(ny)}," and "u(x_(i),y_(i))rarru_(n),x_(i) <= c(ny_(i))u_{n}=\sup \{u(x, y) \mid x \leqq c(n y)\}, \text { and } u\left(x_{i}, y_{i}\right) \rightarrow u_{n}, x_{i} \leqq c\left(n y_{i}\right)
    we can suppose that x i → x x i → x x_(i)rarr xx_{i} \rightarrow x and y i → y y i → y y_(i)rarr yy_{i} \rightarrow y (since { y i } y i {y_(i)}\left\{y_{i}\right\} and therefore { x i } x i {x_(i)}\left\{x_{i}\right\} is bounded). By the upper semi-continuity of c c cc,
    我们可以假设 x i → x x i → x x_(i)rarr xx_{i} \rightarrow x 和 y i → y y i → y y_(i)rarr yy_{i} \rightarrow y （因为 { y i } y i {y_(i)}\left\{y_{i}\right\} ，因此 { x i } x i {x_(i)}\left\{x_{i}\right\} 是有界的）。根据 c c cc 的上半连续性，
    x ≦ lim sup c ( n y i ) = c ( n y ) x ≦ lim sup c n y i = c ( n y ) x <= lim s u p c(ny_(i))=c(ny)x \leqq \lim \sup c\left(n y_{i}\right)=c(n y)
    and by the continuity of u , u ( x , y ) = lim u ( x i , y i ) = u n u , u ( x , y ) = lim u x i , y i = u n u,u(x,y)=lim u(x_(i),y_(i))=u_(n)u, u(x, y)=\lim u\left(x_{i}, y_{i}\right)=u_{n}. Therefore the supremum is attained.
    因此， u , u ( x , y ) = lim u ( x i , y i ) = u n u , u ( x , y ) = lim u x i , y i = u n u,u(x,y)=lim u(x_(i),y_(i))=u_(n)u, u(x, y)=\lim u\left(x_{i}, y_{i}\right)=u_{n} 的连续性使得上确界得以实现。
    2 In other words, we have a correspondence, providing a set of utility maximizing choices for n n nn-men. It arises when the consumption function c c cc coincides with the indifference curve for part of its length. It is convenient nevertheless to use the notation of the text, despite its suggestion that we are dealing with a function.
    换句话说，我们有一个对应关系，为 n n nn -人提供了一组效用最大化的选择。当消费函数 c c cc 与无差异曲线在其部分长度上重合时，就会出现这种情况。尽管文本的符号暗示我们正在处理一个函数，但使用这种符号仍然是方便的。
    3 3 ^(3){ }^{3} It is easy to see that the result is true for a correspondence also.
    很容易看出，这个结果对于对应关系也是成立的。
  4. 1 The analysis and result can be generalized to the utility function u ( x , z , n ) u ( x , z , n ) u(x,z,n)u(x, z, n) where the parameter n n nn can indicate variations in tastes as well as skill. The extension is fairly routine and will not be discussed here.
    分析和结果可以推广到效用函数 u ( x , z , n ) u ( x , z , n ) u(x,z,n)u(x, z, n) ，其中参数 n n nn 可以表示口味和技能的变化。这个扩展相当常规，因此在这里不作讨论。
  5. 1 In [4] a theorem is proved which states that the conditions of Theorem 2 are in fact sufficient (as well as necessary) for an optimum in the special case now being considered.
    在[4]中证明了一个定理，该定理表明定理 2 的条件实际上对于当前考虑的特殊情况是充分（也是必要）的。
  6. 1 See the general assessment by Lydall [3].
    请参见 Lydall 的总体评估[3]。