公共支出纯理论Paul A. Samuelson英文原著及中文翻译

THE PURE THEORY OF PUBLIC EXPENDITURE
Paul A. Samuelson
1. Assumptions. Except for Sax, Wicksell,Lindahl, Musgrave, and Bowen, economists have rather neglected the theory of optimal public expenditure, spending most of their energy on the theory of taxation. Therefore, I explicitly assume two categories of goods: ordinary private consumption goods ()n X X X ,,21 which can be parcelled out among different individuals ()s i ,,,,2,1 according to the relations ∑=s
i i
j j X X and collective consumption goods ()m n n X X ++ ,1 which all enjoy in
common in the sense that each individual's consumption of such a good leads to no subtraction from any other individual's consumption of that good, so that
i j n j n X X ++= simultaneously for each and every i th individual and each collective
consumptive good. I assume no mystical collective mind that enjoys collective consumption goods; instead I assume each individual has a consistent set of ordinal preferences with respect to his consumption of all goods (collective as well as private) which can be summarized by a regularly smooth and convex utility index
()
i m n i i i X X u u +=,,1 (any monotonic stretching of the utility index is of course also an admissible cardinal index of preference). I shall throughout follow the convention of writing the partial derivative of any function with respect to its j th argument by a
j subscript, so that i
j i
i j
X u u ∂∂= etc. Provided economic quantities can be divided into two groups, (1) outputs or goods which everyone always wants to maximize and
(2) inputs or factors which everyone always wants to minimize, we are free to change the algebraic signs of the latter category and from then on to work only with "goods," knowing that the case of factor inputs is covered as well. Hence by this convention we are sure that 0 i
j u always.
To keep production assumptions at the minimum level of simplicity, I assume a regularly convex and smooth production-possibility schedule relating totals of all
outputs, private and collective; or ()0,,1=+m n X X F , with 0 j F 、and ratios n j
F F determinate and subject to the generalized laws of diminishing returns.
Feasibility considerations disregarded, there is a maximal (ordinal) utility frontier representing the Pareto-optimal points —of which there are an (s —I) fold infinity —with the property that from such a frontier point you can make one person better off only by making some other person worse off. If we wish to make normative judgments concerning the relative ethical desirability of different configurations involving some individuals being on a higher level of indifference and some on a lower, we must be presented with a set of ordinal interpersonal norms or with a social welfare function representing a consistent set of ethical preferences among all the possible states of the system. It is not a "scientific" task of the economist to "deduce" the form of this function; this can have as many forms as there are possible ethical views; for the present purpose, the only restriction placed on the social welfare function is that it shall always increase or decrease when any one person's ordinal preference increases or decreases, all others staying on their same indifference levels: mathematically, we narrow it to the class that any one of its indexes can be written ()s u u u U U ,,,21 = with 0 j U .
2. Optimal Conditions. In terms of these norms, there is a "best state of the world" which is defined mathematically in simple regular cases by the marginal conditions
r j i
r i
j
F F u u = ()n j r s i ,2,1,;,,2,1== or ()n j r s i ,2;1;,,2,1=== （1）
r j n s i i
r i j n F F u u +=+=∑1 ()n r m j ,2,1;,,2,1== or ()1;,,2,1==r s j （2）
1=q k
q i k i u U u U ()n k s q i ,2,1;,,2,1,== or ()1;,,2;1===k s i q （3） Equations (1) and (3) are essentially those given in the chapter on welfare economics in my Foundations of Economic Analysis. They constitute my version of
the "new welfare economics." Alone (1) represents that subset of relations which defines the Pareto-optimal utility frontier and which by itself represents what I regard as the unnecessarily narrow version of what once was called the "new welfare economics."
The new element added here is the set (2), which constitutes a pure theory of government expenditure on collective consumption goods. By themselves(1)and
(2)define the (s —Ι) fold infinity of utility frontier points; only when a set of interpersonal normative conditions equivalent to (3) is supplied are we able to define an unambiguously "best" state.
Since formulating the conditions (2)some years ago, I have learned from the published and unpublished writings of Richard Musgrave that their essential logic is contained in the "voluntary-exchange" theories of public finance of the Sax-Wicksell-Lindahl-Musgrave type, and I have also noted Howard Bowen's independent discovery of them in Bowen's writings of a decade ago. A graphical interpretation of these conditions in terms of vertical rather than horizontal addition of different individuals' marginal-rate-of-substitution schedules can be given; but what I must emphasize is that there is a different such schedule for each individual at each of the (s —Ι)fold infinity of different distributions of relative welfare along the utility frontier.
3. Impossibility of decentralized spontaneous solution. So much for the involved optimizing equations that an omniscient calculating machine could theoretically solve if fed the postulated functions. No such machine now exists. But it is well known that an "analogue calculating machine" can be provided by competitive market pricing, (a) so long as the production functions satisfy the neoclassical assumptions of constant returns to scale and generalized diminishing returns and (b) so long as the individuals' indifference contours have regular convexity and, we may add, (c) so long as all goods are private. We can then insert between the right- and left-hand sides of (Ι) the equality with uniform market prices
r j p p and adjoin the
budget equations for each individual
i i n n i L X p X p =++ 11 ()s i ,,2,1 = （1）1
where L' is a lump-sum tax for each individual so selected in algebraic value as to lead to the "best" state of the world. Now note, if there were no collective consumption goods, then (Ι) and (r)‘can have their solution enormously simplified. Why? Because on the one hand perfect competition among productive enterprises would ensure that goods are produced at minimum costs and are sold at proper marginal costs, with all factors receiving their proper marginal productivities; and on the other hand, each individual, in seeking as a competitive buyer to get to the highest level of indifference subject to given prices and tax, would be led as if by an Invisible Hand to the grand solution of the social maximum position. Of course the institutional framework of competition would have to be maintained, and political decision making would still be necessary, but of a computationally minimum type, namely, algebraic taxes and transfers ()s L L L ,,,21 would have to be varied until society is swung to the ethical observer's optimum. The servant of the ethical observer would not have to make explicit decisions about each. person's detailed consumption and work; he need only decide about generalized purchasing power, knowing that each person can be counted on to allocate it optimally. In terms of communication theory and game terminology, each person is motivated to do the signalling of his tastes needed to define and reach the attainable-bliss point.
Now all of the above remains valid even if collective consumption is not zero but is instead explicitly set at its optimum values as determined by (1)，(2)，and
(3).However no decentralized pricing system can serve to determine optimally these Levels of collective consumption. Other kinds of "voting" or "signalling" would have to be tried. But, and this —is the point sensed by Wicksell but perhaps not fully appreciated by Lindahl, now it is in the selfish interest of each person to give false signals, to pretend to have less interest in a given collective consumption activity than he really has, etc. I must emphasize this: taxing according to a benefit theory of taxation can not at all solve the computational problem in the decentralized manner possible for the first category of "private" goods to which the ordinary market
pricing applies and which do not have the "external effects" basic to the very notion of collective consumption goods. Of course, utopian voting and signaling schemes can be imagined. ("Scandinavian consensus," Kant's "categorical imperative," and other devices meaningful only under conditions of "symmetry," etc.) The failure of market catallactics in no way denies the following truth: given sufficient knowledge the optimal decisions can always be found by scanning over all the attainable states of the world and selecting the one which according to the postulated ethical welfare function is best. The solution"exists"; the problem is how to "find" it.
One could imagine every person in the community being indoctrinated to behave like a "parametric decentralized bureaucrat" who reveals his preferences by signalling in response to price parameters or Lagrangean multipliers, to questionnaires, or to other devices. But there is still this fundamental technical difference going to the heart of the whole problem of social economy: by departing from his indoctrinated rules, any one person can hope to snatch some selfish benefit in a way not possible under the self-policing competitive pricing of private goods; and the "external economies" or " jointness of demand" intrinsic to the very concept of collective goods and governmental activities makes it impossible for the grand ensemble of optimizing equations to have that special pattern of zeros which makes laissez-faire competition even theoretically possible as an analogue computer.
4. Conclusion. To explore further the problem raised by public expenditure would take us into the mathematical domain of "sociology" or "welfare politics," which Arrow, Duncan Black, and others have just begun to investigate. Political economy can be regarded as one special sector of this general domain, and it may turn out to be pure luck that within the general domain there happened to be a subsector with the "simple" properties of traditional economics.
汉语翻译
公共支出纯理论
1、假设：除了萨克斯、维克塞尔、林达尔、马斯格雷夫和鲍恩,经济学家们宁愿忽视最优公共支出理论，而将他们的大量精力用于税收理论。

因此，我明确的假设存在两类物品：g 一类是通常的私人消费品()n X X X ,,21，这些私人消费品以∑=s
i i
j j X X 的关系在不同的个人()s i ,,,,2,1 之间分配；另一类物品是公
共消费品()m n n X X ++ ,1，这类物品所有人共同使用，从某种意义上说，这类物品的任一个个人的消费不会减少其它个人对这类物品的消费，以致于对于任何第
i 个个体和每单位共同消费物品同时满足的i j n j n X X ++=。

我假设不存在使用公共消费品的神秘的集体思想。

相反，我假设每个个体对于他所消费的所有物品（公共品或私人品）都有着系列稳定的一致性偏好，这种偏好可以通过一个规则的光
滑的凸效用指数()i m n i i i X X u u +=,,1 加总（效用指数的一些单调性趋势当然也是
适用于偏好的重要指标）。

我将完全按照用下标j 表示第j 个幅角的表示一些函
数的偏导数的传统，从而有i
j i
i j
X u u ∂∂=等等。

如果经济学中的物品量能被分成两组：（1）任何人通常想要最大化的产出或物品和（2）任何人通常想要最小化的投入或要素，我们便可自由的改变后一类物品的代数符号，从此仅使用“物品”来描述，但同时要明确要素投入的情况也被包含在内。

因此，按照传统，我们通常规定0 i
j u 。

为了使关于产品的假设尽可能简单，我假设一条涉及所有产出量（私人或公共）的规则的凸的光滑的生产可能性曲线；或者是()0,,1=+m n X X F ，其满足0 j F 、比率n j
F F 固定并受限于收益递减的一般性规则。

不考虑可行性，这里有一条代表帕累托最优点的效用最大化边界—这是一个（s-1）重无限（无限大）—这些点具有这样的特性，即仅能通过使其他人变差才能使某人变好。

如果我们希望作出关于不同配置下价值意愿的规范判断，这些配置涉及到一些个体处于较高的无差异水平而另一些处于较低的无差异水平，我们必须提出一系列关于人与人之间关系的规则或者在所有可能的系统状态中代
表一致的价值偏好的社会福利函数。

推导这个函数的形式不是经济学家从事的科学性的工作；并且它将有着与可能的价值观点相同数量的模式；对于目前的目标，对社会福利函数唯一的限制是：当任何一个人的一般性偏好增加或减少，而其他人的依然保持相同的无差异水平时，社会福利函数也会增加或减少。

数学上，我们将其局限为这样的类型，那就是它所有指数的任何一个能被写成()
s u u u U U ,,,21 =，同时0 j U 。

2、最优化条件。

在这些模式方面，存在着一个“世界的最优状态”，数学上，这用通常的边际条件情况来定义。

r j i
r i
j
F F u u = ()n j r s i ,2,1,;,,2,1==或者()n j r s i ,2;1;,,2,1=== （1）
r j n s i i
r i j n F F u u +=+=∑1 ()n r m j ,2,1;,,2,1==或者()1;,,2,1==r s j
（2）
1=q k
q i k i u U u U ()n k s q i ,2,1;,,2,1,==或()1;,,2;1===k s i q （3） 方程（1）和（3）在我的《经济分析基础》中的福利经济学一章中被给出了。

它们构成了我 “新福利经济学”的观点。

单独的（1）式代表了关系集合，这种关系集合定义了帕累托最优的效用边际，同时其自身也代表了我视为叫做“新福利经济学”的不必要的狭窄的观点。

在这里，新加入的因素是集合（2），其构成了关于公共消费品的政府支出纯理论。

通过这些公共消费品，（1）和（2）定义了效用边界点的（s-1）重无限;仅仅当一系列相当于式（3）的人与人之间关系的规范条件被给定时，我们才能定义一个模糊的“最优”状态。

自从几年前系统地陈述过条件（2）以来，我从Richard 发表的或未发表的著作中了解到其本质的逻辑包含在萨克斯的公共财政学的“意愿交换”理论中，我也在十年前鲍恩的著作中笔记过鲍恩对这些问题的独立研究。

能够得到关于垂直而不是水平加总不同个体的边际替代率曲线方面的这方面情况的图表理解，但我必须强调的是，在沿着效用边界的相关福利的不同贡献的每个（s-1）重无限，对于每个人存在不同的这种曲线。

3、分散性自发解决的不可能性。

如果满足公理性方程，其涉及的优化方程式将如此之多以致于一台无所不知的计算机也只是在理论上能解决。

事实上，这样的计算机并不存在。

但是，众所周知的是，市场价格能够提供一台“类似的计算机”，（a ）只要生产函数满足规模报酬不变和一般收益递减的新古典主义假设，（b ）只要个人的无差异曲线是规则的凸的。

同时，（c ）只要所有的物品都是私人的。

我们此时能在等式（1）式的左右两边插入不变的市场价格
r j p p ，同时引入
对每个人的预算方程。

i i n n i L X p X p =++ 11 ()s i ,,2,1 = （1）1. 其中i L 对于每个个体而言是一次性赋税，如此在代数学意义的选择将导致世界的“最优”状态。

现在要注意的是，如果没有公共消费品存在，此时，（1）和（1）1能使它们的解决极大的简化。

为什么呢？因为，一方面生产性厂商间的完全竞争将保证物品以最小的成本生产，以合适的边际成本出售，同时所有的要素获得它们合适的边际生产率；另一方面，每个个人，作为一个竞争性的买主试图在给定价格和税收的情况下得到最大水平的无差异水平，其好像被一只无形的手引导来实现社会最优的状态。

当然，必须保持竞争的制度结构，同时政治决策仍然是必要的，但其应是一种可计算的最少的方式。

换句话说，代数的税收和转移支付()s L L L ,,,21 将必须被改变，直到社会达到道德观测者眼中的最优状态。

道德观测者的仆人将不必做出关于每个人具体消费和工作的明确的决定，他仅仅需要决定一般性的购买力，知道每个人能被指望最优化的配置购买力。

在信息理论和博奕论方面，每个人通过被激励来显示需要确定的偏好信息并达到可获得的福利点。

现在，即使公共消费品不为零，但是能设定在满足（1）（2）（3）式的最大化价值上，上面的所有论述将都是有效的。

然而，分散的价格体系并不利于形成最优化的公共物品消费水平。

必须使用其他的“投票”或“信息显示”的方式。

但是，这是维克塞尔意味的点，但可能不会被林达尔完全的理解，例如，个人出于自身的利益而给出错误的信息，假装对提供的公共物品的兴趣比真实的少等等。

我必须强调的是：按照税收受益理论的征税根本不能解决第一类“私人”
品可能的分散性方式的计算问题，“私人”品通常的市场价格体系能够提供并不会对公共消费品的真实的主张产生“外部影响”的基础。

当然，理想化的投票和信息显示制度能够被想像（“斯堪的纳维亚共识”,康德的“绝对命令”,和其他设备有意义的条件下只有“对称”,等等）。

市场交易学的失败没有办法否定以下事实：假设存在足够的知识，最优化的决定通常是通过搜索所有可能取得的市场状态并选择按照要求的道德福利函数是最好的状态来实现。

你能想像社会中的每个人都被教导成像“参量分散的官僚”一样行动，这种人通过对价格作出反映的信号，或对调查表及其它设备的拉格朗日乘数来显示他的偏好。

但是，对于社会经济的整个问题的核心仍然存在基础性的技术区别：通过违背他被教导的规则，任何一个人都希望能以一种在私人品的自制性竞争价格体系中不可能的方式来获得一些个人利益；同时，涉及公共物品和政府行为概念本质的“外部经济性”和“联合需求”，对于全部最优化方程，使那些特殊的零等式变得没有可能，而这种模式使自由放任的竞争至少在理论上可能作为一台相似的电脑。

4、结论
要进一步研究研究由公共支出产生的问题，将使我们进入一个“社会学”和“福利政治学”的数学领域，在这些领域，艾罗，邓肯·布莱克等人已经开始进行了研究.政治经济学能被作为这个一般领域的特殊部分，在这个一般性的领域中，可能碰巧有一个涉及传统经济学“简单”性质的次级部分，这将绝对是一件幸运的事。