THE PURE THEORY OF PUBLIC EXPENDITURE
Paul A. Samuelson
1. Assumptions. Except for Sax, Wickll,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 which can be parcelled out among different individualsaccording to the relations and collective consumption goods which all enjoy in common in the n that each individual's consumption of such a good leads to no subtraction from any other individual's consumption of that good, so that simultaneously for each and every 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 t 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 (any monotonic stretching of the utility index is of cour also an admissible cardinal index of preference). I shall throughout follo
w the convention of writing the partial derivative of any function with respect to its th argument by a subscript, so that 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 ca of factor inputs is covered as well. Hence by this convention we are sure that 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; 避孕环多少钱or, with 、and ratios determinate and subject to the generalized laws of diminishing returns.
Feasibility considerations disregarded, there is a maximal (ordinal) utility frontier reprenting 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 wor 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 prented with a t of ordinal interpersonal norms or with a social welfare function reprenting a consistent t 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 prent purpo, the only 匀速直线运动公式restriction placed on the social welfare function is that it shall always increa or decrea 雨的四季朗读when any one person's ordinal preference increas or decreas, all others staying on their same indifference levels: mathematically, we excellent是什么意思narrow it to the class that any one of its indexes can be written with .
2. Optimal Conditions. In terms of the norms, there is a "best state of the world" which
is defined mathematically in simple regular cas by the marginal conditions
or （1）
or （2）思想工作 or （3）
Equations (1) and (3) are esntially tho given in the chapter on welfare economics in my Foundations of Economic Analysis. They constitute my version of the "new welfare economics." Alone (1) reprents that subt of relations which defines the Pareto-optimal utility frontier and which by itlf reprents what I regard as the unnecessarily narrow version of what once was called the "new welfare economics."
The new element added here is the t (2), which constitutes a pure theory of government expenditure on collective consumption goods. By themlves(1)and (2)define the (s—Ι) fold infinity of utility frontier points; only when a t of interpersonal n
ormative 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 esntial logic is contained in the "voluntary-exchange" theories of public finance of the Sax-Wickll-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 the 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 辣椒炒茄子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 inrt between the right- and left-hand sides of (Ι) the equality with uniform market prices and adjoin the budget equations for each individual
（1）1
where L' is a lump-sum tax for each individual so lected 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? Becau on the one hand perfect competition among productive enterpris 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 eking as a competitive福州游玩攻略 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 cour 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 would have to be varied until society is swung to the ethical obrver's optimum. The rvant of the ethical obrver 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.
