On the Theory of Effective Demand
On the Theory of Effective Demand
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论有效需求理论
DOI：
10.2307/2232265
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发表时间：
1982
期刊：
影响因子：
--
通讯作者：
G. Weinrich
中科院分区：
文献类型：
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作者：
G. Weinrich
在他的文章《论有效需求理论》（Green, 1980）中，J. Green研究了一类随机配给方案，这些方案的分配由个体代理人感知，仅取决于他自己的行为和需求和供给的总价值。在某些假设下，假设有三个或更多的智能体，这种配给函数的期望在每个智能体自己的行为中被认为是线性的（定理，第345页）。然而，正如本文将说明的那样，该定理仅对四个或更多的代理是正确的。关于三个行为人的关键情况，格林自己提出了一个反例（例一，第348/9页）。由于这个例子的配给规则不是线性的，格林得出结论，它不满足他的定理的假设。然而它满足，这意味着格林的证明是不可行的。因此，必须给出（修改过的）定理的一个新的证明。在不丧失一般性的前提下，设总需求和总供给Z+， Z为固定，使得Z+ > Z-。考虑向量z （zl，…）， z1)的个人需求和供给，这样，zi > (I = I)…，n， zi 0可以写成zi = Ai Z+，其中Ai为非负权，使得= 1Ai = i。可行性，即EqiS(zi, Z+, Z-) = X-， E的匿名性和连续性；i暗示一个连续函数f: [0， i] [0， i]由Ec定义；b(Ai Z+, Z+, Z-) = f(Ai) x对于所有i， f有这样的性质：对于所有A1，…， An使得= 1Ai = i, ' f(Ai) = i.由于自愿交易，f(o) = o.当且仅当是恒等映射时，Eqi在zi（正半线上）是线性的。要看这个，请注意，通过匿名，
In his article 'On the theory of effective demand' (Green, i 980), J. Green deals with a class of stochastic rationing schemes whose distribution as perceived by the individual agent depends on his own action and on the aggregate values of demand and supply only. Under certain assumptions, the expectation of such a rationing function is claimed to be linear in each agent's own action, provided there are three or more agents (theorem, p. 345). However, as will be shown in this note, the theorem is correct only forfour or more agents. With respect to the critical case of three agents Green himself presents a counter-example (example I, pp. 348/9). As the rationing rule of this example is not linear, Green concludes that it does not fulfil the assumptions of his theorem Yet it does, which implies that Green's proof cannot be viable. Therefore a new proof of the (altered) theorem must be given, which is done in what follows. Without loss of generality, let aggregate demand and supply Z+, Zbe fixed such that Z+ > Z-. Consider a vector z (zl, ..., z1) of individual demands and supplies, such thatzi > ofor i = i) ...,n, zi o can be written zi = Ai Z+, where the Ai's are non-negative weights such that = 1Ai = i. Feasibility, i.e. EqiS(zi, Z+, Z-) = X-, anonymity and continuity of E;i imply a continuous function f: [o, i] [o, i] defined by Ec;b(Ai Z+, Z+, Z-) = f(Ai) Xfor all i. f has the property that for all A1, ..., An such that = 1Ai = i, `lf(Ai) = i. Because of voluntary trade,f(o) = o. It is immediate that Eqi is linear in zi (in the positive half-line) if and only iff is the identity map. To see this, notice that by anonymity,