Все эти вопросы так или иначе затрагиваются в серии

TEXT 10 WALRAS’ LAW AND THE BUDGET CONSTRAINT In the “equation counting” approach to general equilibrium, we assemble the demand and supply components of the system together into the excess demand relationships, each a function of all prices, to obtain the equilibrium conditions. There are n relationships in variables. Prom the beginning it was, however, correctly realized that the count is not correct because the relationships are homogeneous of degree zero in . Assuming that the are single-valued functions and not point-to-set mappings (this assumption was always made in the neoclassical analysis), they are homogeneous functions and so can be written in the form , where is the vector. There are now relationships in only independent variables, being indeterminate. Thus it is necessary-to show that only of the relationships are in-dependent, if the count is to come out right. Walras found the dependence supplied through the budget constraint. In a pure exchange economy, if every consumer obtains his income from the sale of goods, and spends all that income on goods, it is obvious that the value of purchases equals the value of sales for all trades and so, for the economy as a whole,. The same relationship holds for an economy with production, provided all profits are distributed to consumers. The relationship is, of course, part of the equilibrium condition. If it is assumed to hold everywhere, even when general market equilibrium has not been established, it is known as Walras’ Law. The implication of Walras' Law is that the excess demand functions are not all independent since we can always find one excess demand, say, from . This implication has been used to generate many fruitless arguments, especially in monetary theory. Walras’ Law is neither necessary nor sufficient for the purpose for which it has been used. It is not necessary that we be able to solve properly for in terms of except at the equilibrium point itself, for which the equilibrium condition gives , and the counting method certainly is not sufficient to guarantee the existence of equilibrium, if only because of the implicit nonnegativity constraints. Walras' Law carries important implications for dynamic behavior. It implies that, even when the markets are out of equilibrium, every economic agent acts as though the existing prices are equilibrium prices and plans sales and purchases that balance, even if all those sales and purchases cannot be made. This behavior is related to Walras' theory of tâtonnement, .which represents a dynamic process. Walras' Law acts as a law of conservation and tends to stabilize market behavior. In the equilibrium analysis given in this chapter we do not count equations end unknowns and do not interest ourselves in whether the excess demand functions are independent or not. Indeed, we do not assume that excess demand is a single valued function so that the question of dependence is irrelevant. We do use the budget constraint, in a weak form, in establishing the existence of equilibrium, but we use it in quite a different way from that of the equation counting approach. In showing the existence of equilibrium, we shall be proceeding in the following way. We take an arbitrary set of prices, then consider a set of excess demand vectors that would be consistent with producer and consumer behavior at those prices if they were equilibrium prices. The way in which we construct this set of excess demand vectors is quite artificial and chosen for mathematical reasons. It does not purport to represent any actual of adjustment. We make this construction for all feasible prices, and obtain a mapping from prices into excess demand vectors, which will be a point-to-set mapping. We then draw a fixed point theorem to show that at least one of the price vectors maps into a set of excess demand vectors which includes an excess demand vector satisfying the market equilibrium conditions. Since the manning has been chosen so that the equilibrium conditions for the individual decision makers are always satisfied, we thus prove the existence of at least one point of equilibrium in the economy. Confining our attention to excess demand which satisfies the budget constraint at all prices is quite different from assuming hat excess demand in fact satisfies Walras' law. It is quite consistent with the approach here that individual's actual behavior out of equilibrium does not satisfy Walras' Law. We would till set up out artificial mapping based on the budget constraint and prove that an equilibrium exists, although the actual dynamic processes of the economy may not result in its being attained. Finally we mау note that we require only the weak budget constraint , whereas Walras' Law requires equality if the dependence of the excess demand functions is to be established. I. Прочитайте текст. II. Составьте схему передаваемой текстом информации. III. Перечитайте текст. Проверьте, не нуждается ли ваша первоначальная схема в коррективах. При необходимости внесите нужные исправления. IV. На основе схемы, используя главные ключевые фрагменты, конспективно изложите все существенные положения текста. TEXT 11 THE EXCESS DEMAND THEOREM The equilibrium proofs which follow in the succeeding sections require the use of topological methods at a crucial point in the argument. Since the same analysis can be used for both the models which follow, it seems useful to make a package of the part of the argument which is common to both and present it here. This leaves the later discussion freer to discuss the special properties of the models themselves. Some readers may wish merely to accept the following theorem as proved, then pass directly on to the next section. Although the theorem is mathematical and does not need to be given the particular interpretation here attached to it, we use it only in respect to excess demand, hence the name and the interpretation: Excess Demand Theorem If every semipositive price sector (which we can take to be normalized so that is associated with a set of excess demand vectors such that has certain properties and the mapping has appropriate continuity properties, and if the weak budget constraint is satisfied for all , then there is some price for which for some . Since pz = 0 is a special саsе of , this theorem guarantees the existence of an equilibrium excess demand vector if the conditions of the theorem are met. Both the existence proofs given in this chapter are based on showing that the conditions of the excess demand theorem can be met with the individual decision makers operating in accordance with the appropriate rules. The formal requirements on Z(p) and the mapping are: is nonempty and convex for all ; is upper semicontinuous and the set of 's is bounded. The general sense of the theorem can be illustrated from two-market example. We have everywhere. The theorem is obviously satisfied if are nonpositive everywhere. But suppose for some . Then we must have . It is obvious that, unless is unbounded (which ruled out) cannot remain positive indefinitely as the ratio increases and so must be nonpositive for sufficiently large . The only question is whether may become positive. It may, indeed, do so, but not before becomes negative. Thus there must be some for which both and are nonpositive, and thus some normalized price vector for which , are nonpositive. Formal proof of the theorem is a much more difficult matter. We proceed as follows. Denote by P the set of semipositive normalized prices. This is clearly a compact convex set. Denote by the set of all for [ is the union of the sets]. If is not convex, we replace it by any compact convex set containing , which we denote by . Now define the set as follows: is a maximum for } That is, we choose an arbitrary excess demand vector from the set of all excess demand vectors which are attainable at some prices, then find the price vector for which the value of this excess demand is maximized. It is important to note that the rice vector is any price vector, not necessary the particular which is associated with through the mapping . Clearly is a mapping from into a subset of . Since is convex we know this mapping to be upper semicontinuous (see continuity properties of optimal solutions). is a convex set since it is the intersection of the hyper-plane with . Consider, the set , that is, the set consisting of normalized price vectors paired with excess demand vectors. If we take point , in , then associates a set of excess demand vectors with, and associates a set of price vectors with . In other words, the mapping, , maps a point in into a subset of . We have shown the mapping to be upper semicontinuous, and has been assumed to have the same property, so that the combined mapping is upper semicontinuous also. We have shown that is convex and has been assumed convex, so that x is convex. Thus we have an upper semicontinuous mapping , from the set into a convex subset of itself. These are the conditions for invoking the Kakutani Fixed Point Theorem. The theorem states that there exists some which is a fixed point, that is, for which and. From the construction of , implies that, for , . Using the weak budget condition it follows that, since , Thus , for all . Clearly the last inequality is satisfied for all only if| , thus proving the theorem. We might note that a strong budget condition (Walras' Law) does not strengthen the conclusion of the theorem. I. Прочитайте текст. II. Составьте схему передаваемой текстом информации. III. Перечитайте текст. Проверьте, не нуждается ли ваша первоначальная схема в коррективах. При необходимости внесите нужные исправления. III. На основе схемы, используя главные ключевые фрагменты, конспективно изложите все существенные положения текста.

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