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This paper extends the results on trade and welfare obtained in Ohyama(1972) in the case of a traditional world economy with a finite number of goods to the one of a world economy over a discrete-time infinite horizon with l1 , the space of all bounded sequences, as the underlying commodity space. The case with l1 is a typical special case of economies with infinite number of goods. In this paper, it is shown that the main results ottained in Ohyama(1972) still hold in the world economy over a discrete- time infinite horizon by following the method used in Ohyama(1972). It turns out that Ohyama(1972)'s method is, indeed, very general in a sense that it also applies to more general cases including economies with infinitely many goods.

This paper shows the existence of gains from trade in a dynamic world free trade economy over a discrete-time infinite horizon with using Grandmont-McFadden's(1972) domestic income transfer policy which makes each consumer benefit from world free trade. For this purpose, this paper employs l∞, the space of all bounded sequences, as the underlying commodity space and l1, the space of all summable sequences, as the price space, with following the general equilibrium analysis of infinite dimensional commodity spaces developed by Bewley(1972), Mas-Colell(1986), and Zame(1987). Moreover, since gains from trade requires the comparison of two consumption bundles, one under free trade and the other under autarky, but transitivity of preferences assumed the comparison of three consumption bundles, this paper also drops transitive preferences to establish gains from trade in this dynamic world economy. Furthermore, since this paper uses general consumption sets instead of the positive orthant l∞+, the argument finding an equilibrium price in l1, requires the exclusion condition to consumption sets and the mixture condition to the production set. This paper also drops the cheaper point assumption used in Grandmont-McFadden(1972) and replaces it with a variant of McKenzie's(1959, 1981, 2002) irreducibility assumption and the strong irreducibility assumption of Boy-McKenzie(1993), requiring the interrelatedness of consumers in the world economy.

This paper considers several versions of the infinite dimensional Gale-Nikaido lemma and establishes the existence of competitive equilibrium in a simplified Bewley(1972)’ economy with l∞ as the commodity space with applying an infiinite dimensional Gale-Nikaido Lemma obtained in this paper.

Gale(1955)とNikaido(1956a)は独立にゲール・二階堂の補題を証明し、それを利用してn次元ユークリッド空間Rnを財空間とする経済における競争均衡の存在証明を行った。そして、ドブリュー(1959、第5章)では、ゲール・二階堂の補題を用いるGale(1955)とNikaido(1956a)の議論に沿って、Rnを財空間とする経済における競争均衡の存在証明を行い、一般均衡理論の文献においてゲール・二階堂の補題を著名にした。Nikaido(1956b,57b,59)では、更に、Rnにおけるゲール・二階堂の補題をノルム空間や局所凸線形位相空間にまで一般化した。この、有限次元空間におけるゲール・二階堂の補題を証明した直後に無限次元空間までゲール・二階堂の補題を一般化したという事実は、驚くべき事である。無限個の財がある経済における競争均衡の存在問題はPeleg-Yarri(1970)とBewley(1972)から開始されたのであるが、Debreu(1954)同様に、これらよりも10年以上先に既に無限個の財がある経済を取り上げていたのである。そこで本稿の目的はNikaido(1956b,57b,59)による無限次元空間のゲール・二階堂の補題を再考して、1つの一般化の方向を議論する事である。

本論文ではビューリー(2007、第5章)において取り扱われた、クーン/タッカー的な接近法による古典的な凸有限経済における厚生経済学の基本定理を取り上げる。この経済の消費者が凹な効用関数を持つと仮定する事によって、厚生経済学の基本定理は制約条件付最適化(最大化)問題として表現できる。そこでこの問題に非線形計画法の基本定理であるクーン/タッカー定理を適用する事により、パレート最適な配分では線形社会厚生関数が最大化される事が示される。更に、制約条件付最適化(最大化)問題における最大値関数の性質により、財・サービスに対してその市場価格がその限界社会厚生値になっている事や、最大化された線形社会厚生関数の各係数が市場均衡における各消費者の所得/貨幣の限界効用の逆数になっている事が導かれる。これらの結果は効用関数の微分可能性を前提にすることなく得られている。

マッケンジー教授による『古典的一般均衡理論』(2002,MITPress)の最終章(第7章)では、離散型多部門経済成長モデルの枠組みで既約型一般ラムゼー成長モデルにおける通常ターンパイク定理が扱われていて、将来効用の割引の存在とは関わり無く、最適経路が最適定常経路に収束して行く事が示されている。ただし、この結果の前提である最適経路の存在に関して、ターンパイク定理の証明で必要となる定常的な最適経路の存在以外には示されていない。そこでまず、マッケンジー(1983、86)に従って、将来効用に割引のある既約型一般ラムゼー成長モデルにおける最適経路の存在を示す。次に、マッケンジー(2002,第7章)では、最適経路が最適定常経路の近傍に収束して行く近傍ターンパイク定理で必要となる最適定常経路の存在が、マッケンジー(1982,86)に沿って角谷の不動点定理を応用する形で行われているので、その議論を行う。最後に、マッケンジー(2002,第7章)では、マッケンジー(1982)に沿った形で、最適経路が最適定常経路の近傍に収束して行く近傍ターンパイク定理の証明が行われているが、その証明は複雑なので、それをより分かり易く修正した証明を与える。これらによって、マッケンジー(2002、第7章)における将来効用の割引のあるケースの定常的な離散無限期間の既約型一般ラムゼー成長モデルに関する議論を補完して、より分かり易くする事が本稿の主な目的である。

本稿では,無限次元財空間モデルにおけるゲール・二階堂の補題を扱った二階堂(1956b,57b,59)では,その結果を利用して競争均衡の存在が示される無限次元財空間モデルの例がなかったので,l∞のケースのビューレー(1972)モデルを一例にして,フローレンザーノ(1983)の議論を参考にしながら,二階堂(1956b,57b,59)で示された無限次元のゲール・二階堂の補題を修正し,その結果をこのl∞を財空間に持つ経済に応用してその経済における競争均衡の存在証明を行う。これにより,二階堂(1956b,57b,59)にはなかった,その結果を利用して競争均衡の存在が示される無限次元財空間モデルの例を,与えたことになる。

As shown in Debreu(1959, Ch.5), Gale-Nikaido's lemma of Gale(1955) and Nikaido(1956a) is the key in proving the existence of a competitive equilibrium in classical economies with a finite number of commodities. Nikaido(1956b, 57b, 59) extend the Gale-Nikaido's lemma in economies with a finite number of commodities to the one in some infinite dimensional spaces such as normed spaces and locally convex topological vector spaces. This is surprising since Gale-Nikaido's lemma is generalized to the infinite dimensional spaces just after Gale-Nikaido's lemma in economies with a finite number of commodities is established. Since the literature on the existence of competitive equilibrium in economies with infinite number of commodities stars after Peleg-Yarri(1970) and Bewley(1972) and it was one of the main topics for 80's in general equilibrium theory, Nikaido(1956b, 57b,59) precedes to the literature as Debreu(1954) does. The purpose of this paper is to reconsider the Nikaido(1956b, 57b, 59)'s generalization of Gale-Nikaido's lemma in infinite dimensional spaces from the present state of general equilibrium theory of infinite dimensional spaces.

This paper shows that the theorem on the existence of competitive equilibrium in classical finite economies with ordered preferences but without individual survival is proved with using generalized notion of resource relatedness condition of Arrow-Hahn (1971) as well as with using irreducibility of McKenzie (1959, 1981). (JEL C62, D51)

本稿では、財空間として l∞ を利用する無限期間定常経済モデルにおける競争均衡の存在定理の証明を、根岸アプローチに基づいて行う。そして均衡価格を l1 に見つける。通常のケースでは、消費集合として非負象限を用いるが、本稿では一般的な消費集合を用い、その際にどのような条件が必要なのかを明確にする。また通常の多くのケースでは、消費者が常に十分な水準の所得を実現させる個人強生存条件を仮定するが、本稿では、個人生存条件は仮定するが、個人強生存条件は仮定せずに、有限次元財空間モデルと同様に既約性と経済全体での強生存条件を仮定して、競争均衡の存在定理を証明する。

The South Basin of Lake Biwa is a typical shallw lake (4 km x15 km X 4m) located near theoutlet (Seta River) of the whole lake. Laboratory experiment on the wind-driven current in thelake was conducted by using a toporographical model and the most fundamental law amongthe rotating similarities. The Rossby numbers due to the longitudinal wind and the outlet dischargeand the vertical Ekman number in the experiments are set to be similar to those in the lake.A slight difference in the surface current patterns was observed between rotating and non-rotating cases. Another difference is noted between the surface and the bottom current patternsexcept for the topographic gyre in the shallower part. Numerical results prepared for theexperiment are also discussed.

It is an important problem for us to analyze the three dimensional structure of the current in ashallow lake such as the South Basin of Lake Biwa. Assuming the coefficient of vertical eddyviscosity to be 0.1-10 cm2/sec, the frictional layer depth is evaluated to be within a range of 1-15 meter and is no more than a few times of the mean depth of that basin.We developed the two-dimensional three-level model considering the surface and the bottomEkman layer so as to simulate the currents under weak wind conditions, and conducted severalnumerical calculations under the transient boundary conditions of the measured wind and theoutlet discharges. The calculated current velocity is reasonarably in accordance with the mea-sured one if the relation of the eddy viscosity with wind velocity and water depth is appropriatelygiven. The effect of the non-uniformity of the wind field on the currents is also investigated.Drift currents and resultant topographic gyre patterns in the shallower part of the South Basin, slope currents with longitudinal convective circulation and its deflective revolution under the Cor-iolis effect due to decreasing eddy viscosity after the wind ceased are examined. The distributedwind tends to weaken the the secondary circulation compared with the result under the uniformwind velocity.

The South Basin of Lake Biwa is a typical shallw lake (4 km x15 km X 4m) located near theoutlet (Seta River) of the whole lake. Laboratory experiment on the wind-driven current in thelake was conducted by using a toporographical model and the most fundamental law amongthe rotating similarities. The Rossby numbers due to the longitudinal wind and the outlet dischargeand the vertical Ekman number in the experiments are set to be similar to those in the lake.A slight difference in the surface current patterns was observed between rotating and non-rotating cases. Another difference is noted between the surface and the bottom current patternsexcept for the topographic gyre in the shallower part. Numerical results prepared for theexperiment are also discussed.

It is an important problem for us to analyze the three dimensional structure of the current in ashallow lake such as the South Basin of Lake Biwa. Assuming the coefficient of vertical eddyviscosity to be 0.1-10 cm2/sec, the frictional layer depth is evaluated to be within a range of 1-15 meter and is no more than a few times of the mean depth of that basin.We developed the two-dimensional three-level model considering the surface and the bottomEkman layer so as to simulate the currents under weak wind conditions, and conducted severalnumerical calculations under the transient boundary conditions of the measured wind and theoutlet discharges. The calculated current velocity is reasonarably in accordance with the mea-sured one if the relation of the eddy viscosity with wind velocity and water depth is appropriatelygiven. The effect of the non-uniformity of the wind field on the currents is also investigated.Drift currents and resultant topographic gyre patterns in the shallower part of the South Basin, slope currents with longitudinal convective circulation and its deflective revolution under the Cor-iolis effect due to decreasing eddy viscosity after the wind ceased are examined. The distributedwind tends to weaken the the secondary circulation compared with the result under the uniformwind velocity.