Indivisible commodities and an equivalence theorem on the strong core

We consider a pure exchange economy with finitely many indivisible commodities that are available only in integer quantities. We prove that in such an economy with a sufficiently large number of agents, but finitely many agents, the strong core coincides with the set of cost-minimized Walras allocations. Because of the indivisibility, the preference maximization does not imply the cost minimization. A cost-minimized Walras equilibrium is a state where, under some price vector, all agents satisfy both the preference maximization and the cost minimization.Indivisible commodities, strong core, cost-minimized Walras equilibrium, core equivalence

Core allocations may not be Walras allocations in any large finite economy with indivisible commodities

We consider an exchange economy where every commodity can be consumed only in integer amounts. Inoue [Inoue, T., 2005. Do pure indivisibilities prevent core equivalence? Core equivalence theorem in an atomless economy with purely indivisible commodities only. Journal of Mathematical Economics 41, 571-601] proved that in such an economy with a continuum of agents, the core coincides with the set of Walras allocations. We show that this equivalence holds only in an atomless economy by giving two examples of the sequence of replica economies such that in any replica economy, there exists a core allocation that is not a Walras allocation.indivisible commodities, core, Walras equilibrium, strong core, cost-minimized Walras equilibrium

A utility representation theorem with weaker continuity condition

We prove that a preference relation which is continuous on every straight line has a utility representation if its domain is a convex subset of a finite dimensional vector space. Our condition on the domain of a preference relation is stronger than Eilenberg (1941) and Debreu (1959, 1964), but our condition on the continuity of a preference relation is strictly weaker than theirs.linear continuity, utility representation

Representation of TU games by coalition production economies

We prove that every transferable utility (TU) game can be generated by a coalition production economy. Given a TU game, the set of Walrasian payoff vectors of the induced coalition production economy coincides with the core of the balanced cover of the given game. Therefore, a Walrasian equilibrium for the induced coalition production economy always exists. The induced coalition production economy has one output and the same number of inputs as agents. Every input is personalized and it can be interpreted as agent's labor. In a Walrasian equilibrium, every agent is permitted to work at several firms. In a Walrasian equilibrium without double-jobbing, in contrast, every agent has to work at exactly one firm. This restricted concept of a Walrasian equilibrium enables us to discuss which coalitions are formed in an equilibrium. If the cohesive cover or the completion of a given TU game is balanced, then the no-double-jobbing restriction does not matter, i.e., there exists no difference between Walrasian payoff vectors and Walrasian payoff vectors without double-jobbing.Coalition production economy, transferable utility game, core, Walrasian equilibrium, Walrasian equilibrium without double-jobbing, coalition structure

Strong core equivalence theorem in an atomless economy with indivisible commodities

We consider an atomless exchange economy with indivisible commodities. Every commodity can be consumed only in integer amounts. In such an economy, because of the indivisibility, the preference maximization does not imply the cost minimization. We prove that the strong core coincides with the set of cost-minimized Walras allocations which satisfy both the preference maximization and the cost minimization under the same price vector.indivisible commodities, core equivalence, strong core, cost-minimized Walras equilibrium

Representation of TU games by coalition production economies

Inoue T. Representation of TU games by coalition production economies. Working Papers. Institute of Mathematical Economics. Vol 430. Bielefeld: Universität Bielefeld; 2010.We prove that every transferable utility (TU) game can be generated by a coalition production economy. Given a TU game, the set of Walrasian payoff vectors of the induced coalition production economy coincides with the core of the balanced cover of the given game. Therefore, a Walrasian equilibrium for the induced coalition production economy always exists. The induced coalition production economy has one output and the same number of inputs as agents. Every input is personalized and it can be interpreted as agent's labor. In a Walrasian equilibrium, every agent is permitted to work at several firms. In a Walrasian equilibrium without double-jobbing, in contrast, every agent has to work at exactly one firm. This restricted concept of a Walrasian equilibrium enables us to discuss which coalitions are formed in an equilibrium. If the cohesive cover or the completion of a given TU game is balanced, then the no-double-jobbing restriction does not matter, i.e., there exists no difference between Walrasian payoff vectors and Walrasian payoff vectors without double-jobbing

Hidden photon CDM search at Tokyo

We report on a search for hidden photon cold dark matter (HP CDM) using a novel technique with a dish antenna. We constructed two independent apparatus: one is aiming at the detection of the HP with a mass of $\sim\,\rm{eV}$ which employs optical instruments, and the other is for a mass of $\sim5\times10^{-5}\, \rm{eV}$ utilizing a commercially available parabolic antenna facing on a plane reflector. From the result of the measurements, we found no evidence for the existence of HP CDM and set upper limits on the photon-HP mixing parameter $\chi$.Comment: Contributed to the 11th Patras Workshop on Axions, WIMPs and WISPs, Zaragoza, June 22 to 26, 201

Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras II: Types C_r, F_4, and G_2

We prove the periodicities of the restricted T and Y-systems associated with the quantum affine algebra of type C_r, F_4, and G_2 at any level. We also prove the dilogarithm identities for these Y-systems at any level. Our proof is based on the tropical Y-systems and the categorification of the cluster algebra associated with any skew-symmetric matrix by Plamondon.Comment: 36 pages; minor changes, references update