Mathematical Theory of Exchange-driven Growth

Exchange-driven growth is a process in which pairs of clusters interact and exchange a single unit of mass. The rate of exchange is given by an interaction kernel $K(j,k)$ which depends on the masses of the two interacting clusters. In this paper we establish the fundamental mathematical properties of the mean field kinetic equations of this process for the first time. We find two different classes of behaviour depending on whether $K(j,k)$ is symmetric or not. For the non-symmetric case, we prove global existence and uniqueness of solutions for kernels satisfying $K(j,k)\leq Cjk$. This result is optimal in the sense that we show for a large class of initial conditions with kernels satisfying $K(j,k)\geq Cj^{\beta}$ ($\beta>1)$ the solutions cannot exist. On the other hand, for symmetric kernels, we prove global existence of solutions for $K(j,k)\leq C(j^{\mu}k^{\nu}+j^{\nu}k^{\mu})$ ($\mu,\nu\leq2,$ $\mu+\nu\leq3),$ while existence is lost for $K(j,k)\geq Cj^{\beta}$ ($\beta>2).$ In the intermediate regime $3<\mu+\nu\leq4,$ we can only show local existence. We conjecture that the intermediate regime exhibits finite-time gelation in accordance with the heuristic results obtained for particular kernels.Comment: Mistakes in the uniqueness proofs are fixed. Some typos are corrected. Some references are adde

Money in a theory of exchange

Major problems in monetary economics are to: introduce money into the economy in a way that explains how money arises endogenously; explain why money is preferred to other methods of exchange; and identify the welfare gains from using money. In this paper, Daniel L. Thornton develops a framework for assessing money's role in the economy and identifies the welfare gains associated with its use. In Thornton's framework, money is welfare enhancing not only because it reduces the resources necessary for exchange-thereby increasing both consumption and leisure-but, money further increases welfare by promoting further trade and greater specialization. ; Thornton then discusses the implications of his analysis for several important issues in monetary theory: the existence of fiat money; the role of money and credit in exchange; the asset demand for money; the buffer-stock notion of money demand; the welfare benefits of money; and the welfare costs of inflation.Money ; Monetary theory

Lifting defects for nonstable K_0-theory of exchange rings and C*-algebras

The assignment (nonstable K_0-theory), that to a ring R associates the monoid V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor: (1) There is no functor F, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that VF is equivalent to the identity. (2) There is no functor F, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp., von Neumann regular rings), such that VF is equivalent to the identity. (3) There is a {0,1}^3-indexed commutative diagram D of simplicial monoids that can be lifted, with respect to the functor V, by exchange rings and by C*-algebras of real rank 1, but not by semiprimitive exchange rings, thus neither by regular rings nor by C*-algebras of real rank 0. By using categorical tools from an earlier paper (larders, lifters, CLL), we deduce that there exists a unital exchange ring of cardinality aleph three (resp., an aleph three-separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V(R) is the positive cone of a dimension group and V(R) is not isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0 or a regular ring.Comment: 34 pages. Algebras and Representation Theory, to appea

A Fundamental Theory of Exchange Rates and Direct Currency Trades

In this paper we construct a two-country search model to determine the nominal exchange rate between two fiat monies. Our model allows agents to use any currency to trade for goods in all countries. However, search frictions restrict agents opportunities for instantaneous arbitrage, and hence make the nominal exchange rate determinate. The nominal exchange rate depends on the two countries economic fundamentals, including the stocks and growth rates of the two monies. Direct exchanges between currencies are essential and they imply a nominal exchange rate that is different from the relative price between the two currencies in the goods markets. There are persistent violations of the law of one price and purchasing power parity in equilibrium, despite the fact that prices are perfectly flexible and all goods are tradeable between countries. Nominal and real exchange rates can move together in the steady state in response to money growth shocks.Exchange Rates; Search; Money; Currency Trade.

High-Field Low-Frequency Spin Dynamics

The theory of exchange symmetry of spin ordered states is extended to the case of high magnetic field. Low frequency spin dynamics equation for quasi-goldstone mode is derived for two cases of collinear and noncollinear antiferromagnets.Comment: 2 page

Bargaining, Coalitions, and Competition

We study a Gale-like matching model in a large exchange economy, in which trade takes place through non-cooperative bargaining in coalitions of finite size. Under essentially the same conditions of core equivalence, we show that the strategic equilibrium outcomes of our model coincide with the Walrasian allocations of the economy. Our method of proof makes use of the theory of the core. With respect to previous work, our positive implementation result applies to a substantially larger class of economies: the model relaxes differentiability and convexity of preferences, and also admits an arbitrary number of divisible and indivisible goods.Finite coalitions and Edgworthian theory of exchange; marginal artes of substitution and Jevonsian theory of exchange; matching and bargaining; core; Walrasian equilibrium

A Theory of Exchange Rates and the Term Structure of Interest Rates

The purpose of this paper is to construct a model of exchange rate determination that is consistent with the stylized facts regarding the uncovered interest parity for short term and long term interest rates. This task is especially challenging because of the forward premium anomaly found for short term interest rates and forward exchange rates. With an assumption that investors have a short investment horizon, the model is consistent with these stylized facts even when the degree of risk aversion is low. The model predicts a complicated relationship between exchange rates and the term structure of the interest rates.forward premium anomaly, uncovered interest parity for long term bonds

The quanto theory of exchange rates

We present a new, theoretically motivated, forecasting variable for exchange rates that is based on the prices of quanto index contracts, and show via panel regressions that the quanto forecast variable is a statistically and economically significant predictor of currency appreciation and of excess returns on currency trades. We also test the quanto variable's ability to forecast differential currency appreciation out of sample, and find that it outperforms predictions based on uncovered interest parity, on purchasing power parity, and on a random walk

The quanto theory of exchange rates

We present a new, theoretically motivated, forecasting variable for exchange rates that is based on the prices of quanto index contracts, and show via panel regressions that the quanto forecast variable is a statistically and economically significant predictor of currency appreciation and of excess returns on currency trades. We also test the quanto variable's ability to forecast differential currency appreciation out of sample, and find that it outperforms predictions based on uncovered interest parity, on purchasing power parity, and on a random walk