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We extend the analysis of Kiyotaki and Wright, who study economies where the commodities that serve as media of exchange (or, commodity money) are determined endogenously. Kiyotaki and Wright consider only steady-state, pure-strategy equilibria; here we allow dynamic and mixed-strategy equilibria. We demonstrate that symmetric, steady-state equilibria in mixed-strategies always exist, while sometimes no such equilibria exist in pure-strategies. We prove that the number of symmetric steady-state equilibria is generically finite. We also show, however, that for some parameter values there exists a continuum of dynamic equilibria. Further, some equilibria display cycles.

Content uploaded by Timothy J. Kehoe

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... The formal modeling of equilibria with classical commodity money and classical fiat money is known as the Kiyotaki-Wright model [42], which explores the economic foundation of the emergence of money as a perishable medium of exchange. The model is tested numerically [43][44][45][46] and a mixed strategy case is also proposed [47]. Kiyotaki and Wright initially addressed infinitely durable goods and later their model was extended to goods with a different durability [48,49]. ...

... Note that V i does not explicitly depend on v j and w j ( j = i), but it is related with them via ρ. Before we describe a generic equilibrium and the flow of entangled quantum goods, let us start with an example which includes a review on the original Kiyotaki-Wright model [42,47]. More general case is given in Sect.2.4, ...

... Indeed the classical results [42,47] are reproduced when parameters s i (i = A, B, C) are chosen appropriately. As the quantum game must reproduce the classical results, we generally require ...

The quest of this work is to present discussions of some fundamental questions of economics in the era of quantum technology, which require a treatment different from economics studied thus far in the literature. A study of quantum economic behavior will become the center of attention of economists in the coming decades. We analyze a quantum economy in which players produce and consume quantum goods. They meet randomly and barter with neighbors bilaterally for quantum goods they produced. We clarify the conditions where certain quantum goods emerge endogenously as media of exchange, called quantum commodity money. As quantum strategies are entangled, we find distinctive aspects of quantum games that cannot be explained by conventional classical games. In some situations a quantum player can acquire a quantum good from people regardless of their strategies, while on the other hand people can find quantum strategies that improve their welfare based on an agreement. Those novel properties imply that quantum games also shed new light on theories of mechanism design, auction and contract in the quantum era.

... The formal modeling of equilibria with classical commodity money and classical fiat money is known as the Kiyotaki-Wright model [38], which explores the economic foundation of the emergence of money as a perishable medium of exchange. The model is tested numerically [39][40][41][42] and a mixed strategy case is also proposed [43]. Kiyotaki and Wright initially addressed infinitely durable goods and later their model was extended to goods with a different durability [44,45]. ...

... Note that V i dose not explicitly depend on v j and w j (j = i), but it is related with them via ρ. Before we describe a generic equilibrium and the flow of entangled quantum goods, let us start with an example which includes a review on the original Kiyotaki-Wright model [38,43]. More general case is given in Sec.2.4, ...

... Indeed the classical results [38,43] are reproduced when parameters s i (i = A, B, C) are chosen appropriately. ...

The quest of this work is to present a discussion of some fundamental questions of economics in the era of quantum technology, which require a treatment different from economics studied thus far in the literature. A study of quantum economic behavior will become the center of attention of economists in the coming decades. We analyze a quantum economy in which players produce and consume quantum goods. They meet randomly and barter with neighbors bilaterally for quantum goods they produced. We clarify the conditions where certain quantum goods emerge endogenously as media of exchange, called quantum commodity money. As quantum strategies are entangled, we find distinctive aspects of quantum games that cannot be explained by conventional classical games. In some situations a quantum player can acquire a quantum good from people regardless of their strategies, while on the other hand people can find quantum strategies that improve their welfare based on an agreement. Those novel properties imply that quantum games also shed new light on theories of mechanism design, auction and contract in the quantum era.

... To construct such equilibria is beyond the scope of this paper but could possibly be relevant for interpreting experimental results. See Kehoe et al. (1993), Renero (1998), and Wright (1994) for analysis of non stationary equilibria in the context monetary models similar to the one in this paper. ...

... If the offer is accepted both parties are bound to execute their part of the contract, otherwise the money doesn't change hands and (without loss of generality) player 1 takes the action s. 31 . Then, in the second period, player 1 can transfer a unit of money to player 2 if holding money before player 2 takes an action a 2 in {w, s} . ...

... Following the intuitions of Karl Menger 65 and starting with the Jones' model in the mid-1970's 47 , several seδrch-theoreticδl models have been proposed in order to identify the conditions for money emergence 23,54,55,67,1,56,78,45,50,93,60 . These models belong to the class of micro-founded macroeconomic models with rational expectations. ...

Through three studies, this thesis aims to explore the cognitive micro-foundations of economics. In a first study, I investigate the role of the information for coordination on a unique medium of exchange, that is to say money emergence. Relying on the search theoretical models (Kiyotaki & Wright, 1989, and Iwai, 1996), the goal of this study is to challenge the assumption that an exhaustive information is a necessary condition for money emergence. In a second study, I tackle the role of the information in duopoly competition. Using a model a-la-Hotelling (1929), we test the hypothesis that varying the amount of information available by consumers substantially impacts market’s dynamics. In a third study, I am interested in decision-making under risk in rhesus monkeys. Based on the prospect theory (Kahneman & Tversky, 1989, 1992), the main purpose is to assess to what extent macaques exhibit an asymmetric treatment of gains and losses similar to that of humans.

... I n the last decades, monetary economics has shifted from a purely macroeconomic understanding of money to an analysis of its micro-foundations, both in its game-theoretical and behavioral dimensions. Following the intuitions of Karl Menger (1892) and starting with the Jones' model in the mid-1970's (Jones, 1976), several search-theoretic models have been proposed in order to identify the conditions for money emergence (Diamond, 1984;Wright, 1989, 1991;Oh, 1989;Aiyagari and Wallace, 1991;Kiyotaki and Wright, 1993;Shi, 1995;Iwai, 1996;Kehoe et al., 1993;Wright, 1995;Luo, 1998). They are considered search-theoretic models in the sense that they describe situations where agents need to search for a trading partner before transacting (Nosal and Rocheteau, 2011). ...

Several micro-founded macroeconomic models with rational expectations address the issue of money emergence, by characterizing it as a coordination game. These models have in common the use of agents who dispose of perfect or near-perfect information on the global state of the economy and who display full-fledged computational abilities. Several experimental studies have shown that a simple trial-and-error learning process could constitute an explanation for how agents coordinate on a single mean of exchange. However, these studies provide subjects with full information regarding the state of the economy while restricting the number of goods in circulation to three. In this study, by the mean of multi-agent simulations and human experiments, we test the hypothesis according to which coordination over a unique medium of exchange is possible in the context of information scarcity. In our experimental design, subjects and artificial agents are only aware of the outcome of their own decisions. We provide results for economies with 3 and 4 goods to evaluate to which extent it is possible to generalize results obtained with 3 goods to n goods. Our findings show that in an economy à la Iwai, commodity money can emerge under drastic information restrictions with three goods in circulation, but generalization to four or more goods is not guaranteed.

... Up to now prices were fixed, since every trade involves a one-for-one swap. Beginning the second generation of papers in this literature, Shi (1995) and Trejos-Wright (1995) 7 Other applications of these first-generation models include the following: Kiyotaki and Wright (1989), Aiyagari and Wallace (1991), Kehoe et al. (1993), andWright (1995), among others, allow goods to be storable and discuss commodity money. Wright (1991,1993), , and Camera et al. (2003) endogenize specialization in production and consumption. ...

... Kehoe et al. (1993) build cyclical equilibria in a similar environment under sets of parameters that do not admit pure strategy equilibria. On this point, see alsoRenero (1998). ...

We discuss the emergence of money in a Kiyotaki and Wright (J Polit Econ 97:927–954, 1989) environment through two computational methodologies. One assumes that agents are fully rational and coordinate on Nash equilibria. The other considers boundedly rational agents whose choices are guided by a classifier system à la Marimon et al. (J Econ Dyn Control 14:329–373, 1990). We apply the two methodologies to study the conditions under which individuals can learn to play speculative strategies—to accept a high-storage-cost good as money. Our analysis suggests that, while in both types of systems money can emerge along a dynamic path, boundedly rational agents make conflicting choices at times, even when the classifier system provides clear information about the likely gains of a trade.

Experimental work in monetary economics is usually based on theory that incorporates an infinite horizon. Yet, hard constraints on laboratory sessions lead to finite times when the game must (with probability 1) end, and then simple backward induction implies monetary equilibria cannot exist. Hence, these experiments cannot evaluate subjects’ ability to settle on the use of money as a medium of exchange, that ameliorates trading frictions, as an equilibrium outcome. To address this, we present some finite-horizon games where monetary exchange is an equilibrium outcome, and report some experimental results using these games.

We analyze the rise in the acceptability fiat money in a Kiyotaki–Wright economy by developing a method that can determine dynamic Nash equilibria for a class of search models with genuine heterogeneous agents. We also address open issues regarding the stability properties of pure strategy equilibria and the presence of multiple equilibria, numerical experiments illustrate the liquidity conditions that favor the transition from partial to full acceptance of fiat money, and the effects of inflationary shocks on production, liquidity, and trade.

The belief that the long run equilibrium of a competitive monetary economy that does not experience any exogeneous shocks — whether originating from the external environment or from policy — should be modelled as a state that is stationary or perhaps growing at a constant rate, seems to be deeply rooted in the mind of some economists.

A new proof is offered for the theorem that, in almost all finite games, the number of equilibrium points isfinite andodd. The proof is based on constructing a one-parameter family of games with logarithmic payoff functions, and studying the topological properties of the graph of a certain algebraic function, related to the graph of the set of equilibrium points for the games belonging to this family. In the last section of the paper, it is shown that, in the space of all games of a given size, those exceptional games which fail to satisfy the theorem (by having an even number or an infinity of equilibrium points) is a closed set of measure zero.

Selten's concept of perfect equilibrium for normal form games is reviewed, and a new concept of proper equilibrium is defined. It is shown that the proper equilibria form a nonempty subset of the perfect equilibria, which in turn form a subset of the Nash equilibria. An example is given to show that these inclusions may be strict.

In this paper we develop a regularity theory for stationary overlapping generations economies. We show that generically there is an odd number of steady states in which a non-zero amount of nominal debt (fiat money) is passed from generation to generation and an odd number in which there is no nominal debt. We are also interested in non-steady state perfect foresight paths. As a first step in this direction we analyze the behavior of paths near a steady state. We show that generically they are given by a second order difference equation that satisfies strong regularity properties. Economic theory alone imposes little restriction on those paths: With n goods and consumers who live for m periods, for example, the only restriction on the set of paths converging to the steady state is that they form a manifold of dimension no less than one, no more than 2nm.