On the evolution from barter to fiat money

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1 On the evolution from barter to fiat money Ning Xi a, Yougui Wang,b a Business School, University of Shanghai for Science and Technology, Shanghai, , P. R. China b Department of Systems Science, School of Management, Beijing Normal University, Beijing, , P. R. China Abstract In this paper, we propose an evolutionary model to investigate how an economy can evolve from a system of barter to monetary trade, while regarding the latter as involving reciprocal altruistic behavior. We use a replicator equation to describe how bounded rational agents shift their strategies, and in doing so, present an alternative solution to the start problem raised by Dowd (2001). Key words: Fiat money, Search model, Altruism, Replicator dynamics JEL: E40, C73 1. Introduction Why do selfish agents trade real commodities for intrinsically worthless forms of money? This question is one of the fundamental problems in monetary economics. Many attempts have been made to understand this issue using the framework of a monetary search model (e.g., Kiyotaki and Wright, 1993; Ritter, 1995). In this kind of model, the random interaction between rational agents results in both a barter and monetary equilibrium. Using this model, however, a path leading the economy from the barter equilibrium to the monetary equilibrium is not possible. Suppose that the economy is initially in a barter equilibrium. Any one agent does not expect that other agents will accept money, so he will also refuse money. Consequently, an Corresponding author. address: ygwang@bnu.edu.cn (Yougui Wang) Preprint submitted to Economics Letters January 4, 2009

2 economy in a barter equilibrium will not transition to a monetary equilibrium. Since Dowd (2001) raised the so-called start problem, no scholar has offered a good solution. In this paper, we propose an evolutionary model to account for the movement from a barter system to a system that uses fiat money. Monetary trade is, in essence, a reciprocal altruistic behavior. In monetary trades, sellers offer real commodities and, in return, receive intrinsically worthless money. The reverse is true for buyers. Thus, monetary trade can be thought of as an altruistic trade where sellers are donors and buyers are recipients. This kind of altruistic behavior can be explained by reciprocity that includes both direct reciprocity and indirect reciprocity. The former is captured in the principle: You scratch my back and I ll scratch yours, while the latter is captured in the principle: You scratch my back and I ll scratch someone else s (see Nowak and Sigmund, 2005). Obviously, monetary trade is based on indirect reciprocity. Nowak and Sigmund (1998) proposed a theoretical framework to investigate the role of indirect reciprocity. In their model, there are two types of agents: discriminators, who only help agents who previously helped someone else, and defectors, who never help any agent. The agents produce offspring that inherit parents types proportional to their payoffs, which come from the interaction among them. It is found that discriminators ultimately prevail. Our model is based on this framework. 2. The Environment The environment in our model is similar to the environment in the monetary search model (see Shi, 2006). In the economy, there is a [0, 1] continuum of infinitely lived agents and a variety of non-storable consumption goods. The economy is specialized; that is, each agent produces and consumes specific goods, and cannot produce his own consumption goods. The unit production cost for any agent is c (which is greater than zero) and the benefit from each unit of consumption is b (and b > c). Suppose that for any agent, his consumption goods can be produced by a fraction α of the agents. Other things being equal, as α becomes smaller, the economy becomes more specialized and exchange becomes more difficult. In contrast to the monetary search model with a complete-rationality hypothesis, we assume that the agents are rationally bounded. The agents either accept money or refuse money. Correspondingly, the population is 2

3 divided into two groups, acceptors and rejecters, and the proportion of each is denoted by x and 1 x, respectively. In the economy, the total supply of fiat money is fixed at M per capita, where 0 < M < x. Initially, fiat money is distributed randomly among acceptors, with each holding one unit. Thus, a fraction, M, of the agents holds money. For convenience, we make the following assumptions (which are the same as in the monetary search model): (1) money and goods are indivisible; (2) consumption goods are produced in units of one; and (3) the agents who hold money cannot produce. Under these assumptions, a commodity must trade one-for-one against other commodities or money. Each acceptor holds either 0 or 1 unit of money, and acceptors without money and rejecters can produce commodities, but acceptors with money cannot. Time is discrete. In each round, the agents randomly interact with each other and then obtain their payoffs. The agents prefer the strategy with a higher payoff and imitate this strategy in each round, so that this strategy gradually spreads in the population. We use the continuous replicator equation to describe this dynamic process (see Weibull, 1995), ẋ = (P a P )x, (1) where P a and P denote acceptors payoff and the average payoff in the population, respectively. As a limiting case of discrete dynamics, this continuous equation does not have a conclusion that varies qualitatively. 3. The Transition to the Monetary Equilibrium As deduced from equation (1), we can monitor the share x by the difference between the payoff for acceptors and the payoff for rejecters. For an acceptor with money, monetary trade occurs only when such an agent meets an acceptor without money that can produce his consumption goods. The probability of this case is α(x M). In this case, the acceptor with money obtains the benefit b and pays a unit of money with no intrinsic value. Thus, the payoff for acceptors with money is Pa 1 = α(x M)b. For an acceptor without money, there are two possible trades; barter and monetary trade. When he meets an acceptor without money (or a rejecter) and the agents can produce consumption goods for each other, barter occurs. The probability of this case is α 2 (1 M) and the payoff is b c. When an acceptor without money meets an acceptor with money who needs his products, monetary trade occurs. The probability of this case is αm and the payoff is c. 3

4 Thus, the payoff for acceptors without money is P 0 a = α 2 (1 M)(b c) αmc. Because the share of acceptors with money is M/x, while that of acceptors without money is (x M)/x, among all the acceptors the average payoff can be expressed as P a = M x x M α(x M)(b c) + α 2 (1 M)(b c). (2) x Barter occurs for a rejecter only when he meets an acceptor without money, or another rejecter, and they can produce consumption goods for each other. The probability of this case is α 2 (1 M) and the payoff is b c. Thus, rejecters payoff is given by Combining equations (2) and (3), we obtain P r = α 2 (1 M)(b c). (3) P a P r = M x (b c)[α(x M) α2 (1 M)]. (4) The difference between α(x M) and α 2 (1 M) decides how x varies. The quantity α(x M) represents the probability that acceptors with money will trade, and the quantity α 2 (1 M) represents the probability that rejecters will barter. Only when α(x M) > α 2 (1 M), that is, when x > α(1 M)+M, will the payoff for acceptors be higher than that for rejecters. In this case, rejecters will vary their strategies and become acceptors. Consequently, the economy will eventually reach monetary equilibrium where every agent accepts money. Consider the economy in the barter equilibrium, where no one trades the commodity for fiat money. For rational agents, they can calculate their future payoffs accurately. Any one rational agent finds that his payoff will decrease if he alone deviates from the barter equilibrium. Thus, an economy consisting solely of rational agents cannot ever deviate from the barter equilibrium. However, for bounded rational agents, who cannot calculate their future payoffs, the appearance of fiat money offers a new strategy which may improve their payoffs. Thus, some agents will try to accept money. Denote this share of agents by x 0. The above analysis shows that only when this share exceeds a threshold, that is, when x 0 > α(1 M) + M, (5) 4

5 will the economy start its transition from barter trade to monetary trade. In our model, the way in which fiat money is introduced into the economy ensures x 0 > M. α is a parameter that describes specialization and it becomes progressively smaller with the progress of specialization. Thus, when α is sufficiently small, condition (5) is satisfied and the transition from barter to fiat money begins. 4. Conclusion The transition from barter to monetary trade cannot be inferred from a monetary search model. The cause of this start problem is attributed to the complete-rationality hypothesis. From the viewpoint of reciprocal altruism, we build an evolutionary model with bounded rational agents to address this problem, and find an alternative way to link barter and monetary trade. Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC) under grants of and References [1] Dowd, K., 2001, The Emergence of Fiat Money: A Reconsideration, Cato Journal 20, [2] Kiyotaki, N. and R. Wright, 1993, A Search-Theoretic Approach to Monetary Economics, American Economic Review 83, [3] Nowak, M.A. and K. Sigmund, 1998, Evolution of Indirect Reciprocity by Image Scoring, Nature 393, [4] Nowak, M.A. and K. Sigmund, 2005, Evolution of Indirect Reciprocity, Nature 437, [5] Ritter, J.A., 1995, The Transition from Barter to Fiat Money, American Economic Review 85, [6] Shi, S., 2006, Viewpoint: A Microfoundation of Monetary Economics, Canadian Journal of Economics 39, [7] Weibull, J.W., 1995, Evolutionary Game Theory. (The MIT Press, Cambridge). 5