# Models of emergence of money

### 1. Kiyotaki and Wright (1989)

The expected discounted lifetime utility is defined as:

where Ui  is the utility from consuming good i, Di*  is the disutility from producing good i* (i* ¹ # 1), and β is the discount factor. cij  denotes the cost of storing good j for type i. is equal to one if the agent consumes good i, and it is zero oth- erwise.  is one if the agent produces i* and it is zero otherwise. Likewise,  if agent i stores any good j, it is zero otherwise.

The direct utility of consuming i for type i and producing i* is: ui = Ui – Di. In addition to this, there is an indirect utility of storing i*. Thus, the expected discounted utility for type i for this occasion may be defined as:

Vi(i) = ui + Vi(i*)

The indirect utility of storing good j ¹ i is:

where  is the expectation of Vi at next period’s random state j’, condi- tional on j, and the maximisation is over strategies.

Lastly, the distribution of potential matches is characterised by the time path of P(t) = [. . . pij(t) . . .], where pij(t) is the proportion of type i agents holding good j in inventory at time t. Yet it is assumed that p(t) = p for all t. And finally, b = β/3. Given these definitions,

#### Theorem 1

In Model A, under the maintained assumptions,

1. if c13c12 > 0.5bu1, then there is a unique equilibrium in which all agents use fundamental strategies and good 1 serves as the unique commodity money;
2. if , then there is a unique equilibrium in which type II and type III agents use fundamental strategies while type I agents speculate, and both goods 1 and 3 serve as commodity monies;
3. these are the only equilibria.

(Kiyotaki and Wright 1989: 939)

#### Theorem 2

In model B, under the maintained assumptions, there always exists an equilibrium in which all agents play fundamental, with goods 1 and 2 serving as commodity money; for parameter values implying

there also exists an equilibrium in which types II and III speculate while type I agents play fundamental, with goods 2 and 3 serving as commodity money;

these are the only equilibria.

(Kiyotaki and Wright 1989: 941)

### 1. Fiat-money equilibrium

Aiyagari and Wallace (1991) generalise the Kiyotaki–Wright (1989) model to N goods + 1 fiat money and show that there always exists an equilibrium where the lowest-storage-cost commodity serves as a medium of exchange. And if ‘fiat money’ is such an object, it becomes a medium of exchange. Similarly, Kiyotaki and Wright (1991) show that fiat-money equilibrium exists. Briefly, they pick up the unsettled questions about the existence of fiat-money equilibrium in the Kiyotaki–Wright (1989) model. They (1991: 222) inquire whether ‘a worthless paper or shell’ may serve as a medium of exchange ‘merely because’ individuals believe that others will accept it in exchange, or not. They show that if agents believe that the others would exchange their commodities for the fiat good, then there exists an equilibrium where an intrinsically useless object is used as a me- dium of exchange, that is, as fiat money.

Kiyotaki and Wright (1993) provide a simpler and more tractable version of the Kiyotaki–Wright (1991) model, where they discuss the welfare implications of their model in addition to the proof of the existence of pure monetary equilibrium. The Kiyotaki–Wright (1993) model is defined with a large number of infinitely lived agents and a large number of (indivisible) consumption goods, which they call real commodities. There is also a good with no intrinsic value, which we may call as fiat money if agents accept it in their exchange. Initial stage of the model economy consists of a fraction M of agents who are endowed with the intrinsi- cally valueless good, and a fraction (1 – M) of agents who are endowed with real commodities (1 > M ≥ 0). Agents who are endowed with real commodities are called ‘commodity traders’, and those who are endowed with fiat good are called ‘money traders’. The fiat good cannot be produced, and real commodities can. In accordance with the Kiyotaki–Wright (1989) model, agents meet randomly at the marketplace, and trade entails one-for-one swap of goods; they cannot consume what they produce, and they cannot produce if they do not consume. In contrast to the Kiyotaki–Wright (1989) model, in Kiyotaki–Wright (1993) commodities can be stored with no costs; yet real commodity barter entails transaction costs (e), while there are no transaction costs for monetary exchange – that is, for the fiat good. (However, they show that this assumption may be relaxed.) In a similar fashion to the Kiyotaki–Wright (1989) model, ‘agents maximize their expected discounted utility from consumption net of transaction costs, given strategies of others’ (Kiyotaki and Wright 1993: 66).

Under these conditions, agents always accept a barter offer if one of their consumption goods is offered, and it is assumed that a commodity trader never accepts a real commodity if it is not one of his consumption goods (note here that this assumption rules out the existence of a commodity money in this economy). A parameter x is defined to characterise the level of differentiation of commodities and tastes in this model economy, that is, in terms of what agents produce and consume. x may be considered as the probability that a real good will be accepted in exchange. Accordingly, x2 may be considered as the probability that a one- for-one swap of real commodities occur. Similarly, let P denote the probability that a commodity trader A accepts money, which represents the trading strategies of others. Therefore, in the Kiyotaki–Wright (1993) model, whether individuals accept money or not depend on the trading strategies of others (i.e. on whether others will accept the fiat good in exchange, or not) and on the initial endowment of the fiat good in the economy, which is represented by M.

Given these specifications and assumptions, Kiyotaki and Wright (1993) prove that there are three kinds of equilibria: non-monetary equilibrium (π = 0), pure- monetary equilibrium (π = 1), and mixed-monetary equilibrium (π = x). The in- tuition behind these equilibria is as follows. If the fiat good is accepted with lower probability than a barter offer (i.e. if π < x), then it is a good idea to never accept the fiat good in exchange. If the fiat good is accepted with a higher probability than a barter offer (i.e. if π > x), then it is a good idea to use fiat good in exchange. And if the probability that fiat good and real commodities can be exchanged is the same (i.e. if π = x), then individuals would be indifferent between using fiat good in exchange, or not. In addition to the specification of these equilibria, Ki- yotaki and Wright (1993) show that introducing fiat money to a barter economy enhances welfare, and that an economy with multiple currencies is possible (i.e. an economy where several type of intrinsically valueless goods serve as a medium of exchange).

### 2. Marimon et al. (1990)

Economy A1 (A1.1 and A1.2) converges to a fundamental equilibrium, that is, the lowest-storage-cost commodity emerges as a medium of exchange. But Marimon et al. (1990: 359) report that the convergence is lower when ‘initial classifiers are randomly generated’ – that is, for A1.2.

Economy A2 is similar to Economy A1, but only the utilities of the com- modities are higher. Marimon et al. (1990: 360) convey that the results for this economy are ‘fairly inconclusive’. The simulations for Economy A2.2 do not support Kiyotaki–Wright argument that the economy would converge into the speculative equilibrium. Remember that speculative equilibrium is an equilibrium where some agents use a high-storage-cost commodity as a medium of exchange, for they believe that it is more marketable. They argue that the ‘trading patterns were closer to fundamental equilibria’. But it is also argued that fundamental equilibrium only exists if the discount rates are sufficiently low. (Marimon et al. (1990: 362) report that Marimon and Miller’s (1989) simulations for Economy A2 converged to speculative equilibrium.) Results for Economy B support the idea that fundamental equilibrium is selected. Yet in early stages of the simulation, economy B1 converges to a speculative equilibrium, before moving to the funda- mental equilibrium in later stages. Although the Economy B2 does not converge, it is closer to the fundamental equilibrium.

In economy C, a fiat good (good 4) is introduced to the economy, which does not provide any utility. Good 4 has no storage costs, but it decreases the storage capacity of the agents. It is assumed that some agents hold good 4 in period 0. Marimon et al. (1990: 366) report that economy C2 converges to the fundamental equilibrium quite fast. Yet economy C1 does not converge probably because of the low storage costs of good 1 (see Table 6.2), which also provides utility. The simulation of this economy implies that if the storage costs of non-fiat commodi- ties are sufficiently high and if at least some of the agents hold no-utility goods at the initial stage, fiat money may be brought about. Yet, note here that there is no reason for the agents to hold no utility goods and accept it in exchange; rather, it is assumed that agents hold fiat goods. Note that the results presented here are for randomly generated classifiers. Marimon et al. do not report what happens when agents know all the available strategies. Obviously, if some of the AI agents consider using good 4 in exchange by chance – which corresponds to believing that it will be accepted – then it is possible that fiat money emerges in the end of the process. In Marimon et al.’s simulations, whenever the economy converges to fiat-money equilibrium this happens quickly; if it does not converge quickly (i.e. in the first stages), it does not converge at all. This supports the idea that if agents are able to trade with fiat goods accidentally in the first periods, fiat money emerges; if not, the strength of the rule ‘trade with the fiat good’ decreases and fiat money does not emerge. Of course, this also suggests that when agents are not fully rational they may explore different strategies and create new opportunities. But in general, there seems to be no reason for the agents to accept a fiat good in exchange if they do not believe that it will be accepted by others – hence the com- mon belief assumption. Finally, economy D depicts a more complex economy with no fiat good. Marimon et al. report that although the trading patterns are close to a fundamental equilibrium, the simulations are inconclusive. That is, although agents accept a good in exchange if its storage cost is lower than the storage cost of the good in their inventory, no single medium of exchange emerges.

### 3. Gintis (1997, 2000)

An alternative to Marimon et al. (1990) is presented by Gintis (1997, 2000) (also see Dawid 2000). Gintis (1997: 24) reports that he replicated Marimon et al.’s simulations and found that they were highly sensitive to the choice of parameters. (This sensitiveness supports the argument that Marimon et al. only show what is possible under specific conditions.) Gintis’s simulation is different in that it is based on Darwinian notions, such as natural selection, mutation and adaptation, and dispels AI agents. Every agent is characterised as having a ‘genotype’, which determines his strategy. That is, he replaces the rational agents of the Kiyotaki– Wright model, or AI agents of Marimon et al., with a description of strategies. To ensure variation, Gintis assumes that genotypes are randomly assigned in period 0. In period 0 there are five goods with different storage costs. As it is in the Kiyotaki–Wright (1989) model, there are different types of agents who cannot consume what they produce. (Hence) there are twenty types of agents. In period 0, every agent has an empty inventory and zero wealth. Beginning from period 1, agents are matched in pairs randomly and whether they trade or not is dictated by their genomes. Successful trade increases the wealth of trading agents. After some time the least fit agents of each type die. That is, the least successful strategies become extinct after some time and new agents are introduced into the economy. New agents are offspring of successful agents and they take over the successful genomes. Yet there is mutation. For this reason, new agents’ genomes are different from their parents’ genomes.

Gintis shows that usually the lowest-storage-cost commodity emerges as a me- dium of exchange out of this process (i.e. fundamental equilibrium). He reports that, other than storage costs, frequency of use can be considered as an important factor in the process of the emergence of money. That is, if there is a good that is more likely to be accepted by all other agents in trade than the lowest-storage-cost good, then this good emerges as a medium of exchange (i.e. speculative equi- librium). Gintis also shows that unless we assume at the outset that a very high percentage of agents accept a fiat good, fiat-good equilibrium will not emerge. That is, unless there is a common belief that a fiat good will be accepted by all others, fiat-money equilibrium cannot be reached even if one exists:

using fiat money involves a self-fulfilling prophecy, in the sense that if enough of the population expects a fiat good to be accepted as money, then everyone will accept it.

(Gintis 2000: 228)

Gintis’s results are generally in accordance with the previously examined models. Yet he introduces an evolutionary mechanism that selects the successful strategies, and which does not give much role to rational decisions of the agents. This may suggest that the emergence of a commodity money equilibrium does not need much intelligence or rationality. Yet the introduction of new agents who inherit the successful strategies suggests that there must be some role for the indi- vidual learning or imitation.

Source: Aydinonat N. Emrah (2008), The Invisible Hand in Economics: How Economists Explain Unintended Social Consequences, Routledge; 1st edition.