7. The Evolution of Money as a Medium of Exchange in a Primitive Economy

The editor has drawn my attention to another related paper written recently by Sethi (1999) which analyzes the Kiyotaki–Wright model using an evolutionary game theoretic framework.

Within the context of a dynamic programing model where the expected discounted utility is maximized by traders, Renero (1998) also finds that initial conditions matter; however, his conclusions, unlike those of this chapter, are that many of the conventional equilibria found in the literature (e.g., Kiyotaki and Wright (1989), Aiyagari and Wallace (1991)) are not stable.

The coexistence of more than one commodity serving as a medium of exchange is well documented (e.g., cattle, goats, and cloth of the Wabena of Tanganyika Territory in Africa in the early twentieth century; tobacco and sugar in seventeenth century North America; and wadmal (spun from the fleece of the sheep) and fish in fifteenth century Iceland (Quiggin 1949).

Just as Aiyagari and Wallace (1991) generalize the Kiyotaki–Wright model to N goods, the following model can be generalized to N commodities and N agents. Further discussion of this generalization is provided later.

The model is basically that of Kiyotaki and Wright (1989) with two differences: all goods are perishable and trade occurs over the two trading sessions (the morning and afternoon). The above assumptions considerably reduce the complexity of calculating the distribution of agents’ holdings over time and thus the computation of equilibria.

This assumption prevents trade from occurring in the afternoon session when only one party wants to trade. This assumption can be justified if any trade incurs an arbitrarily small transaction cost, this paper’s results would remain basically unchanged. Nevertheless, it is interesting to consider an alternative model where the framework remains the same, but traders make trades in the afternoon as long as one party wants to trade. In this case, it is easy to show that, in terms of the notation of the latter part of this section, \(\frac{{P}_{i(2)}^{t}(s)} {{\overline{P}}_{i(2)}^{t}} = \frac{{P}_{i(1)}^{t}(s)} {{\overline{P}}_{i(1)}^{t}}\) for all t, and the economy converges to the fundamental equilibrium where the most storable good is the unique medium of exchange.

This assumption is reflected in the following probability calculations. The framework could be altered by not allowing successful morning traders to participate in the afternoon. While the essence of the results would remain the same, the following probability calculations would have to be slightly altered.

Notice that λ _i(k) is a model parameter and reflects relative storability among commodities. More precisely, the λ _i(k) is a function of relative storability and does not depend on time. The justification for this is as follows. Since the behavior of agents i(1) and i(2) differs only with respect to meeting agent type i + 1 in the morning and since in the afternoon both subtypes of agents behave the same in that they only trade for their own consumption goods, it follows that an alternative way of viewing agents’ subtypes in the morning of each day is to see agent i as deciding on whether, in the event of meeting agent i + 1 in the morning (which occurs with a constant probability of P _i + 1), he or she will trade with agent i + 1 (i.e., choose strategy i(2)) or will not trade with agent i + 1 (i.e., choose strategy i(1)). Suppose such factors as inertia and portability, that may influence λ _i(k), are set aside and λ _i(k) is hypothesized to be only a function of storability. Since the expected storage costs, conditional on not meeting agent i + 1, incurred by agents i(1) and i(2) are identical, what only matters in determining the λ _i(k) (the probability of switching strategies) is the relative storage costs incurred as a result of meeting i + 1. Suppose good i costs c _i to store from morning to afternoon. In the event of meeting agent i + 1, agent i(1) would not trade and would only incur a storage cost of c _i + 1 in storing good i + 1 until the afternoon. Agent i(2) would trade with i + 1 and would incur a storage cost of c _i − 1 until the afternoon. Therefore, λ _i(k), the probability of agent i switching from strategy k to k ^′ (k≠k ^′ ), is a parameter which is an increasing function of \({(-1)}^{k}({c}_{i-1} - {c}_{i+1})\).

If, instead, an alternative dynamic, the best reply dynamic (see Kandori et al. (1993)) is applied to the selection of strategies, then not all combinations of pure strategies would be steady states. However, all of the asymptotically stable equilibria in this paper are the same asymptotically stable equilibria under the best reply dynamic. Furthermore, the open balls characterized in all propositions of this paper would also support the respective asymptotically stable equilibria under the best reply dynamic. Property (i), as outlined in the definition of the basin of attraction in Theorem 1 of Appendix A, indicates that for all points in the basin of attraction, the strategy corresponding to the respective asymptotically stable equilibrium is the strategy with the higher relative payoff. But this precisely describes a basin of attraction for the corresponding asymptotically stable equilibrium of the best reply dynamic model (since in the best reply dynamic model players play the strategy with the higher relative payoff).

I have also explored an extension of the model which allows for mixed strategy equilibria. Definition 1 would have to be altered slightly. In addition to properties (1), (2), and (3), an additional property is added: \({f}_{i}\big{(}{P}_{1(2)}^{t},{P}_{2(2)}^{t},{P}_{3(2)}^{t}; \lambda \big{)} = 0\) if \(\frac{{P}_{i(2)}^{t}(s)} {{\lambda }_{i(2)}{\overline{P}}_{i(2)}^{t}} = \frac{{P}_{i(1)}^{t}(s)} {{\lambda }_{i(1)}{\overline{P}}_{i(1)}^{t}}\) for P _i(2) ^t  ∈ (0, 1), and for i = 1, 2, 3. Mixed strategy equilibria only exist under specific restrictions on the parameters. For example, the equilibrium (x, 1, 0) for 0 < x < 1 identified by Kehoe et al. (1993) exists only if \(2({P}_{1} + {P}_{3}) = \frac{{\lambda }_{1(2)}} {{\lambda }_{1(1)}}\). For this reason, this paper focuses only on pure strategy equilibria.

If the focus is not in finding basins of attraction (described by the open balls B _j ) and the interest is only in determining the asymptotic stability of equilibria, for asymptotic stability, one only has to show that the equilibrium is a strict Nash equilibrium.

In the above, mutations produce mistakes in the sense that the imitation is irrational. An alternative interpretation of μ _i(k) is that it represents noise in imitation. The more extensive the noise in the information structure regarding the success of the mimiced agent, the higher would be μ _i(k) relative to λ _i(k). Agents imitate others, regardless of the success of others. Another type of mistake would occur if agents arbitrarily moved away from steady states (i.e., arbitrary perturbations of the steady state). However, as noted by Weibull (1995), this type of mutation in the form of small perturbations is indirectly taken care of by the way of dynamic stability criteria. By the definition of asymptotic stability, such arbitrarily small movements imply that the economy returns to the steady state. However, with such perturbations away from an asymptotically unstable equilibrium or very large perturbations away from any equilibrium, the economy could evolve into another steady state.

In a model of N goods and N agents, where agent i consumes only good i and produces only good i + 1 (modulo N), one could specify s _i ^j as the probability that agent i is willing to trade his or her production good for j where j≠i and j≠i + 1, when agent i meets an agent holding good j.

Suppose traders only play pure strategies. That is, s _i ^j  ∈ { 0, 1}. If s _i ^j  = 0 then agent i is not willing to trade for consumption good j. Agent i is willing to trade for commodity j when s _{i{ }} ^j  = 1. The proportion of agent i choosing s _i ^j  = 1 at time t could be referred to as P _ij(2) ^t and the proportion of agent i choosing s _i ^j  = 0 could be referred to as P _ij(1) ^t . One could further define μ _{i{ }} ^j as the probability of agent i imitating strategy s _i ^j regardless of the success of s _i ^j and λ _{i{ }} ^j as the incremental probability of agent i imitating a successful strategy s _i ^j . Furthermore, the sizes of λ _{i{ }} ^j and μ _{i{ }} ^j could be modeled as a function of the degree of storability. That is λ _{i{ }} ^j  > λ _{i{ }} ^{j ′} and μ _{i{ }} ^j  > μ _{i{ }} ^{j ′} if commodity j is more storable than commodity j ^′ . The number of dynamic equations of motion would then equal to N ×(N − 2). Thus, the model becomes more intricate as N goes beyond 3. While the number of possible asymptotically stable equilbria undoubtedly grows, similar to the conclusions of this paper, increasing the size of the mutation rates further selects a reduced set of long run equilibria.

It should be noted that if agents make large enough mistakes, under different restrictions with respect to mutation and storability parameters, the system could move from one basin of attraction to another. Nevertheless, Proposition 7.6 focuses on the set of restrictions under which mutation rates become sufficiently large to produce a unique fundamental equilibrium.

If we interpret the μ _i(k) as representing noise in imitation and since all agents perceive the relative storability of commodities in the same way, then Proposition 7.6 could be loosely interpreted as saying: “If agents know very little about what other agents are doing, then the fundamental equilibrium is the most likely to occur.”