Directory UMM :Data Elmu:jurnal:J-a:Journal of Economic Behavior And Organization:Vol41.Issue1.Jan2000:

Vol. 41 (2000) 27–53

On the emergence of exchange and mediation in a

production economy

Herbert Dawid

∗

Department of Economics, University of Southern California, Los Angeles, CA 90089, USA

Received 14 October 1997; accepted 21 August 1998

Abstract

In this paper we simulate the behavior of a population of boundedly rational agents in a two good economy where all agents can spend their time budget for the production of one or both goods or trading. Agents update their strategies according to a simple imitation type learning rule with noise. It is shown that in several different setups both direct trade and trade via mediators who specialize in trading can emerge. Both increasing returns to scale in production and heterogeneity of production technologies facilitates the development of trade. For heterogeneous production technologies we can also observe the transition from a pure production economy via direct trade to an economy with mediated trade. ©2000 Elsevier Science B.V. All rights reserved.

JEL classification: D83; F10

Keywords: Bounded rationality; Learning; Trade; Mediation

1. Introduction

Mediated trade is a phenomenon which can be observed in many different markets in the real world economy. Examples reach from retail stores to real estate agents and stock brokers. In all those cases ‘producers’ do not directly deal with the ‘consumers’ of their goods but there are middlemen in-between who facilitate the trades. The basic role of these middlemen is to reduce the search costs of buyers and sellers needed to find an appropriate trading partner. The middlemen profit from these transactions by marking up their selling prices but it is quite obvious that trading via mediators nevertheless pays off for all agents in the economy.

∗_{Tel.: +1-213-740-8842; fax: +1-213-740-8543} E-mail address: dherbert@usc.edu (H. Dawid)

In most economic models the problem of search and transaction costs in trading is ne-glected and it is assumed that producers and consumers have no problems finding each other and agreeing on the equilibrium price to carry out the trade1. There are several exceptions like Rubinstein and Wolinsky (1987), Day (1994) or Pingle (1997) where middlemen are ex-plicitly introduced into a model of trade. The perspectives taken in these three contributions vary significantly. Whereas Rubinstein and Wolinsky deal with middlemen in an equilib-rium theoretic framework, Day studies the out of equilibequilib-rium behavior of an economy with market mediators by specifying adaptation rules for the agents. Pingle, on the other hand, carries out econometric analyses of data obtained by experiments in a model where agents can choose between producing and opening a store (i.e. mediating). The data presented in Pingle (1997) shows that mediation emerges as an important part of the economy also in laboratory experiments.

In this paper we follow a similar line as Day (1994) and deal with the question of the emergence of trade and mediation in a dynamic off-equilibrium framework. We do not assume that all agents have rational expectations and know their optimal decision but rather consider a learning model where agents are boundedly rational and slowly acquire the experience and knowledge to improve their performance. It has been argued in several places (see e.g. Sargent (1993) or Simon (1983)) that the relaxation of the very demanding rationality and knowledge assumptions underlying general equilibrium theory might allow new insights and the study of more realistic economic models. We adopt this point of view here and study the behavior on the path towards equilibrium rather than equilibrium behavior itself. Contrary to Day and Pingle’s work we do not represent agents by mathematical equations or experiment with human agents, but we simulate the adaptive behavior of a population of boundedly rational economic agents on a computer. Computer simulations have recently gained high importance also in economic research (see e.g. Routledge, 1995; Arifovic, 1996; Dawid, 1996; Curzon Price, 1997; McFadzean and Tesfatsion, 1997) and chances are they will become even more important in the near future. Although such an approach does not permit exact general results like analytical studies do, it enables us to study complex interactive models which could not be dealt with analytically. The main question posed in this article is: can boundedly rational agents who all start off as pure producers organize in a way to use the possible profits of production specialization and trading? How do certain parameters like the transactions costs of trade, the inertia of agents or the returns to scale in production influence the emergence of trade? We will also study the question whether increasing returns to scale in production lead to direct trade without middlemen or to mediation, respectively.

The learning rule we use to describe the adaptive behavior of the agents is in the spirit of imitation dynamics often used in the evolutionary game theory literature. The underlying assumptions about the knowledge and computational ability of the agents are weak but we will see that the massive interaction of the agents nevertheless leads to quite efficient behavior involving mediation and direct trading.

The paper is organized as follows. In Section 2 we present the basic simulation model where all agents in the population have the same production technology and produce with

1_{Note however that there is some literature dealing with the emergence of a medium of exchange in trade; see}

increasing returns to scale. In this section we also provide some considerations about what outcomes to expect for different parameter setups. Simulation results for this model are shown in Section 3 where also the influence of parameter variations on the simulation results are explored. We finish with some concluding remarks in Section 4.

2. The model

In this section we describe the basic model we will use in our simulations. All agents are equal here and the attraction of specialization and trade entirely stems from increasing returns to scale in production. Within the basic model we consider two different setups. In the first setup the agents cannot hold stocks and thus only engage in direct trade whereas the second setup gives the agents the possibility of becoming a mediator, building up stocks and exclusively concentrating on trading.

2.1. The agents

We model the evolution of a system ofninteracting economic agents who may split their available time on production and trading. All agents have identical technology and preferences but may differ in their actions. Each agent has a fixed time budget in each period which is normalized to one. There are two goods in the economy and he may spend this time on producing good 1, producing good 2 or trading. The behavior of agentiis described by two variables governing his production (spi,pri) and one variable(si) determining the amount of time he invests in trading. The variables may vary with time but we omit the time argument in our notation. The variable spi _∈ [0,1] denotes the degree of specialization in the production decision of the agent, whereas pri ∈ {1,2}determines which of the two products the agent specializes in. If we denote byx_gi the fraction of time the agent invests in producing goodgwe have

Why we use this special representation of the production decision will become clear when we describe the learning process. This representation allows the agents to change directly from specialization in good 1 to specialization in good 2 without going through states of lesser specialization in-between.

Production functions of the goods are identical for all agents and read

where increasing returns to scale are assumed(αg>1). The preferences of the agents are

The trading behavior of the agents is determined as follows: given a pricepof good 1 (expressed in units of good 2) the agent has to determine whether he likes to buy or sell good 1. Assuming that an agent currently holdsγi units of good 1,δi units of good 2 and maximizes the utility gained by consuming all the goods he is holding after this trade he wants to sell

units, respectively. Whenever two agents are matched for trading (the mechanism governing this matching is described below) they exchange goods where the pricepand the quantityy

of good 1 exchanged are determined such that excess demand for good 1 equals the excess supply. It is easy to see that with the supply and demand functions given above there is always a unique pair(p, y)with this property.

2.2. Transactions within a period

In each periodtthree different stages occur: 1. Production

2. Trade 3. Consumption

In the first stage all agents produce according to their production variablesx_gi. After pro-duction they might trade the good. We denote byγi andδi the amount of goods 1 and 2 agentiis holding during the trading period. The holdings vary during the trading period and these variables always denote the current value. Initially, we have

γi ₌f1(x1i), δ

i

=f2(x2i).

sacrifice time for trading — go all the way and loose twice as much time. Of course he can also keep producing the whole time and wait for some of his neighbors to come all the way and trade with him.

In our simulations we use the following procedure to determine the trading partners. Two agentsiandj can only trade with one another if the sum of the time invested in trading exceeds some given thresholdχ > 0. The initial trading time budget for each agent is in each period given bySi =si and is reduced step by step during the trading period by the amount of time which has already been used for trading. Interpreting the effort invested in trading as search costs this restriction means that the agents have to invest time for searching for partners if they like to trade. Note, however, that all effort might be invested by one party allowing ‘professional’ traders who can be reached by others without any costs. We use the following matching algorithm for trading: Defining the ‘trading pool’ as the set of all agents who might trade in the rest of periodt the algorithm can be described as follows:

1. Choose randomly an agenti from the trading pool where each agent in the pool is chosen with the same probability.

2. IfSi ₌0 return agentito the trading pool and goto 1.

3. Choose randomly (again uniformly) an agentj from the rest of the trading pool. 4. IfSi₊Sj < χ goto 5, else determine amount and price for the trade betweeniand

j. Ifδiγj < δjγi good 1 is traded from agentito agentj, if the inequality holds the other way round good 1 is traded from agentj to agenti. In the case of equality no trade takes place. Let us assume that good 1 is traded fromitoj. The amount traded is determined by the intersection of the trade functions of the two agents

yij =

them has to invest trading time in the amount of χ₂ to find the partner; however, if an agent does not actively search for a partner or does so only for a short time (Si < χ /2) he is harder to find and his partner has to invest more time. Note also that this scheme implies that a trader leaves the market as soon as he has invested all his trading time.

Note that the trading scheme described above assumes that passive traders (agents with

si ₌0) only trade once and leave the market afterwards.

After the trading period all agents consume their current holding and receive a utility of

Ui =b

The fact that we did not allow the agents to build up stocks in the previous model rules out the possibility of the emergence of agents who exclusively concentrate on trading without producing any goods themselves. We call these agents mediators and will now extend the model by allowing the agents to decide to mediate in the market rather than to produce. This is done by adding three more decision variables to the existing three decision variables of each agent. The first of the three additional variables, idi describes the identity of agent

i. Whenever this variable has value 0 the agent is a producer and behaves in exactly the same way as the agents described above. If idi ₌1 the agent is a mediator. This implies thatsi ₌1 and no time is invested in producing. Furthermore, a mediator has a different kind of trading behavior than the producing agents. Whereas the producing agents trade in a way to maximize their utility from consumption the meditator rather sells and buys good 1 at fixed prices. He sells one unit of good 1 forpi_s units of good 2 and buys it for

p_bi units of the second good. These two prices are decision variables of the agent and again might change over time. Of course, mediators always markup selling from buying prices and we havepi_s > pi_b. To be able to mediate, an agent has to possess some stocks of both goods. Thus, we assume that whenever an agent changes due to imitation or innovation from production to mediation he initially produces without trading for four periods (two periods for each good) and afterwards completely stops production and starts trading. The stock of goodgagentiholds is denoted byl_gi. If an agent switches from mediation to production he leaves his stock untouched and is able to use this stock again if he decides to switch back to mediation at some time.

Mediators trade only with producers but never with other mediators. When a mediatori

is matched for trading with a producerj an amount of

zero and no trade occurs. The minimum operator used in these expressions ensures that no mediator sells a higher amount of a good than he has on stock. After the trade the current holdings of the producer and the stock of the mediator are updated.

The mediators try to keep their overall size of stock after consumption constant over the periods. However decreasing marginal utilities of both goods makes it profitable for the mediators to smooth their consumption and consume equal amounts of both goods. Thus, they consume the same aggregate amount of goods they have gained by trading in the current period but split consumption equally between the two goods in order to increase utility. Denoting byl_g,ti ₋₁the stock held at the end of periodt −1 and by l˜i_g,t the stock held after trading in periodtwe defineci _{= ˜}l₁i_{,t+ ˜}l₂i_,t−l₁i_,t−1−l₂i_,t−1. In every period the consumption of mediatoriin periodt

γi ₌min

The decisions of an agent whether he should concentrate entirely on production or invest some time in the search for trading partners and the determination of the production plan are rather complex problems. How much time should be invested in trading and which goods should be offered depends crucially on the decisions of the other individuals. The payoff of a certain strategy is also influenced by the fact which potential trading partners are randomly matched with an agent. Analytically determining the optimal strategy requires the exact knowledge of all other agents’ actions in the next period and, even with this knowledge, quite complex calculations of the expected payoff of all available strategies. As pointed out above we do not assume that the agents have the information and capability to carry out these calculations but rather use a simple learning rule to determine their strategy. The proposed learning algorithm describes imitation based adaptation of the agents strategies. The individuals do not build any expectations in order to optimize anticipated payoffs but just consider the past success of other individuals and try to imitate the ones with above average utility. Imitational learning rules have been analyzed in several mainly game theoretic contexts (e.g. Schlag (1998), Vega-Redondo (1995) or Björnerstedt and Weibull (1996)) and it has been shown that in certain environments proportional imitation constitutes an optimal learning rule (Schlag, 1998). Although such a kind of adaptation underestimates the complexity of the actual decision making process of economic agents in most contexts it can nevertheless provide interesting insights into the evolution of a population of boundedly rational agents who do not have enough information about their environment to be able to predict future developments in a sensible way or to determine optimal responses to expected future developments. In such situations the reliance on strategies which worked well in the past may indeed be rational behavior (see also Pingle (1995)).

We consider a learning process where all agents review their current strategy everyτ

strategy of an agentjincreases with the past payoff ofj. LetU¯j denote the average payoff of agentj in the previousτ periods. With

πj(i)_{= ¯}

Uj j ₆₌i wU¯j j =i

the probability that agentiadopts the strategy of agentjis given by

5j(i)= π

j_(i)

Pn

k=1πk(i) ,

The parameterw ≥ 1 governs the inertia of the agent by increasing the probability that he uses his own strategy again in the nextτ periods. After all agents have adopted their new strategies these strategies are disrupted by stochastic shocks. These shocks might incorporate implementation errors of the agents but also intended innovations. With some small probabilityµv > 0 an amount ξv generated by a normal distribution N (0, σv2)is

added to the variablev_{∈ {}sp, s_}of agenti. If the shock drives a variable out of [0,1] the variable is set to 0 or 1, respectively. The variable determining the good to specialize in, pr is changed from 1 to 2 or vice versa with some small probabilityµpr. This completes

one learning step and all agents use their new strategies for the nextτ periods. Afterwards another learning step takes place and so on.

In the model with mediators innovation effects also the three additional decision variables idi, pi_sandp_bi. The variable idiis inverted with some small probabilityµidand there is some

probability that normally distributed noise is added to the prices. In case that innovations would lead to a violation ofpi_s> p_bi these innovations are neglected.

The learning dynamics we use may be interpreted as a stochastic version of the well known replicator dynamics (Taylor and Jonker, 1978) which was thoroughly analyzed in the biological and economic literature (e.g. Cressman, 1992; Hofbauer and Sigmund, 1998). Of course several variations of this rule could be considered. It would be especially interesting to consider a model where agents can only observe the strategies of the individuals they meet for trading. However, here we assume that information spreads faster than the goods and that agents can also get information about individuals they do not meet themselves. Further, we could include memory in our model and assume that agents remember also past payoffs or payoffs of trading with certain partners in the past. These extensions might make the model slightly more realistic. However, we stick with the simple model presented above because it seems that this model captures the general main properties of imitation learning and thus allows qualitative insights into the dynamics.

From a mathematical point of view the evolution of the population state can be described by a time homogeneous Markov process on a continuous state space. Due to the extremely complicated structure of the transition functions a rigorous mathematical analysis seems to be impossible and even approximation results for decreasing mutation probabilities are out of reach2. Thus we have to rely on simulations in order to study the dynamic behavior of the system. In the next subsection we will carry out a very loose comparative static

2_{Kandori and Rob (1995) derive a general theory characterizing the long run outcome of learning dynamics with}

analyses pointing out some properties of the static model with completely rational agents. The dynamic properties will be studied in the next section.

2.5. Analytical considerations

Having presented the model we will now shortly consider the case that the agents in the model are completely rational. A rigorous derivation of all equilibria is quite complicated even in this static framework and we abstain from doing this here. However, we will get some clues from these rather loose considerations which effects a variation of the parameters might have on the simulation results. In particular we are interested in the value parameter

αgoverning the increasing returns to scale in production. First of all we would like to note that an increase ofαhas two basic effects in our model. The first effect is that a highα

facilitates the specialization of the agents on producing only one good. This is quite obvious since specialization allows the agent to produce in a region where the marginal productivity is larger than if he would produce both goods. If we consider an economy without trading this effect is of course (at least partly) neutralized by the fact that also the marginal utility of a good decreases with increasing consumption. Which of these effects is stronger depends on whetherα >2 or not. The second effect of an increase ofαstems from the fact that the opportunity costs of trading increases with increasingα. The producer always has to invest his most efficient production time in trading and the faster the efficiency of production increases the more costly this is. Thus, we roughly might expect the following picture. For values ofαonly slightly larger than one specialization of production does not pay off and accordingly there is also no trade in the economy. In such a case the average population payoff is

Upr =2b r

a

2α.

For larger values ofαspecialization with direct trade becomes the most efficient way of organizing the production process, but only as long as direct trade does not become too expensive for the producers. In a population where half of the agents produce good 1 and the others produce good 2, but all agents invest a fraction ofχ in trading (which implies that every agent expects to meet one agent producing the other good per period) the average population payoff is3

Utr =bp2a(1−χ )α_.

Ifαis large, it is rather expensive for the producers to invest time in trading since they have to sacrifice their most efficient production time. If no mediation exists the optimal choice of the agents in such a situation would be to specialize in the production of one good without trading. This means that agents can only consume one good, and accordingly have a utility of

Usp=b √

a.

It is easy to see that this expression is larger thanUpr wheneverα >2 and larger thanUtr

ifα > ln 0.5/ln(1₋χ ). This shows that in the model without mediators trade is only attractive for intermediate values ofα.

At this stage we would like to stress that these considerations only imply that trading would pay off for certain values ofαafter it has been established. Of course this does by no

means imply that trade will indeed emerge in a population with boundedly rational agents. In a state where almost all agents spend all their time for production the probability to meet a trading partner is very small for an agent who starts trading and thus his expected payoff might be much smaller thanUtr. However, if several agents who start trading are matched by accident and their payoff is significantly larger than that of the rest of the population they might eventually take over the whole population and establish trade in the economy. For which values ofαtrade does indeed occur in the long run cannot be answered analytically but we have to rely on simulations here.

If we take into account the possibility of mediated trade, the effect, that trading becomes more expensive the higherαis, disappears. In such a case we might expect that professional traders emerge and do all the trading whereas the producers do not spend any time on trading at all. In order to obtain exact analytical expressions characterizing the range ofαfavoring direct trade, respectively, mediated trade we have to take into account the rather complicated matching algorithm. Accordingly these calculations become quite involved and we restrain from carrying them out here. Again, we will gain some insights from the simulations.

Let us now consider a state where there are mediators in the market and determine which number of mediators we should expect. For sake of simplicity we assume that a fraction of

r/2 of the agents only produce good 1(x1 =1, x2 =s =0), a fraction ofr/2 produces good 2(x2=1, x1=s=0)and a fraction of 1−rof the agents does not produce at all but trades(x1=x2=0, s=1). The number of producers a trader can deal with is constrained

by the fact that each producer exchanges goods only with one trader per period and the time constraint. Denote byµ ₌ min [r/(1₋r),2r/(χ (1₊r))] the expected number of trading partners per trader. For the sake of simplicity let us assume that exactly half of these partners produce good 1 and half of them produce good 2. Let us further assume that the trader buys good 1 at a price ofpand sells it at a price of 1/p(such a behavior is optimal due to symmetry) and that he can always deliver the quantity of goods he likes to trade. Given these simplifying assumptions it is easy to see that the income per period of a trader is given by

Umed=2b

r _µap

2(1₊p)(1−p).

In order to optimize this expression the trader should choosep=√2−1 which gives him a utility of

Umed= √

2(√2₋1)b√µa.

Upr₌b√a

In an equilibrium the utility of traders and producers must be equal. Under the assumption that the time constraint is not binding for the traders (i.e.µ =r/(1−r)) this yields the equation

These considerations indicate that if all agents were completely rational, completely coor-dinated andχis sufficiently small there would be about 20% mediators in a heterogeneous state consisting of producers and mediators. A question to be answered in the next sec-tion is whether a similar degree of organizasec-tion can actually be reached by a populasec-tion of boundedly rational agents and, if it can, what kind of transient behavior can be observed. Also, the effect of the parameters in the learning rule andχ(which is neglected here) on the emergence of mediation will be studied by the means of simulations.

3. Simulation results

In all our simulations we use a population of sizen₌100. The population is initialized homogeneously such that the agents decision variables are given byx₁i ₌x₂i ₌1/2, si ₌0. The parametersaandbare both set to 1. We always assume that the agents update their strategies every 10 periods (τ ₌10). The threshold for trading is given byχ ₌ 0.1 and the inertia parameter for the imitation process was always set tow =2. Concerning the innovations a normally distributed noise term with expectation 0 and varianceσ =0.1 is added to the continuous decision variables with probabilityµ=0.02. The probability that the production focus changes from good 1 to good 2 or vice versa is in all these simulations

µpr =0.05. The simulations were run forT =3000 generations. Most of our results appear

The purpose of these simulations in not only to demonstrate the potential for self or-ganization a population of boundedly rational agents has — this has also been shown in several other frameworks — but, on one hand, to study how trade emerges with time and, on the other hand, how certain features of the model like search costs or the agents’ inertia influence the dynamics of the different variables.

3.1. Model without mediators

First, we study the effect of an increase ofαin the model without mediators. Our ana-lytical considerations suggest that for small values ofαthe state where all agents produce both goods without trading is an equilibrium. For intermediate values ofαdirect trade is profitable for the agent if it can be established but if the returns to scale increase very fast the individuals cannot gain from trade but should rather restrict themselves to the production and consumption of a single good.

In order to explore the effects of a variation ofαwe increasedαfrom 1 to 4 with a stepsize of 0.2 and performed 10 simulation runs for each value. In Fig. 1 we show the average values of consumption, trade (measured in units of good 1), degree of specialization4 and time invested in trading after 3000 generations (the superscript av always indicates that the depicted values are average values of 10 runs).

It can be seen that forα _∈ [1,2] the population state more or less stays in the initial state where all agents produce both goods and do not trade. If we look at the trajectories of the variables in such a setup we cannot detect any significant changes throughout the run. Due to some kind of random drift the degree of specialization and the time invested in trade increases a little bit with time. The average utility is slightly smaller than the equilibrium value. However, there occur no relevant amounts of trade and the population remains a set of isolated producers.

This picture changes significantly for faster increasing returns to scale. Here agents learn to specialize in the production of only one good and also the amount of time invested in trading goes up reaching the value ofχ =0.1. The largest trade volume can be observed for values ofαclose to 2.55. Thus we show the trajectories of the crucial variables in a run with this parameter setting in Fig. 2 .

Even with these parameter values no significant trade emerges up to period 1500. How-ever, afterwards we can observe a sharp increase insp_¯,tr andU¯ and the market very quickly changes from a regime of isolated production to one of specialized production and direct trade. The trading volume is on average slightly below 10 units of good 1 per period. This is substantially lower than in an optimally organized market — as described in the last section — where 19.2 units of good 1 would be traded per period. This implies that also the payoff is smaller than in equilibrium, namelyU¯ ₌10.5 compared toUtr=12.4. These

differences are due to the fact that, contrary to our simplifying assumption that every agent

4_{Since values of sp close to 0 and close to 1 both express a high degree of specialization but would give an}

average value of about 0.5 indicating a very low degree of specialization, we use the absolute deviation of sp from 0.5 as a measure of specialization in production.

5_{Such large values ofαseem to be unrealistic, however, the ‘peak’ of trade could be shifted to the left by}

Fig. 1. (a) Mean value (averaged over 10 runs) of the average utility in the population after 3000 periods for values ofα∈[1,4](n=100, a =b=1, τ =10, T =3000, χ =0.1, w=2, σ =0.1, µ=0.02, µpr =0.05).

(b) Mean value (averaged over 10 runs) of the amount of trade in the population after 3000 periods for values ofα∈[1,4]. (c) Mean value (averaged over 10 runs) of the average degree of specialization in the population after 3000 periods for values ofα∈[1,4]. (d) Mean value (averaged over 10 runs) of the average amount of time invested for trading after 3000 periods for values ofα∈[1,4].

trades once every period which was used for calculatingUtr, the agents in the simulation fail to trade in some periods.

Considering again Fig. 1 we realize that a further increase ofαleads to a reduction of the amount of good 1 traded. The agents still specialize in the production of one good but do not trade anymore. They rather keep the time invested in trading low and just consume the goods they produce. This results in average long term payoffs slightly smaller than

Usp=10 which is the highest possible utility for large values ofα. Note, however, that in

Fig. 1 (Continued).

utility than specialization without trade forα <6.6. Fig. 1 shows that even for much smaller values ofαtrade cannot be established in a population of myopic individuals. This is easy to understand because trading requires much more time in the beginning as long as there are only a few other agents who invest time in trading. Thus, the gains from trade must be rather large to allow agents who are involved in trading to be competitive and to establish trade in the population.

Fig. 2. (a) Evolution of the average population utility in a simulation run with parameter values

α= 2.5, n =100, a = b= 1, τ = 10, T =3000, χ =0.1, w =2, σ = 0.1, µ= 0.02, µpr = 0.05. (b)

Fig. 2 (Continued).

only one of the two goods. It is also quite remarkable how fast the amount of goods traded increases if we decreaseχfrom 0.1 to 0.05. For this value the average trading volume in our simulations is even larger than in a perfectly organized economy with direct trade (in such an economy we would have a trading volume of tr_≈22). The average utility of the agents decreases fast with increasingχas long as this threshold is rather small but is more or less independent from this parameter in the region aboveχ₌0.15 where only very little trade occurs. It is very interesting to note that the amount of time invested in trading does not differ much between the cases whereχ ₌0.05 andχ ₌0.1. In both cases the value is approximately 0.1. This value seems to be a slightly larger than necessary forχ₌0.05 and a little bit too small forχ₌0.1 (judging from the trading volumes in both cases). The values₌0.1 seems to be some kind of generic level of the time fraction for trading in cases where trade is established in the population. Unfortunately, we lack a sound explanation for this phenomenon.

To estimate the importance of the agents’ inertia in our model we have also carried out simulation for different values of the parameterwwhich governs the degree of inertia in the learning rule. In Fig. 4 we show average utility and trade volume forw∈[1,3]. Note thatw=1 amounts to the absence of any inertia in the population.

Obviously the influence of this parameter on the long run behavior of the population is very small. We can detect a slight increase both in utility and in the amount of goods traded for increasingwbut this effect is rather insignificant. A very similar picture can be obtained if we depict utility and trade volume in dependence of the innovation probabilityµv. As

long as the probability is larger than some very small threshold and does not get too large the level of innovation does not influence the simulation results very much. In particular, we checked that values ofµv∈[0.03,0.15] yield almost identical results. However,µ≥0.02

is necessary to establish trade in the population6.

Fig. 3. (a) Mean value (averaged over 10 runs) of the average utility in the population after 3000 periods for values ofχ∈[0.05,0.3](α=2.5, n=100, a=b=1, τ=10, T =3000, w=2, σ =0.1, µ=0.02, µpr=0.05).

(b) Mean value (averaged over 10 runs) of the trading volume in the population after 3000 periods for values of

χ∈[0.05,0.3]. (c) Mean value (averaged over 10 runs) of the average amount of time invested in trading after 3000 periods for values ofχ∈[0.05,0.3].

3.2. Model with mediators

Now let us consider the model where agents might decide to mediate rather than pro-duce. The parameter settings are the same as in the simulations presented above and, ad-ditionally, we specified the probability to change from mediation to production or vice versa asµid = 0.01, the probability that the selling and buying prices of a mediator are

perturbed by noise is µ ₌ 0.02 and the variance of the normally distributed noise is

Fig. 3 (Continued).

Again, we start by considering the effects of a change of the parameterαon the outcome of an average simulation. In Fig. 5 we show the average utility of the agents, the amount of trade and the number of mediators in the population forα∈[1,4].

Looking at this figure we get the following picture: for smallα — like in the model without mediation — no trade emerges but the agents keep producing and consuming both goods. Ifαis larger than 1.6 trade occurs in the population and it follows from the number of mediators and the average time non-mediators invest in trading that this trade is mediated. However, forα_∈[1.6,2.2] mediators earn on average less than the producers and thus the number of mediators and accordingly the trade volume is rather small. Nevertheless, the proportional imitation rule of the agents allows a small number of mediators to survive. Most probably this would not be the case if we would use a stronger rule like the ’imitate the best’ rule. Ifα >2.2 the long run outcome of the simulations very much looks like the equilibrium with mediation we considered in the last section. We have a large degree of specialization of the producers and they do not invest in trading. There are between 10 and 14 mediators in the market and the utility of producers and mediators is almost equal. Even the amount of goods traded is very close to the 24 units of good 1 which would be traded in an equilibrium with mediation. The utility of the agents is almost constant forα >2.2 and significantly larger than in the model without mediation.

Fig. 5 suggests that, if we allow the agents to become middlemen in trade, in the long run we always get mediated trade if we get trade established at all. An interesting question in this context is whether the evolution of mediated trade needs a stage with direct trade as step in-between or whether mediation emerges directly from a population of pure producers. To answer these questions we have to look at the time evolution of the population in the model with mediators. A typical simulation result forα₌2.5 is depicted in Fig. 6 . The population evolves towards a state which is very similar to an equilibrium with mediation. The average utility in the equilibrium with mediation isUmed=11.89 (note that this value

Fig. 4. (a) Mean value (averaged over 10 runs) of the average utility in the population after 3000 periods for values ofw∈[1,3](α=2.5, n=100, a=b=1, τ=10, T =3000, χ=0.1, σ=0.1, µ=0.02, µpr=0.05). (b)

Mean value (averaged over 10 runs) of the trading volume after 3000 periods for values ofw∈[1,3].

oscillates close to this value in our simulation. Furthermore, we get strong specialization of all agents and between 10 and 15 mediators in the population facilitating trade. Interestingly enough, the average price charged by the mediators is significantly smaller than the optimal price would be. This inefficiency seems to be the reason why less mediators than predicted are in the market and the overall utility in the population is smaller than the value calculated analytically.

Fig. 5. (a) Mean value (averaged over 10 runs) of the average utility in the population after 3000 periods in the model with mediators and values ofα∈[1,4](n =100, a= b=1, τ =10, T = 3000, χ =0.1, w =2, σ=0.1, µ=0.02, µpr=0.05, µid=0.01). (b) Mean value (averaged over 10 runs) of the trading volume after

3000 periods in the model with mediators and values ofα∈[1,4]. (c) Mean value (averaged over 10 runs) of the number of mediators in the population after 3000 periods for values ofα∈[1,4]. (d) Mean values (averaged over 10 runs) of the average utility of mediators (dotted line) and producers (solid line) after 3000 periods for values ofα∈[1,4].

oscillations. However, the average payoffs of producing and mediating is very close (con-sidering the average payoff between period 1000 and 3000 we get 11.34 for the producers and 11.1 for the mediators).

Fig. 5 (Continued).

of the model with mediation where the population in the long run traded via mediators but developed direct trade before7.

Using Fig. 3, above we analyzed the effect of an increase of the trading time threshold

χon the emergence of trade in the model without mediators. How does this effect change if mediation is an option for the agents? In order to analyze this question we present the average utility, the trade volume and the number of mediators forχ_∈[0.05,0.3].

Comparing Fig. 7 with Fig. 3 we see that the effect of an increase ofχ on the average utility is smaller in the model with mediators. Utility still decreases with increasingχbut

7_{Contrary to this we demonstrate in Dawid (1999) in a similar model with decreasing returns to scale in production}

Fig. 6. (a) Evolution of the average population utility in the model with mediators and parameter values

α=2.5, n=100, a=b=1, τ=10, T =3000, χ=0.1, w=2, σ =0.1, µ=0.02, µpr=0.05, µid=0.01.

Fig. 7. (a) Mean value (averaged over 10 runs) of the average utility in the population after 3000 periods in the model with mediators and values ofχ ∈[0.05,0.3](α= 2.5, n= 100, a=b= 1, τ =10, T =3000, w=2, σ =0.1, µ=0.02, µpr =0.05). (b) Mean value (averaged over 10 runs) of the trading volume after

3000 periods in the model with mediators and values ofχ∈[0.05,0.3]. (c) Mean value (averaged over 10 runs) of the number of mediators after 3000 periods forχ∈[0.05,0.3]. (d) Mean value (averaged over 10 runs) of the average time invested in trading after 3000 periods in the model with mediators and values ofχ∈[0.05,0.3].

Fig. 7 (Continued).

conjecture and could indeed observe such outcomes in different simulation runs with this parameter constellation, however not in all of them.

4. Conclusions

allow mediation we can observe massive direct trade. Our simulations show that the learning behavior is rather insensitive with respect to variations of the parameters in the learning rule like the level of inertia or the mutation probability. Furthermore, we have shown that the amount of time which is needed to find a trading partner has significant effects on the average utility and the trading volume in the model without mediators. However, this effect is strongly dampened if mediation is an option. The fact that the agents can organize the economy such that direct or even mediated trade gets established are particularly interesting because our simple learning rule does not involve any foresight whatsoever but only imitation based on past performance coupled with random innovations. Loosely speaking, in this framework a shop has to be profitable right from the start in order to survive. However, we have to keep in mind that as well the number of mediators as the average utility in our simulations were also in the long run significantly smaller than in a case where trading and production is optimally organized. Thus, our boundedly rational agents do not organize in an optimal way, but still are able to establish trade in the economy and profit from this innovation.

In a historic context these simulations could be considered as an extremely stylized model of the effects of the emergence of trade between ancient villages. Our findings pretty much match the characterization of the effect of these transitions recently given in Day (1999):

The emergence of empires about 500 B.C. based on widespread trading networks, made possible a great increase in specialization and a pronounced expansion again in produc-tivity (p. 296).

Several extensions of the model presented here seem to be worth studying. An increase in the number of goods may change the results decisively. In particular, if not all agents can produce all goods the emergence of trade should be facilitated and mediation might become more useful since finding an appropriate trading partner will become more difficult in this setup. Another interesting way to proceed would be to increase the capability of the agents to perceive and forecast the state of the population. Of course there are several different ways to model this and we will not provide any concrete suggestions here. The goal of this paper was to show in a very simple framework how direct and mediated trade emerges in a population of agents who do not base their production decisions on optimization but learn by a simple imitation rule.

Acknowledgements

The author would like to thank the Department of Economics at the University of Southern California for its hospitality and the Austrian Science Foundation for financial support under contract No. J1281-SOZ. This research has highly profited from the suggestions and remarks of Richard Day. Helpful comments of James Robinson and two anonymous referees are gratefully acknowledged.

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