Coexistence of several currencies in presence of increasing returns to adoption
CCoexistence of several currencies in presence ofincreasing returns to adoption
Alex Lamarche-Perrin (1) , André Orléan (2) , Pablo Jensen (1)(1)
Univ Lyon, Ens de Lyon, Univ Claude Bernard, IXXI, CNRS,Laboratoire de Physique, F-69342 Lyon, France (2)
CNRS EHESS ENPC ENS, UMR 8545, Paris-JourdanSciences Economiques (PjSE) 48 Boulevard Jourdan, 75014 Paris,France
January 15, 2018
Abstract
We present a simplistic model of the competition between differentcurrencies. Each individual is free to choose the currency that minimizeshis transaction costs, which arise whenever his exchanging relations havechosen a different currency. We show that competition between curren-cies does not necessarily converge to the emergence of a single currency.For large systems, we prove that two distinct communities using differ-ent currencies in the initial state will remain forever in this fractionalizedstate.
In general, a currency is useful - and therefore sought by an individual - onlyinsofar as it can be used to buy goods. This implies that the currency is widelyaccepted as payment by her suppliers. In a world where several currenciesexist, the attractiveness of a currency for an individual can be measured by itsgreater or less acceptance in the group of individuals with whom it is used toexchange goods. It follows that the more a currency is accepted, the greaterits attractiveness becomes. From this point of view, money is akin to language[1]. The assumption that in cross-currency competition, increasing returns toadoption play a fundamental role has been present in economic theory sincethe famous article by Carl Menger [2]. The same idea has been examined byvarious authors in the Economics litterature [3, 4, 5, 6, 7, 8, 9, 10] as well asin Physics journals [11, 12, 13]. Recent models have pushed further Menger’sbasic idea either by trying to introduce social aspects of money [8], or providinga unified framework able to explain at the same time the emergence of a singlecurrency and some other economic phenomena. Yasutomi’s[11] model, furtherrefined by [12, 13], can also describe the collapse of a currency, while Donangeloand Sneppen’s links the emergence of money to its anomalous fluctuation invalue [14]. Duffy and Ochs [7] tested some predictions of Kiyotaki and Wrightmodel[4] in laboratory experiments. Another stream of literature connects the1 a r X i v : . [ q -f i n . GN ] J a n mergence of money to the more general question of the emergence of socialnorms using game theory[5, 7, 6, 10].In this article, we propose a model that differs from all these approaches bytwo characteristics : its exchange mechanism is simpler and it takes into ac-count the specificity of local situations through the introduction of a network ofexogenously fixed bilateral links, for example because of constraints originatingin the social division of labor. On the first point, most previous work assumesome more realistic exchange mechanism. For example, models inspired by Ya-sutomi’s model [11, 12, 13] take, as elementary interaction between two agents,the ”transaction”, which "consists of several steps including search of the co-trader, exchange of particular goods, change of the agent’s buying preferencesand finally the production and consumption phase". On the second point, un-like most models, we do not assume a completely connected interaction network,which leads to simpler analytical treatments but obscures the local aspects ofeconomic transactions. We demonstrate that, under such conditions, compe-tition between currencies does not necessarily converge to the emergence of asingle currency. Even if an equilibrium with a single currency remains possi-ble, the most frequent stable configuration is the division of the trading spacebetween different currencies. For large systems, we prove that two distinct com-munities using different currencies in the initial state will remain forever in thisfractionalized state. We consider an economy composed of N agents, numbered i = 1 , ..., N , eachstarting with its own currency s i . A currency will be referred to as an integerin [1 , N ] , and we assume that each agent begins with its own currency s i = i .The agents are disposed on the vertices of a random graph [15], whose edgesrepresent commercial links between the agents. After choosing a density of links p ∈ [0 , , for each pair of agents ( i, j ) , we create a link between i and j withprobability p , or let the agents disconnected with probability (1 − p ) .The interaction dynamics is set in the following way. Each time step, wechoose an agent at random. First, this agent is allowed to change the currencyit uses. Then, it trades with all its neighbours, i .e. with all the agents he sharesa link with. The profit an agent gets is defined according to the following idea:if two agents share the same currency, their trading business is made easier andno cost has to be paid; conversely, if they use different currencies, they musttrade through the help of some "moneychanger" who gets a commission for itswork: we will then consider that those agents have to afford some fixed cost(which we will take as a unit cost) in order to complete their trade. As thereis no other constraint, we translate the profit of each particular trade to theorigin so that a successful trade is worth and a trade which has to resort to amoneychanger is worth − .We define a simple utility function U i ( t ) for an agent i at period t , as theopposite of the sum of all trading costs an agent has to pay when realising itstrades at period t : U i = − (cid:88) j ∈V ( i ) (1 − δ s j s i ) = − Card (cid:8) j ∈ V ( i ) | s j (cid:54) = s i (cid:9) V ( i ) is the set of the neighbours of the agent i in the graph and δ s j s i is theKronecker symbol, ie δ s j s i = (cid:26) if s i = s j if s i (cid:54) = s j .We also define a social utility function for the economy as a whole, as thesum of the utilities of all agents: U = N (cid:88) i =1 U i = − (cid:88) i,j neighbours (1 − δ s j s i ) where the factor accounts for the fact that each pair of neighbours is countedtwice in the sum.As we assume that agents are fully rational and maximize their own utilityfunction, the rule for currency adoption is the simplest mimetic one: an agentadopts the most common currency among its neighbours; if several currenciesare used by the same maximal number of neighbours, two cases appear: eitherthe agent already uses one of them, and keeps using it by default, or she wasusing another one, in which case she picks at random one of the most popularcurrencies among her neighbours. Agents do not take into account neither theanticipated cost of future trades (for t (cid:48) > t ) nor the influence their choice couldhave on other agents.From the evolution rule we have defined, we can infer that social utility canonly increase with time; more, it increases strictly when an agent changes itscurrency. Indeed, if agent i switches from currency s to currency s (cid:48) at time t ,social utility increases with ∆ U = ∆ U i + (cid:88) j ∈V ( i ) ∆ U j = (cid:88) j ∈V ( i ) δ s j s (cid:48) − δ s j s + (cid:88) j ∈V ( i ) δ s (cid:48) s j − δ ss j = 2 (cid:18) Card (cid:8) j ∈ V ( i ) | s j = s (cid:48) (cid:9) − Card (cid:8) j ∈ V ( i ) | s j = s (cid:9)(cid:19) where we write s j = s j ( t −
1) = s j ( t ) for all j ∈ V ( i ) . By definition, agent i switches from s to s (cid:48) at time t if and only if Card (cid:8) j ∈ V ( i ) | s j = s (cid:48) (cid:9) > Card (cid:8) j ∈ V ( i ) | s j = s (cid:9) , which yields ∆ U > . (1)An interesting consequence of equation (1) is that the system can not comeback to a state it has already visited: There exists no loop in the phase diagram.Because the number of states the system can visit is finite, we can infer that fromany initial configuration, the system will reach an equilibrium with probability as time tends to infinity, equilibrium being defined as any state in which noagent can change its currency. Rather than studying the precise dynamics ofthe system, we will hence be more interested in finding the different equilibriathat this economy could reach.An obvious equilibrium corresponds to the whole economy using a singlecurrency. It is also clearly a social optimum, as no agent has to pay any change3igure 1: A metastable state where social utility is stuck at − .Figure 2: At equilibrium for N = 100 , average number of currencies (circles)and of connected components (squares). Insert: average number of currenciesper connected component against mean number of neighbours.cost. However, equation (1) does not imply that social utility will necessarilyreach its maximum, which is zero. As figure 1 shows, we can imagine a situa-tion where separated communities appear, within which agents share the samecurrency, but where several different currencies coexist on the overall economy.Such an equilibrium is clearly not optimal as at least one agent, although suf-fering losses from the trades with agents from other communities, is nonethelessunable to adopt another currency, as this would entail it to suffer even morelosses during some time steps. This contradiction between the individual andcollective optima can be found in many economic simple models [16, 17, 18]. In order to find what kind of equilibria can be reached by the system, we ran , simulations of the model. In each simulation, we consider an economy4igure 3: Social utility function for N = 100 , p = 0 . and a single connectedcomponent. At equilibrium, 14 different currencies are present.of N = 100 agents, each beginning with its own currency. Note that two differ-ent sources of randomness can influence the results: the topology of the graphand the order in which we pick the agents. Hence, we draw a new distribu-tion of edges for each simulation. We then iterate the model, repeating theelementary time steps until equilibrium is reached. We count the number ofcurrencies displayed by the overall economy, and calculate its mean over the , simulations.We plot the results for varying link densities p , together with the meannumber of connected components in the graphs (see figure 3.1). Indeed, we seethat when p is close to , there still are different currencies in the economy:this was to be expected as in this case, many agents have no neighbour at all.They have consequently no other currency to adopt and keep their own eachtime they are selected to update their situation.When the graph cohesion is weak (ie for small values of p ), we remark thatthe number of its connected components remains significantly below the averagenumber of currencies remaining at the end of the simulations. For p > . , whichcorresponds to an average number of neighbours per agent, we can reliablyconsider that there remains only one connected component and that an economyof agents succeeds in agreeing on a single currency as a mean of exchangefor all trades.We also observe the evolution of social utility through these simulations.Figure 3 shows this evolution in a typical simulation where each agent has onaverage neighbours ( p = 0 . ). At equilibrium, which was reached after , time steps, social utility culminates at − , which is quite far from the socialoptimum . Indeed, in this simulation, there remains different currencies atequilibrium, even if only a single connected component exists.We ran the same , simulations with two slightly different dynamics.Instead of randomly picking the agents one by one, the second dynamics con-sisted in computing the optimal choice of each agent in a given state of the5 a ) ( b ) ( c ) Figure 4: Some examples of two-communities graphs with p intra = 0 . , forweak [ p inter = 0 . , figure ( a ) ], medium [ p inter = 0 . , figure ( b ) ] and strong[ p inter = 0 . , figure ( c ) ] interconnection.economy and then updating their currency choice simultaneously; in the thirdcase, we still picked the agents one by one, but always in the same order. Allthese variants yielded comparable results. Using the same general framework, we now study the case where two differentcommunities composed of N agents are trading. We consider as a communitysome pre-defined set of agents sharing a strong intra-connection, while agentsbelonging to different communities share weaker links. Formally, for an economyconsisting of N agents, with p inter and p intra being respectively the probabilitiesof internal and external links ( p inter < p intra ), we select all pairs of agents ( i, j ) and create a link with probability p inter if they belong to the same communityand with probability p intra otherwise. The mean link density of the overallgraph is, for large N , p av = 2 N ( N − (cid:32) × N ( N − p intra + (cid:18) N (cid:19) p inter (cid:33) ≈
12 ( p intra + p inter ) , which might be useful to determine the influence of the precise topology onthe outcome. In the specific case where p inter = 0 , there exist two separatedcommunities without intra links; conversely, if p intra = p inter , we actually findthe previous case of a single community set on a random graph with uniform linkprobability p intra . Figure 4 shows some examples of such graphs for p intra = 0 . ,from weak inter-connection to strong inter-connection.Two different cases can be studied: as in the previous one-community case,every agent can begin with its own currency, or each community could be alreadyunified — which means that its agents all share the same currency.6e begin with the case where each agent is initially given its own cur-rency. Just as in the one community case, the question we are concerned withis whether, and on what conditions, the economy as a whole adopts a singlecurrency or if, on the contrary, each community adopts its own currency, lead-ing to a changing cost when trading across them. The result can be expectedto depend on the precise system history. If an agent of the first communityadopts a currency of the second community and starts spreading its use amongits neighbours, the two communities might end using the same currency. But as,on average, an agent shares more links with agents from its own community, sheis more likely to adopt a currency from her own community, leading to distinctcurrencies.We test the model with N = 100 and p intra = 0 . . As this is far from theconvergence threshold exhibited by experiments with one community, we canfigure out what will happen for extreme values of p inter . If p inter = 0 , eachcommunity will end with its own currency, as our choice for p intra ensures thata convergence will take place inside each community, and the absence of externallinks makes impossible the adoption of a currency from the other community.Instead, if p inter = 0 . p intra , we fall back on the one community case, sothat a single currency will spread in the whole economy.In order to see how the transition takes place, we run , simulationsfor several values of p inter and plot the probability that a single currency isadopted in the whole economy at equilibrium. For each simulation, we drawa new random graph as before. The results are represented by the circles onfigure 5. The curves agree with our predictions: for low values of p inter , eachcommunity uses a different currency; but as p inter increases, the economy tendsto adopt a single currency. Here again, we explore what happens when all theagents update their choice simultaneously instead of one by one (represented bythe squares in figure 5), but the two curves are almost identical.However, this result must be compared to the one-community case studiedbefore: even for p inter = 0 , the link density of the overall graph is still p intra =0 . . For an equivalent link density, we saw in section 3.1 that when the graphwas purely random, with such a mean link density the economy almost alwaysunited on a single currency.We now turn to the case where each community has already been unifiedon a single currency. The problem is then to know whether the attractivenessof the other community is strong enough to counter the intra-communitarianbonds.We ran again , simulations for an economy with N = 100 and p intra =0 . , and plotted the results for varying p inter (represented by the triangles onfigure 5). As was to be expected, the transition from two to one currency needsa much higher value of p inter to happen. Indeed, the the intra-communitarianbonds are already effective when agents are offered to switch currency; whereasin the previous case, the bonds required to be activated by the — possible —adoption of a community-dependent currency to become real constraints, andhence was left open the possibility that the agents chose a currency used in bothcommunities. 7igure 5: Probability that a single currency ermerges on a two-communitiesgraph with N = 100 and p intra = 0 . . The circles (one agent randomly pickedeach time step) and the squares (all agents changing their choice simultaneously)represent the case when each agent begins with its own currency as in the one-community case. The triangles shows the transition when each community hadpreviously been unified on a single currency. We now show rigorously that for two very large communities ( p inter < p intra , N →∞ ), a transition to a single currency never happens. Intuitively, the transitionmay be triggered by a single agent having more external than internal neighbors,which is unlikely but not impossible when p inter < p intra , because of randomfluctuations. One expects that fluctuations in the links’ distributions becomesmaller as N increases, leading to a more stable advantage of inner links. How-ever, since the number of nodes increases, the probability that at least one nodehas more external than internal links may also increase, leading to a cascade.Our calculations prove that the decrease in the fluctuations is stronger than theincrease in the number of nodes. More precisely, if P N is the probability that,for a given two-community graph with N agents, there exists a single-currencyequilibrium, we demonstrate that, when p inter > p intra , P N → as N → ∞ .An obvious necessary condition for the existence of such an equilibrium isthat there exists at least one agent which might change its currency. We thusassume p inter < p intra and begin by studying the probability, for a given agent,to have less neighbours in its own community than in the other.For a given N , we randomly choose one of the agents and define X a,N as itsnumber of intra communitarian links and X r,N as its number of inter communitarianlinks. For random graphs, X a,N and X r,N are two independent random vari-ables following binomial laws with N number of trials and respective parameters8 intra and p inter . Let φ Na and φ Nr respectively denote their cumulative distribu-tion function.We thus study the probability P ( X a,N ≤ X r,N ) = ∞ (cid:88) k =0 P ( X r,N = k and X a,N ≤ k )= ∞ (cid:88) k =0 P ( X r,N = k ) P ( X a,N ≤ k )= ∞ (cid:88) k =0 P ( X r,N = k ) φ Na ( k ) . Let p av = p intra + p inter ; as p inter < p intra , then p inter < p av < p intra . Foreach N , we define k N = (cid:98) p av N (cid:99) the floor of p av N . We obtain: P ( X a,N ≤ X r,N ) = k N (cid:88) k =0 P ( X r,N = k ) φ Na ( k ) ≤ φ Na ( k N ) + ∞ (cid:88) k = k N P ( X r,N = k ) φ Na ( k ) ≤ ≤ φ Na ( k N ) k N (cid:88) k =0 P ( X r,N = k ) + ∞ (cid:88) k = k N P ( X r,N = k ) ≤ φ Na ( k N ) + (1 − φ Nr ( k N )) (2)We will now show that φ Na ( k N ) −→ N → + ∞ and − φ Nr ( k N ) −→ N → + ∞ , and show that these limitsfollow at least a geometrical decay.By definition: φ Na ( k N ) = k N (cid:88) k =0 (cid:18) Nk (cid:19) p k (1 − p ) N − k , where we use p instead of p intra to simplify notations. Moreover, if k ≤ p av N , (cid:0) Nk (cid:1) = NN − k (cid:0) N − k (cid:1) ≤ − p av (cid:0) N − k (cid:1) . Hence: φ Na ( k N ) ≤ k N (cid:88) k =0 − p av (cid:18) N − k (cid:19) p k (1 − p ) N − k ≤ ... ≤ k N (cid:88) k =0 (cid:18) − p av (cid:19) N − k N (cid:18) k N k (cid:19) p k (1 − p ) N − k = (cid:18) − p − p av (cid:19) N − k N k N (cid:88) k =0 (cid:18) k N k (cid:19) p k (1 − p ) k N − k =1 ≤ ρ Na (3)9here ρ a = (cid:16) − p intra − p av (cid:17) − p av < .Similarly, if we now use p for p inter , we get: − φ Nr ( k N ) = 1 − k N (cid:88) k =0 (cid:18) Nk (cid:19) p k (1 − p ) N − k = N (cid:88) k = k N +1 (cid:18) Nk (cid:19) p k (1 − p ) N − k = N − k N − (cid:88) k (cid:48) =0 (cid:18) Nk (cid:48) (cid:19) p N − k (cid:48) (1 − p ) k (cid:48) Moreover, if k ≤ N − k N − N − (cid:98) p av N (cid:99) − , then k ≤ (1 − p av ) N and (cid:0) Nk (cid:1) ≤ p av (cid:0) N − k (cid:1) . From this point, − φ Nr ( k N ) ≤ N − k N − (cid:88) k (cid:48) =0 (cid:18) p av (cid:19) k N +1 (cid:18) N − k N − k (cid:48) (cid:19) p N − k (cid:48) (1 − p ) k (cid:48) = (cid:18) pp av (cid:19) k N +1 N − k N − (cid:88) k (cid:48) =0 (cid:18) N − k N − k (cid:48) (cid:19) p N − k N − − k (cid:48) (1 − p ) k (cid:48) ≤ ρ Nr (4)where ρ r = (cid:16) p inter p av (cid:17) p av < .Putting equations (2), (3) and (4) together, we get P ( X a,N ≤ X r,N ) ≤ ρ Na + ρ Nr , (5)which means that the probability for a single agent to have more neighbours inthe other community than in its own geometrically decreases to as N tendsto infinity.For i = 1 , ..., N , let us now call X ( i ) a,N and X ( i ) r,N the random variables repre-senting, respectively, the number of intra communitarian and of inter communi-tarian neighbours of agent i . Recall that P N is the probability that, for a giventwo-community graph with N agents, there exists a single-currency equilibrium.A necessary condition is that there exists some agent in the initial configurationwhich may change its currency. Using equation (5), we can then write: P N ≤ P (cid:16) X (1) a,N ≤ X (1) r,N or ... or X ( N ) a,N ≤ X ( N ) r,N (cid:17) ≤ P (cid:16) X (1) a,N ≤ X (1) r,N (cid:17) + ... + P (cid:16) X ( N ) a,N ≤ X ( N ) r,N (cid:17) = N P ( X a,N ≤ X r,N ) ≤ N (cid:0) ρ Na + ρ Nr (cid:1) which shows that P N −→ N →∞ . 10 Agents heterogeneity
Until now, we have supposed that every agent has the same influence on itsneighbours: the loss due to using different currencies is . We now relax thisassumption by introducing heterogeneity among the agents. We assume thateach agent j has a weight ρ j ; when an agent i trades with j using a currency s i (cid:54) = s j , it now has to afford a trading cost ρ j , so that we can define a newindividual utility function for the agent i : ˜ U i = − (cid:88) j ∈V ( i ) ρ j (1 − δ s j s i ) We first set ρ i = deg ( i ) where deg ( i ) is the degree of agent i in the graphof commercial links. We then run the same simulations as before (for one com-munity, and for two communities either unified or not) and find no qualitativedifference in the transition curve. However, introducing weights has an im-portant impact on which currency will eventually be chosen: is the case wheneach agent begins with its own currency, the average degree (hence the averageweight) of the agent whose currency will be adopted is much higher than whenthe agents are unweighted.We then attribute randomly the ρ i , independently of agent’s degree, butkeeping the same Poisson distribution as in the random graph. We obtain thesame results: the transition curves for currency adoption remain unchanged, butthe average weight of the first owner of the adopted currency is much higherthan the average weight. We have studied a simplistic model showing coexistence of several currencies,even in presence of increasing returns to adoption. Agents exchange only witha limited number of neighbors, through local exogenous commercial links, andseek to minimize their transaction costs by adopting the most common currencyamong these. The main interest of our work is to provide a model that is atthe same time very simple in its structure (exchange network and transactionmechanism) but is able to recover as possible equilibria both the existence of asingle currency or several of them, while most previous models only found thesingle currency equilibrium [3, 5, 6, 8, 10, 11, 12, 13]. This last point remindswork by Brian Arthur [19], where increasing returns lead to the existence ofmultiple equilibria. In further explorations, it would be interesting to investigateto which extent these fractionalized stationary states are robust to noise in thedecision process, for example by introducing a trembling hand, the possibilityfor an agent to choose, with a small probability a currency that is not the mostcommon among his neighbors.
References [1] Orléan André. The empire of value, a new foundation for economics.
M.I.T.Press , 2014.[2] Carl Menger. On the origin of money.
Economic Journal , 2:233–255, 1892.113] Jones Robert. The origin and development of media of exchange.
Journalof Political Economy , 84:757–775, 1976.[4] Nobuhiro Kiyotaki and Wright Randall. On money as a medium of ex-change.
Journal of Political Economy , 97:927–954, 1989.[5] Dowd Kevin et David Greenaway. Currency competition, network external-ities and switching costs: Towards an alternative view of optimum currencyareas.
The Economic Journal , 103:1180–1189, 1993.[6] Sethi R. Evolutionary stability and social norms.
Journal of EconomicBehavior and Organisation , 29:113–140, 1996.[7] John Duffy and Jack Ochs. Emergence of money as a medium of exchange:An experimental study.
The American Economic Review , 89:847–877, 1999.[8] Costas Lapavitsas. The emergence of money in commodity exchange, ormoney as monopolist of the ability to buy.
Review of Political Economy ,17:549–569, 2005.[9] Masaaki Kunigami, Masato Kobayashi, Satoru Yamadera, Takashi Ya-mada, and Takao Teraano. A doubly structural network model: Bifur-cation analysis on the emergence of money.
Evolutionary and InstitutionalEconomics Review , 7:65–85, 2010.[10] S. Gangotena. A toy model of the emergence of money from barter.
Preprint , 2016.[11] Yasutomi A. The emergence and collapse of money.
Physica D , 82:180–194,1995.[12] A. Z. Górski, S. Drożdż, and P. Oswiecimka. Modelling Emergence ofMoney.
Acta Physica Polonica A , 117:676–680, 2010.[13] P. Oświ¸ecimka, S. Drożdż, R. G¸ebarowski, A. Z. Górski, and J. Kwapień.Multiscaling Edge Effects in an Agent-based Money Emergence Model.
Acta Physica Polonica B , 46:1579, 2015.[14] R. Donangelo and K. Sneppen. Self-organization of value and demand.
Physica A Statistical Mechanics and its Applications , 276:572–580, Febru-ary 2000.[15] E. N. Gilbert. Random graphs.
Ann. Math. Statist. , 30(4):1141–1144, 121959.[16] H Hotelling. Stability in competition.
Economic Journal , 39:41, 1929.[17] Hernan Larralde, Juliette Stehle, and Pablo Jensen. Analytical solution ofa multi-dimensional hotelling model with quadratic transportation costs.
Regional Science and Urban Economics , 39(3):343–349, 2009.[18] Sebastian Grauwin, Eric Bertin, Remi Lemoy, and Pablo Jensen. Com-petition between collective and individual dynamics.
Proceedings of theNational Academy of Sciences , 106(49):20622–20626, December 2009.[19] Brian Arthur. Competing technologies, increasing returns, and lock-in byhistorical events.