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  Complexity has come from abundant subjects of thought,moreover,has reacted upon them,from mathematics to physics,from computer science to social sciences.Meanwhile,with the development of economics and the emergence of new way of trade,economics is no longer rife with linearity,continuity and a variety of phenomena that are easily predicted or understood.These phenomena have been labeled as complexity economics.The price system is a typical example of the application of complexity in economics.In this system,there are many similar and interacting parts(individual producers,agents),simple rules to obey(cost-benefit analysis)and aggregate patterns form from individual behavior(price).This report will first introduce the characteristics of neoclassical economics and come to illustrate the definition and ideas of complexity economics,which is helpful to comprehend the complexity in price system.And before moving to concrete examples,it will interpret a relative ideology——the evolution of walrasian behavior.Then it will demonstrate several examples as concrete applications about price system which embody the operation principle of complexity in it.After these examples,an overview and a conclusion based on the illustration above will be stated.This report is aimed to introduce a new model and ideology of price system,and then a new ideology about economics,by illustrating and analyzing several representative examples.
  2.From neoclassical economics to complexity economics从新古典经济学到复杂性经济学
  i)Definition of complexity economics
  According to Richard H.Day(1994),the definition of complexity in economics in terms of dynamic outcomes is that“an economic system is dynamically complex if its deterministic endogenous processes do not lead it asymptotically to a fixed point,a limit cycle,or an explosion.”(as cited in Rosser,1996).But this definition is in a broad sense so that some systems that others would argue should not be included are included.To define it in a narrow sense,we need more specific characteristics and they will be stated in next paragraph.
  ii)A comparison between the two types
  Complexity economics seems to be an inversion of neoclassical theory.Axel Leijonhufvud remarks that neoclassical economics“smart people in unbelievably simple situations,”whilst the real world involves“simple people with incredibly complex situations.”(as cited in Gintis,2006).According to Gintis(2006),there are five main aspects which the two types differ from each other.The first one is dynamics:the neoclassical economics is static,linear and thermodynamically closed so that it can be interpreted by algebraic geometry;while the complexity economics is dynamic,nonlinear and thermodynamically open,which lead itself to be far from equilibrium in general.The second one is agents:in the former,agents have“perfect information”and can optimize the information and surplus naturally;while in the latter,agents have“limited information”and face an obstacle of high price in information processing.This characteristics can be associated with the third one.The third one is networks:in neoclassical economics,agents face impersonal price system structure respectively without interaction;however,in complexity economics,agents have to participate in complex overlapping networks so that they can avoid the disadvantages of limited information and high costs in information processing as much as possible.In this way,under appropriate circumstance,agents in complexity economics can form non-optimal but high-efficient model for operating in complex environments.The forth one is emergence:in neoclassical economics,all the macro properties can be derived from its micro properties(for example,the fundamental theorems of welfare);but in complexity economics,macro patterns are emergent properties derived from micro interactions and behaviors,in the same sense that the chemical properties of a complex molecule,such as various carbon of simple substance,is an emergent property derived from its nuclear and electronic structure.In this case,we cannot analytically derive the macro-level properties from micro-level ones(its component parts),although there might be some undetected connections.Now we only can apply novel mathematical techniques to illustrate the emergent properties to some degree.The last one is evolution:there is no conditions or necessity for mechanism to create novelty or growth in complexity in neoclassical economics;while in the complexity economics,the evolution of differentiation,selection and amplification contributes to the novelty of system and the growth of complexity.
  3.The Evolution of Walrasian Behavior Walrasian Behavior的进化
  In neoclassical economics,Walrasian equilibrium is the main concept in price system,which determines the price in markets according to linear supply-demand relationship.It is undeniable that walrasian theory still plays an irreplaceable part in nowadays economics.However,this theory builds upon a central hypothesis which excludes strategic behavior of manipulating prices directly or indirectly in agents’own advantages.In Complexity and Artificial Markets(Schredelseker and Hauser,2008),specific computations are made to illustrate the evolutionary model in price system.It shows the results of simulation experiments about an economy in which agents may have different behavioral rules on price determination.
  As we know,agents in our economy environment will compare the profits gained in each iteration to those gained by otherfirms in that iteration so that they can choose a better strategy in the long-run operation.Below is the terse and concise summary of the computations from Schredelseker and Hauser(2008).Assume a set of N firms by i={1,2,…,N}competing in a market.For every output supplied to the market,this demand function has a clearing price P(Q(t))for market at which it is sold.Assume all firms are“ex-ante symmetric”with typical cost function C(q)=c1q(i)c2,where q(i)is the production of each firm i={1,2,…,N},and the parameters c1 and c2 are positive.The evolutionary dynamics,which follows t=0,1,2,..,proceed in discrete time.The principle that profits induced by current output is P(Q(t))qi−C(qi),i={1,2,…,N}.When the profits are realized,firms can choose a better strategy in the long-run operation by comparison and iteration.In this way,the individual profit function can be presented:
  And the relative profit is:
  From the two functions,we can see the effect on prices that one firm changes its output(quantity)is completely offset by another firm as there is no externalities in the product.And the resulting equation,after maximization and without iteration,simply:P,which means that price is equal to marginal cost in the Walrasian allocation.So a conclusion can be drawn that only if agents maximize relative profits with no imitation,the Walrasian equilibrium can be reached.And the above equations shows that the relative to the average measure is equivalent to the absolute difference in the profits between any two identical firms.Hence,in the real markets,agents imitating the most successful firm from the past round performance so that those strategies that do not perform as well as the average firm will be eliminated before coming to next round.(Schredelseker and Hauser,2008)

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