Prof. Ping CHEN, <pchen@ccer.pku.edu.cn>
Submitted to Fang Jian at <jason_fang@sohu.com>
Anytime between June 12 (for students graduated this Spring) to August 22, 2003

Introduction:
From linear (exponential) growth model to nonlinear (logistic) growth model is a revolution in economic thinking, which generated chaotic trajectory from stable equilibrium path and unstable (J-shaped or U-shaped) distribution from one-humped distribution. The later has no definition of mean in equilibrium sense. The picture of a representative agent may be radically different from a macro system with many agents. We are not only challenging the conventional program of optimization for rational agents, but also mainstream macro and finance theory based on Brownian motion and representative agent model such as Lucas (1972).

There are six problems in our open set for final exam. You may choose two problems for your final exam: one deterministic model (marked by D) and one stochastic model (marked by S). Your score is judged from competitive basis, i.e. if you solve a difficult problem alone, you would have a better chance to go ahead of the class. Format:

(a) Raise the math problem from economic issue;

(b) find analytical solution and stability condition;

(c) if there is no analytical solution, find numerical solution and asymptotic value for extreme cases;

(d) discuss its economic implications of your mathematical results, comparing your results with orthodox linear theory;

(e) summery: what is your finding, what is new and why it is interesting, discuss unsolved problems for future research. Requirement: Have nice, simple, and clean math result and deep economic insight. You may feel free to write in English or Chinese. Note: there may not have close or unique solution. You should make up your own mind if your math solutions are relevant to economic problem of growth and fluctuations.

Warning: You may copy solutions from existing literature. But be careful to fulfill all the intermediate steps to let me know you really know how to work out the solutions. Do not jump to your solutions while keep your writing style simple and clean. All answer sheets should be written by Microsoft Word document with the formula written by MathType Equation. Name your file by your name initial and date such as CPaed052203 for Chen Ping AED(exam) dated 05/22/03.

Suggested readings during exam (that can be downloaded from my web site on <陈平推荐阅读论文（下载）> at：
http://pchen.ccer.edu.cn/homepage/Homepage%20Chinese/cn.htm
The relevant references are listed at the end of each problem.

(a) R.M.May & G.F.Oster, “Bifurcations and Dynamical Complexity in Simple Ecological Models,” American Naturalist, 110(974), 573-599 (1976).

(b) R.M.May, “Stability in Randomly Fluctuating versus Deterministic Environment,” American Naturalist, 107(957), 621-650 (1973).

(c) P.Chen, "Origin of Division of Labor and Stochastic Mechanism of Differentiation", European Journal of Operational Research, Vol. 30, No.3, pp. 246-250 (1987).

(d) P.Chen, "Imitation, Learning, and Communication: Central or Polarized Patterns in Collective Actions", Self-Organization, Emerging Properties and Learning, pp. 279-286, Plenum, New York (1991).

(e) P.Chen, “Microfoundations of Macroeconomic Fluctuations and the Laws of Probability Theory: the Principle of Large Numbers vs. Rational Expectations Arbitrage,” Journal of Economic Behavior & Organization (Nov. 2002).

(f) 陈平,“劳动分工的起源和制约－从斯密困境到广义斯密原理，”经济学（季刊）， 第1卷，第2期（2002年1月）.:

(g) Ping Chen & Huajun Li, Endogenous fluctuations and persistent mode in economic growth: a challenge to the geometric Brownian motion in economic dynamics, Draft paper for internal distribution only (May 2003).

Problems

(1D) Replacing Cobb-Douglas function with logistic function, resolve the optimization problem for consumer demand and firm supply, find the form of demand and supply curve under equilibrium. Compare your result with that of standard neo-classical economics textbook, such as Variant. Find empirical examples that people’s utility has upper limit rather than concave function. If this type model is true, can we still hold that greed or maximizing utility is the driving force of invisible hand? What is its implications in individual behavior and market regulations? (Reference: Variant: Microeconomic Analysis).

(2D) Solve continuous-time and discrete-time logistic model of growth. Find the stability conditions for them. Discuss the situation when equilibrium state become unstable and chaotic. Draw bifurcation diagram for chaotic regime. Discuss the empirical issue to observe logistic growth. What is the impact to econometric modeling in empirical analysis? (Reference: a).

(3D) Solve the two-species competition model with logistic growth. Find their equilibrium solutions and stability conditions. Compare your result of dynamic competition under limited resource with perfect competition and monopolistic competition model in neo-classical economics. Which model is capable of explaining recent recession in US caused by over-investment in tele-communication. Do you have any suggestion to define a “proper competition” with “excess-competition?” (Reference: c,f).

(4S) Solve distribution function for stochastic model of logistic growth in continuous-time. Find the conditions when the distribution is a one-humped, J-shaped, or U-shaped distribution. Discuss its economic implications of the three types of distributions. Specifically, can you have equilibrium pricing by marginal analysis? What would happen for a product cycle during rising and flattering stage? Do you think a firm’s profit margin could be the same during different phase of logistic curve? (Reference: b,d).

(5S) One-dimensional model of random walk in discrete-time. Assume the probability moving to right is p, the probability of staying unchanged q, the probability of moving to left (1-p-q). For modeling economic growth, we must have p>(1-p). Calculate the first and second movements of displacement, and its relative deviation. Comparing your result with the random walk model with q=0, and the model of linear birth-death process. Also comparing your result with linear diffusion model or Brownian motion. Can you judge the observed macroeconomic data or stock index series are generated by external or internal instability? (Reference: g).

(6S) Calculate nonlinear birth-death process with quadratic terms. Comparing your result with the random walk model with q=0, and the model of linear birth-death process. There is a fierce debate between efficient market theory and behavioral finance theory (or social psychology). Comparing your result with empirical features of macro and fiancé data, can you judge if the Lucas model of microfoundations of macro fluctuations is right at the level of order of magnitude? (Reference: d, e, g, and Yuanhui Ji, thesis 2003).

Any changes caused by CCER or PKU regulation would be informed at CCER web site and athttp://pchen.ccer.edu.cn/BBS/