经济物理的基本问题 陈平 北京大学国家发展研究院 退休教授 复旦大学新政治经济学中心 高级研究员 清华大学高等研究院 讲座 2013 年 11 月 21 日

“ 经济物理 ” 的历史由来 法国社会学家孔德 (Auguste Comte, ): “social physics” 首次出书叫 “ 社会物理 ” （ 1856 ）。 陈平博士论文第一稿原名 “ 社会物理引论 ” ，描写具 有文化因子（有别于蚂蚁）的劳动分工的演化（ 2012 改名 “ 代谢增长论 ” ），被导师 Ilya Prigogine 否 定，改为 “ 非线性经济动力学 ” （ 1987 ）。国际会议 改名为 “ 演化经济动力学 ” （ 1989 ，陈平 2004 ） 美国波士顿大学的物理学家 H. Eugene Stanley (1995) 为 power law 的发现制造了一个新词：经济物 理 econophysics ，被物理学界接受，成为文献分类 的新领域

物理学家设想的理论模型或计算机模拟 瑞士数学家与物理学家贝奴力（ Daniel Bernoulli ）： gambling and utility function(1738) John von Neumann ( ): game theory, growth model Ilya Prigogine ( ): Brussletor > BZ reaction> dissipative structure, chemical reaction and population dynamics, self-organization(1971, 1977), complex systems (1987) Hermann Haken (1927- ): synergetics (1973), adiabatic approximation, laser > Hamiltonian system Wolfgan Weidlich (1931- ) : Ising model for public opinion, social dynamics Santa Fe Institute (SFI, 1984-), Philip Anderson (spin glass model), Murray Gell-mann (complex adaptive system), edge of chaos, artificial life ， automata system ， interacting agents

经济物理的基本问题 经济复杂（非线性，非均衡，非稳态，多体，多层 次，开放系统）现象能否简化为相对简单的科学模 型？ （ 1 ）能否找到相对优越参照系》经济学的哥白尼问 题 （ 2 ）能否找到普适的基函数作为建模的基石》经济 学的开普勒问题 （ 3 ）能否找到普遍的动力学方程，具有某种普适不 变量》牛顿 - 爱因斯坦 - 薛定谔问题 【目标】：超越简单的经验观察的统计描述

耗散系统的哥白尼问题 : 经济指数的最佳趋势 - 波动分离问题 保守系统的优化框架和耗散系统的演化框架 非线性动力学与非平衡态统计力学在经济学的应 用 观察宏观与金融指数的非稳态运动 - - FD vs. HP (von Neumann) filter 白噪声还是色混沌？ Louis Bachelier (1900) 的 Brownian motion 还是 Schumpeter 的生物钟？ 软边界振子与延时微分方程（类似神经元模型）

基函数与经济波动的本质： 外来噪声驱动 vs. 内生（非线性）振子 描写经济动力学的基函数 白噪声，谐振波，与 wavelets Wigner transform, Gabor wavelet, and logistic wavelet 经济波动 (business cycles) 的本质：内生震荡还是 外来噪声驱动？ Frisch model of noise driven harmonic cycles (1933), Harmonically bounded Brownian motion (Unlenbeck & Orstein 1930, Unlenbeck & Wang 王承 书 1945)

噪声与混沌的模式识别

Math Representation in Frequency Domain Fourier spectrum E(f) for color cycle and white noise

Spectrum E(f) of Harmonic Wave, White Noise, and Color Noise (Chaos) Wave Noise

Stationary (Discrete Fourier Transform) vs. Non-Stationary (Joint-Time-Frequency Analysis) Time Series Analysis 钱世锷，陈大庞 (1994)

S&P500 Stock Price Index FSPCOM with log-linear (LL) and HP growth trends

Detrended cycles with varying lengths

Different images of auto-correlations

Uncertainty Principle in Signal Processing

Gabor (Gaussian) Wavelet with Minimum Uncertainty

Gabor Space (1946) in discrete time-frequency space Minimum uncertainty in time- frequency space Un- orthogonal base function

The Coherent State (Gaussian Packet, Gabor Wavelet) The Uncertainty Principle in Quantum Mechanics & Information Theory

Separating noise and signal in Gabor space Time section in Gabor space (signal peak is far above the noise level) Time-dependent band filter in time-frequency space

Separating signals and noise in Gabor (time-frequency) space

Phase portraits of HP cycles White noise vs. color chaos

Stock Price Indexes (Standard & Poor 500) correlation dimension of color chaos in =2.5 variance of color chaos = 69 %

Structural instability of linear dynamical models

Structural Stability (Resilience) and Regime Switch in Nonlinear Dynamics

Linear Demand-Supply Curves > Unique Equilibrium 》 均衡幻象： “ 看不见的手 ” 的自稳定市场

为什么经济学家不能接受混沌？ 经济学自由主义的自稳定的理想市场 = 均衡态 + 白噪声 Frisch 噪声驱动谐振子模型（ 1933 ）》 Friedman 外生货币论》 Lucas 理性预期模型 》真实经济周期 (RBC) 模型》 Bernanke 金融 加速器模型 》 宏观经济学 + 金融经济学 + 计量经济学》全是 噪声驱动（永动机）模型

非线性供求曲线 》 多均衡态 》 经济复杂性 + 市场内生不稳定性的起源

Noise-Driven Cycles （ Frisch 1933 ） vs. Harmonic Brownian Motion （ Unlenbeck & Orstein, 1930 ） Wang 王承书 & Unlenbeck (1945) Frisch model for American business cycles ， which would be damped in 5-20 years （ Chen 1999,2004)!

Frisch model ： Perpetual Motion Machine of Second Type ？ Frisch was not the FIRST: G.E.Uhlenbeck and L.S. Ornstein, "On the Theory of Brownian Motion," Physical Review, 36(3), (1930). Frisch’s Informal conference paper: R. Frisch, “Propagation Problems and Impulse Problems in Dynamic Economics”, in Economic Essays in Honour of Gustav Cassel, George Allen & Unwin, London (1933). Frisch's promised paper, "Changing harmonics studied from the point of view of linear operators and erratic shocks," was advertised three times under the category "papers to appear in early issues" in Econometrica, including Issue No. 2, 3, and 4 of Volume I (April, July, and October 1933) but never appeared in Econometrica since Frisch never mentioned a word about his prize-winning model in his Nobel speech in 1969 (Frisch 1981).

Equilibrium Illusion through First-Differencing (High Frequency Noise Amplifier = Whitening Filter) in Econometric Analysis 经济学滤波器 = 高频噪声放大器》制造均衡幻象 Frequency response for the FD filter X(t) = FD[S(t)]=S(t+1)-S(t)

经济学的基本方程 力学系统：波动方程与扩散方程 化学，人口，与生态系统：化学反应方程和 人口动力学方程（ Lotka-Volterra 模型） 经济学的统一基础 : 生态人口动力学》市场 份额竞争为主（策略定价）》压倒价格竞争 （只是策略竞争的一种手段） 三层次论：微观 - 中观（金融与产业） - 宏观 经济学

薛定谔的启示：如何同时理解 基因（金融市场）的稳定性和变异性？ Principle of Large Numbers 如何理解金融市场在危机冲击下的柔韧性 (resilience) ？ 经济增长与波动过程中的不变量与普适方程 ： relative deviation （ RD ） 和 birth and death process

Principle of Large Numbers for Positive Variables S N =X 1 +X X N Observed relative deviation of US real GDP is 20 times larger than Lucas model （ Chen 2002) ！

Relative Deviation (RD) and the Implied Number (N*) from Empirical Observation Relative deviation Implied number

Large Numbers in US Economy

Weak microfoundations in labor and producer market The observed implied numbers predicted by the Lucas model is 400~ 500 times smaller (RD is 20 times larger ) than in US real economy. Possible microfoundations in financial markets and industrial organization not in labor or producer market Implication > Three layer structure (micro, meso, macro) rather than two layer structure (micro, macro) in economies

Implications from the Failure of Lucas Model of Microfoundations Roots of persistent business cycles: Financial intermediates and industrial organization From two layer model (micro-macro) > Three layer model (micro-meso-macro) Nature of macro fluctuations > social movements

Stable RD in US Macro Indexes observed under HP filter

RD Behavior for Stochastic Models Random walk is damping over time Brownian motion is exploding over time Only the Birth-death process is stable in time, which is a statistical model of endogenous fluctuations

Economic diagnosis

经济危机的诊断和预警 生灭过程的含时转移概率 理解危机的不同途径： Fat tail distribution (Levy, fractal Brownian motion, power law), regime switch between calm and turbulent market 高阶矩，生灭过程的主方程，与时间变 化的转移概率

Linear (BS) & Empirical (S&P 500) Transition Probability in Birth-Death Process from S&P data (Tang & Chen 2010)

Higher Moment Deviations (=System Risk) during Crisis (4-5 th Moments are 1000 times larger than 2 nd Moment > Failure of Portforlio Diversification)

Source of Financial Instability ： Nonlinear Trend & Higher Moments in Birth-Death Process Option pricing model based on nonlinear birth-death process Stability condition ： Under stability: finite first & second moment 》 Ito 》 construct arbitrage portfolio 》 Black-Scholes model

Corrections of Option Pricing Model for Interacting Trend & Diffusion No such thing as risk-free interest rate > only exist moving trend in interest rate Multiple frequency co-exist in financial market > interacting between market trend & group players > complex dynamics

逻辑斯蒂增长，市场份额竞争， 与知识的新陈代谢 李约瑟问题, Wallerstein paradox, 和东西方文 明的分岔 Lotka-Volterra model with culture factor Logistic wavelets 和产业、文明的兴衰 复杂性与稳定性的 trade-off: 理论生物学与演 化经济学 一般斯密原理：劳动分工受市场规模、资源 种类、和环境涨落的限制

Metabolic Growth (Life Cycles): 知识的代谢增长和 技术发展的逻辑斯蒂小波

Physics Foundation of Biology Schrödinger (1948): What is life? (Quantum biology) Large molecule, meta-stable state, non-periodic structure, information as negative entropy Prigogine (1972): Thermodynamics of evolution Dissipative structure with matter flow, energy flow, and information flow in open system Constructive role of positive feedback (catalytic reaction) Self-organization (order out of chaos)

非平衡物理与经济复杂系统： Haken vs. Prigogine Haken synergetics 协同学 Laser 》 Hamiltonian system Adiabetic approximation > take out slow variables (language) Prigogine 》 dissipative systems 》 no potential function 》 slef-organization 自组织 Chemical reaction = population dynamics 》 nonlinear evolutionary dynamics 》 nonlinear economic dynamics

经济物理有待解决的问题 经济学与心理学、脑科学的交叉 货币理论：可伸缩流体力学？ 社会科层与边界的起源 ~ 细胞壁与半透膜 ， 经济结构的对称性与对称破却的机制 》 企业和组织的分类， GDP 与经济熵的测 量

References 陈平, 文明分岔, 经济混沌, 和演化经济动力 学, 北京大学出版社 (2004). Chen, Ping. Economic Complexity and Equilibrium Illusion, Routledge, London (2010).

Readings in Complexity Science Wienner, N. Cybernetics: or Control and Communication in the Animal and the Machine, MIT Press, MA: Cambridge (1948). Schrödinger, E. What is Life? Cambridge University Press, Cambridge (1948). Prigogine, I. From Being to Becoming: Time and Complexity in the Physical Sciences, Freeman, San Francisco (1980). May, R. M., Stability and Complexity in Model Ecosystem, Princeton University Press, Princeton (1974).