Section 1 – Ec1818 Jeremy Barofsky jbarofsk@hsph.harvard

1. Section 1 – Ec1818Jeremy Barofskyjbarofsk@hsph.harvard.edu February 4 and 5, 2010

2. Outline • Introduction and motivating examples • Trying to define complexity • Logistic Curve • Cobweb Model • TimurKuran’s Propagation Model (if time)

3. Motivation: What is discontinuous change? • For motivation we focus on explaining three empirical regularities in social science that theory does not clarify well • Financial market crises • Changes in crime rates • Distribution of city populations and firm sizes

4. The Efficient Markets Hypothesis • With the assumptions of Walrasian Equilibrium: • Prices equal stock value (all information that could change is instantaneously incorporated) • Prices follow a random walk (Bachelier 1900) • Price prediction is impossible • Market allocates resources efficiently “A market is efficient with respect to a particular set of information if it is impossible to make abnormal profits by using this set of information to formulate buying and selling decisions” (Sharpe, 1995)

5. Real Markets Exhibit 1. Excess Volatility (Shiller, 1997) 2. Large price changes without significant news (Cutler, et al 1989) 3. Price movements are temporally correlated- clustered volatility- and have fat tails (Mandelbrot 1963, Farmer 1999) 4. If prices equal value, no trading should occur at all. Yet trade in foreign exchange markets exceeds $1 trillion daily, 50 times greater than world GNP per day (Farmer 2000) 5. Modern Markets do not trade in equilibrium- Limit Order Book stores the excess supply and demand

6. Crime Rates • Crime rates vary enormously over time and space. Ie: 123rd precinct in NYC has 0.022 serious crimes per capita and 1st precinct has 10 times higher rate of 0.21 serious crimes per capita (Glaeser, et al 1996). • Homicides dropped by 50% from 1933-1961, increased rapidly through the 1970s-1980s, and then dropped precipitously again until we have rates again similar to the 1960s. • Econometric models of city characteristics can explain no more than 30% of the variation in cross-city or cross-precinct differences in crime. • In other words, cities with similar characteristics often have very different crime rates. Why?

7. City and Firm Size Follows a Power Law • City population in the U.S. over time and in other nations follows a Zipf distribution, meaning that the number of cities whose population exceeds S is proportional to S-α, where α = 1 for U.S. cities. • 40 cities > 1 million, 20 cities > 2 million, 9 cities > 4 million. • Also, 2nd largest city has ½ the population of the 1st and the 10th largest city has 1/10th the population of the largest city. • Striking empirical pattern that persists even when some cities (Las Vegas) experience rapid growth and others (Boston) exhibit little change at all.

8. Complexity: The Edge between Order and Chaos • These three examples demonstrate that we need models which can both explain long-term equilibrium and occasional big changes. • Kaufman defines complexity as the edge between chaos (where no system behavior repeats itself) and order (where the system is static). • Complex systems tend to exhibit emergent behavior – an aggregate outcome unexpected based on the individual agents. Also exhibit self-organization from randomness (Schelling, Krugman’s Edge City Model).

9. Why haven’t I heard about this before? • Most models in microeconomics model behavior change that is marginal. Key assumption is ceteris paribus, all else being equal, ie: assuming an individual is a price taker so one person’s behavior doesn’t influence others • Not all new: Fundamental Theorem of Welfare Economics • Agent-based modeling says that generating a macro-pattern is a necessary condition to explain the pattern (Epstein, 2005). • New: Order from randomness, no ceteris paribus assumption, and emphasis on process to achieve equilibrium.

10. Complexity Models • Top-down approach: • These models use systems dynamics to specify the overall structure of an entire system using difference or difference equations which describe how the systems changes over time (Cobweb model, systems dynamics models). • Bottom-up approach: • Use agent-based simulations. Specify individual behaviors and environment, then let interactions occur and observe aggregate outcomes (Schelling model). • THE FINE PRINT – many of these models were going to the “next wave” in science, but these waves never hit

11. Logistic curve • Model of discontinuous change that allows us to still do calculus and model phase transitions: p = α / (1+ e -βx ) • Such curves have a range [0, α] where α = 1 usually, derivative is a parabola = αβ(1-p)p, which reaches a maximum at 1/2. • If p represents a probability, then we have a linear log-odds ratio ln[ p/(1-p)] = -βx

12. Logistic Curve

13. Cobweb Model • CGt = St + aWt-1and Wt = Xt – e(CGt), where W = wage, CG = college graduates, S = exogenous supply shift, X = exogenous demand shift. • Solve: CGt = St + aXt-1 – aeCGt-1 • High wages last period raises supply of CG today (positive feedback), but many CG’s today reduce today’s wages (negative feedback) • Can have three possible behaviors: • 1) spiral in until we reach an equilibrium of CG’s, • 2) fluctuate around equilibrium, • 3) spiral out until the system explodes.

14. Schelling Segregation Model • Decision to move or not based on binary utility function – are you happy or not? • One decision rule: each individual prefers 3/8 of neighbors to be own group in “Moore neighborhood.” • Key points: • Order or an equilibrium not necessarily desirable • Behaviors and preferences can be interdependent • Cannot simply extrapolate individual preferences from aggregate outcomes

15. Schelling’s Model: Types are # and 0, left figure shows the initial configuration, right figure shows unhappy pieces that will move (figure from Schelling’s book, chap. 4, p. 149)Rules:1 neighbor, must be of the same color2 neighbors – 1 must of same color3, 4, 5 neighbors – 2 must be of the same color6, 7, 8 neighbors – 3 must be of the same color

16. Propagation Model – threshold effects (Kuran, p.66-68) • Public preference determined by trade-off between own beliefs and social pressure. Social pressure is a function of Y = mean public opinion. • Preferences range [0,100] where 0 = pro-choice, 100 = pro-life. If person has preference x = 20, then will be pro-choice, but cost for public preference of 0 increases as Y increases. • t = political threshold where if Y < t then support 0 (pro-choice) and Y > t support 100 (pro-life). • Models heterogeneous level of t’s in population which determines the propagation dynamics of public preferences.