Adaptive Expectations and Stock Market Crashes

A theory is developed that explains how stocks can crash without fundamental news and why crashes are more common than frenzies. A crash occurs via the interaction of rational and naive investors. Naive traders believe that prices follow a random walk with serially correlated volatility. Their expectations of future volatility are formed adaptively. When the market crashes, naive traders sell stock in response to the apparent increase in volatility. Since rational traders are risk averse as well, a lower price is needed to clear the market: the crash is a self-fulfilling prophecy. Frenzies cannot occur in this model.

Dynamic Equilibrium Selection: A General Uniqueness Result

This paper shows that in a dynamic context, under weak assumptions, the presence of payoff shocks can shrink the equilibrium set to a singleton. We study a model with a continuum of fully rational agents who switch between two actions or states over time (e.g., working in different sectors, employment vs. unemployment, etc.). An agent's incentive to pick a given action is greater if others do the same. Agents receive chances to change actions at random times and may influence the rate at which these chances arrive. Payoff shocks may follow any of a large class of stochastic processes that includes both seasonal and mean-reverting processes. In this general setting, payoff shocks give rise to a unique equilibrium. One implication is that the introduction of aggregate shocks leads to a unique equilibrium in two well-known macroeconomic search models with multiple equilibria (Diamond and Fudenberg, Howitt and McAfee).

Adaptive Expectations and Stock Market Crashes

A theory is developed that explains how the stock market can crash in the absence of news about fundamentals, and why crashes are more common than frenzies. A crash occurs via the interaction of rational and naive investors. Naive traders believe in a simple (but reasonable) statistical model of stock prices: that prices follow a random walk with serially correlated volatility. They predict future volatility adaptively, as a weighted average of past squared price changes. In a crash, the naive traders lower their demand in response to the apparent increase in volatility. This lowers the risk bearing capacity of the market, so that the lower crash price clears the market. Unlike other explanations of market crashes, this mechanism is fundamentally asymmetric: the stock price cannot rise sharply, so frenzies or bubbles cannot occur.Stock market crashes; adaptive expectations; volatility feedback; excess volatility

Insuring customers of a unionized firm against loss of network benefits

We study how optimally to insure customers of a unionized firm, such as an auto maker, against the loss of network benefits that occurs when other consumers abandon the firm. The union first announces a wage. A random demand shock is then realized. The firm then chooses its price and, finally, consumers decide whether or not to buy from the firm. Common knowledge of payoffs is perturbed slightly in order to obtain a unique outcome. In this outcome the union chooses an excessive wage, leading consumers to abandon the firm too often. The first best can be costlessly attained by providing consumers with countercyclical insurance

Scale-Invariant Measures of Segregation

We characterize measures of school segregation for any number of ethnic groups using a set of purely ordinal axioms that includes Scale Invariance: a school district's segregation ranking should be invariant to changes that do not affect the distribution of ethnic groups across schools. The symmetric Atkinson index is the unique such measure that treats ethnic groups symmetrically and that ranks a district as weakly more segregated if either (a) one of its schools is subdivided or (b) its students in a subarea are moved around so as to weakly raise segregation in that subarea. If the requirement of symmetry is dropped, one obtains the general Atkinson index. The role of Scale Invariance is illustrated by studying segregation among U.S. public schools from 1987/8 to 2005/6, a period in which ethnic groups became distributed more similarly across schools. While the Atkinson indices declined sharply, most other indices either rose or declined only slightly.Segregation, segregation indices.

Measuring Segregation

We define a segregation ordering as a ranking of cities from most segregated to least segregated. \ We propose a set of basic properties that any reasonable segregation ordering should have. \ We then fully characterize the class of segregation orderings that satisfy these basic properties. \ We prove that every such ordering is representable by a segregation index that has a particular simple form. \ Finally, we show that with one rarely used exception, each index defined in the literature violates one or more of our basic properties.segregation, evenness, isolation, dissimilarity, axiomatic method

Measuring segregation

We propose a set of axioms for the measurement of school-based segregation with any number of ethnic groups. These axioms are motivated by two criteria. The first is evenness: how much do ethnic groups’ distributions across schools differ? The second is representativeness: how different are schools’ ethnic distributions from one another? We prove that a unique ordering satisfies our axioms. It is represented by an index that was originally proposed by Henri Theil (1971). This “Mutual Information Index” is related to Theil’s better known Entropy Index, which violates two of our axioms

An axiomatization of the multigroup Atkinson segregation indices

This paper gives an axiomatic characterization of the multigroup Atkinson indices of segregation relying entirely on ordinal axioms. We show that the Symmetric Atkinson index represents the unique ordering that treats ethnic groups symmetrically, that is invariant to population growth rates that differ among ethnic groups, that regards school districts as more segregated when schools in them are subdivided (unless the new schools have the exact same ethnic distribution), and that satisfy an independence property. If symmetry among ethnic groups is dropped, one obtains the family of orderings that are represented by the Asymmetric Atkinson indices. The latter result requires the addition of a continuity axiom

Measuring Segregation

We propose a set of axioms for the measurement of school-based segregation with any number of ethnic groups. These axioms are motivated by two criteria. The first is evenness: how much do ethnic groups’ distributions across schools differ? The second is representativeness: how different are schools’ ethnic distributions from one another? We prove that a unique ordering satisfies our axioms. It is represented by an index that was originally proposed by Henri Theil (1971). This “Mutual Information Index” is related to Theil’s better known Entropy Index, which violates two of our axioms.Segregation; measurement; schools; education; indices; peer effects; equal opportunity

Scale-Invariant Measures of Segregation

We characterize measures of school segregation for any number of ethnic groups using a set of purely ordinal axioms that includes Scale Invariance: a school district?s segregation ranking should be invariant to changes that do not a¤ect the distribution of ethnic groups across schools. The symmetric Atkinson index is the unique such measure that treats ethnic groups symmetrically and that ranks a district as weakly more segregated if either (a) one of its schools is subdivided or (b) its students in a subarea are moved around so as to weakly raise segregation in that subarea. If the requirement of symmetry is dropped, one obtains the general Atkinson index. The role of Scale Invariance is illustrated by studying segregation among U.S. public schools from 1987/8 to 2005/6, a period in which ethnic groups became distributed more similarly across schools. While the Atkinson indices declined sharply, most other indices either rose or declined only slightly.