When the probability of measuring a particular value of some quantity varies inversely as a power of that value, the quantity is said to follow a power law, also known variously as Zipf's law or the Pareto distribution. Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. For instance, the distributions of the sizes of cities, earthquakes, forest fires, solar flares, moon craters and people's personal fortunes all appear to follow power laws. The origin of power-law behaviour has been a topic of debate in the scientific community for more than a century. Here we review some of the empirical evidence for the existence of power-law forms and the theories proposed to explain them.

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Power laws, Pareto distributions and Zipf's law

The author systematically examines one of the important issues of information — establishing the market price. He introduces the concept of «search» — where a buyer wanting to get a better price, is forced to question sellers. The article deals with various aspects of finding the necessary information.

Economics of Information

We examine the pricing of aggregate volatility risk in the cross-section of stock returns. Consistent with theory, we find that stocks with high sensitivities to innovations in aggregate volatility have low average returns. In addition, we find that stocks with high idiosyncratic volatility relative to the Fama and French (1993) model have abysmally low average returns. This phenomenon cannot be explained by exposure to aggregate volatility risk. Size, book-to-market, momentum, and liquidity effects cannot account for either the low average returns earned by stocks with high exposure to systematic volatility risk or for the low average returns of stocks with high idiosyncratic volatility.

The Cross-Section of Volatility and Expected Returns

We examine the pricing of aggregate volatility risk in the cross-section of stock returns. Consistent with theory, we find that stocks with high sensitivities to innovations in aggregate volatility have low average returns. Stocks with high idiosyncratic volatility relative to the Fama and French (1993, Journal of Financial Economics 25, 2349) model have abysmally low average returns. This phenomenon cannot be explained by exposure to aggregate volatility risk. Size, book-to-market, momentum, and liquidity effects cannot account for either the low average returns earned by stocks with high exposure to systematic volatility risk or for the low average returns of stocks with high idiosyncratic volatility. IT IS WELL KNOWN THAT THE VOLATILITY OF STOCK RETURNS varies over time. While considerable research has examined the time-series relation between the volatility of the market and the expected return on the market (see, among others, Campbell and Hentschel (1992) and Glosten, Jagannathan, and Runkle (1993)), the question of how aggregate volatility affects the cross-section of expected stock returns has received less attention. Time-varying market volatility induces changes in the investment opportunity set by changing the expectation of future market returns, or by changing the risk-return trade-off. If the volatility of the market return is a systematic risk factor, the arbitrage pricing theory or a factor model predicts that aggregate volatility should also be priced in the cross-section of stocks. Hence, stocks with different sensitivities to innovations in aggregate volatility should have different expected returns. The first goal of this paper is to provide a systematic investigation of how the stochastic volatility of the market is priced in the cross-section of expected stock returns. We want to both determine whether the volatility of the market

https://www0.gsb.columbia.edu/faculty/rhodrick/crosssection.pdf

‡§ ¶ Humanity has just crossed a major landmark in its history with the majority of people now living in cities. Cities have long been known to be society’s predominant engine of innovation and wealth creation, yet they are also its main source of crime, pollution, and disease. The inexorable trend toward urbanization worldwide presents an urgent challenge for developing a predictive, quantitative theory of urban organization and sustainable development. Here we present empirical evidence indicating that the processes relating urbanization to economic development and knowledge creation are very general, being shared by all cities belonging to the same urban system and sustained across different nations and times. Many diverse properties of cities from patent production and personal income to electrical cable length are shown to be power law functions of population size with scaling exponents, , that fall into distinct universality classes. Quantities reflecting wealth creation and innovation have 1.2 >1 (increasing returns), whereas those accounting for infrastructure display 0.8 <1 (economies of scale). We predict that the pace of social life in the city increases with population size, in quantitative agreement with data, and we discuss how cities are similar to, and differ from, biological organisms, for which <1. Finally, we explore possible consequences of these scaling relations by deriving growth equations, which quantify the dramatic difference between growth fueled by innovation versus that driven by economies of scale. This difference suggests that, as population grows, major innovation cycles must be generated at a continually accelerating rate to sustain growth and avoid stagnation or collapse. population sustainability urban studies increasing returns economics of scale

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Growth, innovation, scaling, and the pace of life in cities

This paper develops a simple equilibrium model of CEO pay. CEOs have different talents and are matched to firms in a competitive assignment model. In market equilibrium, a CEO’s pay changes one for one with aggregate firm size, while changing much less with the size of his own firm. The model determines the level of CEO pay across firms and over time, offering a benchmark for calibratable corporate finance. The sixfold increase of CEO pay between 1980 and 2003 can be fully attributed to the six-fold increase in market capitalization of large US companies during that period. We find a very small dispersion in CEO talent, which nonetheless justifies large pay differences. The data broadly support the model. The size of large fi rms explains many of the patterns in CEO pay, across firms, over time, and between countries. (JEL D2, D3, G34, J3)

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Why has CEO Pay Increased So Much

Zipf ’s law is a very tight constraint on the class of admissible models of local growth. It says that for most countries the size distribution of cities strikingly fits a power law: the number of cities with populations greater than S is proportional to 1/S. Suppose that, at least in the upper tail, all cities follow some proportional growth process (this appears to be verified empirically). This automatically leads their distribution to converge to Zipf ’s law.

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Zipf's Law for Cities: An Explanation

Insights into the dynamics of a complex system are often gained by focusing on large fluctuations. For the financial system, huge databases now exist that facilitate the analysis of large fluctuations and the characterization of their statistical behaviour. Power laws appear to describe histograms of relevant financial fluctuations, such as fluctuations in stock price, trading volume and the number of trades. Surprisingly, the exponents that characterize these power laws are similar for different types and sizes of markets, for different market trends and even for different countries--suggesting that a generic theoretical basis may underlie these phenomena. Here we propose a model, based on a plausible set of assumptions, which provides an explanation for these empirical power laws. Our model is based on the hypothesis that large movements in stock market activity arise from the trades of large participants. Starting from an empirical characterization of the size distribution of those large market participants (mutual funds), we show that the power laws observed in financial data arise when the trading behaviour is performed in an optimal way. Our model additionally explains certain striking empirical regularities that describe the relationship between large fluctuations in prices, trading volume and the number of trades.

A theory of power-law distributions in financial market fluctuations

Bayesian consumers infer that hidden add-on prices (e.g., the cost of ink for a printer) are likely to be high prices. If consumers are Bayesian, firms will not shroud information in equilibrium. However, shrouding may occur in an economy with some myopic (or unaware) consumers. Such shrouding creates an inefficiency, which firms may have an incentive to eliminate by educating their competitors' customers. However, if add-ons have close substitutes, a "curse of debiasing" arises, and firms will not be able to profitably debias consumers by unshrouding add-ons. In equilibrium, two kinds of exploitation coexist. Optimizing firms exploit myopic consumers through marketing schemes that shroud high-priced add-ons. In turn, sophisticated consumers exploit these marketing schemes. It is not possible to profitably drive away the business of sophisticates. It is also not possible to profitably lure either myopes or sophisticates to nonexploitative firms. We show that informational shrouding flourishes even in highly competitive markets, even in markets with costless advertising, and even when the shrouding generates allocational inefficiencies.

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Shrouded attributes, consumer myopia, and information suppression in competitive markets

This paper incorporates a time-varying intensity of disasters in the Rietz-Barro hypothesis that risk premia result from the possibility of rare, large disasters. During a disaster, an asset’s fundamental value falls by a time-varying amount. This in turn generates time-varying risk premia and thus volatile asset prices and return predictability. Using the recent technique of linearity-generating processes (Gabaix 2007), the model is tractable, and all prices are exactly solved in closed form. In the “variable rare disasters” framework, the following empirical regularities can be understood qualitatively: (i) equity premium puzzle (ii) risk-free rate-puzzle (iii) excess volatility puzzle (iv) predictability of aggregate stock market returns with price-dividend ratios (v) value premium (vi) often greater explanatory power of characteristics than covariances for asset returns (vii) upward sloping nominal yield curve (viiii) a steep yield curve predicts high bond excess returns and a fall in long term rates (ix) corporate bond spread puzzle (x) high price of deep out-of-the-money puts. I also provide a calibration in which those puzzles can be understood quantitatively as well. The fear of disaster can be interpreted literally, or can be viewed as a tractable way to model time-varying risk-aversion or investor sentiment. (JEL: E43, E44, G12)

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Xavier Gabaix

Papers

Why has CEO Pay Increased So Much

Zipf's Law for Cities: An Explanation

A theory of power-law distributions in financial market fluctuations

Shrouded attributes, consumer myopia, and information suppression in competitive markets

Variable Rare Disasters: An Exactly Solved Framework for Ten Puzzles in Macro-Finance