ON THE ROBUSTNESS OF ECONOMIC MODELS TO HEAVY-TAILEDNESS ASSUMPTIONS

Abstract

Many economic models have a structure that depends on majorization phenomena for convolutions of distributions. The present paper develops a unified approach to the analysis of majorization properties of linear combinations of random variables. It further studies the robustness of these properties and the implications of a number of important economic models to heavy-tailedness assumptions. The paper shows that majorizations for log-concave distributed random variables established in the seminal work by Proschan (1965) continue to hold for not extremely thick-tailed distributions. However, the majorization properties are reversed in the case of distributions with extremely heavy-tailed densities. This is the first probabilistic result that shows that majorization properties of log-concave densities are reversed for a wide class of distributions and is the key to reversals of properties of many economic models built upon the popular log-concavity assumption. In a series of applications of the main probabilistic results, we study robustness of monotone consistency of the sample mean, value at risk (VaR) analysis for financial portfolios, growth theory for firms that can invest in information about their market, as well as that of bundling theory for sellers of baskets of goods with an arbitrary degree of complementarity or substitutability. The main results show that many economic models are robust to heavy-tailedness assumptions as long as the distributions entering these assumptions are not extremely thick-tailed. But the implications of these models are reversed for distributions with sufficiently long-tailed densities. The following list summarizes some of the main results. i) Using the general majorization results obtained, the paper shows, for the first time in the literature, that the stylized fact that portfolio diversification is always preferable is reversed for a wide class of distributions of risks. Namely, in the case of risks with extremely thick-tailed distributions, diversification of a portfolio always leads to an increase in riskiness of the portfolio’s return. The paper further demonstrates that the stylized facts on diversification are robust to thick-tailedness of risks or returns as long as their distributions are not extremely long-tailed. Moreover, we show that, in the world of not extremely heavy-tailed risks, the value at risk satisfies the important condition of coherency. However, coherency of the value at risk is always violated if distributions of risks are extremely thick-tailed. In addition, the paper shows that the sample mean exhibits monotone consistency in the case of data from not extremely thick-tailed populations. Thus, an increase in the sample size always improves performance of the sample mean. 1Job market paper. I am indebted to Donald Andrews, Peter Phillips and Herbert Scarf for all their support and guidance in all stages of the current project. I also thank Aydin Cecen, Brian Dineen, Darrell Duffie, Xavier Gabaix, Philip Haile, Samuel Karlin, Alex Maynard, Ingram Olkin, Ben Polak, Gustavo Soares, Kevin Song and the participants at the International Conference on Stochastic Finance, 2004 and the 18th New England Statistics Symposium at Harvard University for helpful comments and discussions. The financial support from the Yale University Graduate Fellowship and the Cowles Foundation Prize is gratefully acknowledged.

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