Parabolic Catalan numbers count flagged Schur functions and their appearances as type A Demazure characters (key polynomials)

Fix an integer partition lambda that has no more than n parts. Let beta be a weakly increasing n-tuple with entries from {1,..,n}. The flagged Schur function indexed by lambda and beta is a polynomial generating function in x_1, .., x_n for certain semistandard tableaux of shape lambda. Let pi be an n-permutation. The type A Demazure character (key polynomial, Demazure polynomial) indexed by lambda and pi is another such polynomial generating function. Reiner and Shimozono and then Postnikov and Stanley studied coincidences between these two families of polynomials. Here their results are sharpened by the specification of unique representatives for the equivalence classes of indexes for both families of polynomials, extended by the consideration of more general beta, and deepened by proving that the polynomial coincidences also hold at the level of the underlying tableau sets. Let R be the set of lengths of columns in the shape of lambda that are less than n. Ordered set partitions of {1,..,n} with block sizes determined by R, called R-permutations, are used to describe the minimal length representatives for the parabolic quotient of the nth symmetric group specified by the set {1,..,n-1}\R. The notion of 312-avoidance is generalized from n-permutations to these set partitions. The R-parabolic Catalan number is defined to be the number of these. Every flagged Schur function arises as a Demazure polynomial. Those Demazure polynomials are precisely indexed by the R-312-avoiding R-permutations. Hence the number of flagged Schur functions that are distinct as polynomials is shown to be the R-parabolic Catalan number. The projecting and lifting processes that relate the notions of 312-avoidance and of R-312-avoidance are described with maps developed for other purposes.Comment: 27 pages, 2 figures. Identical to v.2, except for the insertion of the publication data for the DMTCS journal (dates and volume/issue/number). This is two-thirds of our preprint "Parabolic Catalan numbers count flagged Schur functions; Convexity of tableau sets for Demazure characters", arXiv:1612.06323v

Cognition and Wealth: The Importance of Probabilistic Thinking

This paper utilizes a large set of subjective probability questions from the Health and Retirement Survey to construct an index measuring the precision of probabilistic beliefs (PPB) and relates this index to household choices about the riskiness of their portfolios and the rate of growth of their net worth. A theory of uncertainty aversion based on repeated sampling is proposed that resolves the Ellsberg Paradox within a conventional expected utility model. In this theory, uncertainty aversion is implied by risk aversion. This theory is then used to propose a link between an individual’s degree of uncertainty and his propensity to give “focal” answers of “0”, “50_50” or “100” or “exact” answers to survey questions and the validity of this interpretation is tested empirically. Finally, an index of the precision of probabilistic thinking is constructed by calculating the fraction of probability questions to which each HRS respondent gives a non-focal answer. This index is shown to have a statistically and economically significant positive effect on the fraction of risky assets in household portfolios and on the rate of growth of these assets longitudinally. These results suggest that there is systematic variation in the competence of individuals to manage investment accounts that should be considered in designing policies to create individual retirement accounts in the Social Security system.

Who Becomes a Stockholder? Expectations, SUbjective Uncertainty, and Asset Allocation

We develop a model of portfolio selection with subjective uncertainty and learning in order to explain why some people hold stocks while others don’t. We model heterogeneity in information directly, which is an alternative to the existing explanations that emphasized heterogeneity in transaction costs of investment. We plan to calibrate the model to survey data (when available) on people’s perception about the distribution of stock market returns. Our approach also leads to a model of learning with new implications such as zero optimal risky assets, or ex post correlation of uncorrelated labor income and optimal portfolio composition. It also points to two factors in probabilistic thinking that should have a major impact on stock ownership. These are the level and the precision of expectations. We construct proxy measures for the two parameters from the 1992-2000 waves of the Health and Retirement Study (HRS). We use a large battery of the subjective probability questions administered in each wave of HRS to construct an overall “index of optimism” (the correlated factor between all subjective probabilities) and “index of precision” (the fraction of nonfocal probability answers, following Lillard and Willis, 2001). We also construct measures for how people forecast the weather, their cognitive capacity, wealth, and basic demographics. Our results indicate that stock ownership and the probability of becoming a stockholder are strongly positively correlated with the indices of the level and precision of expectations. Interpretation of the former is quite challenging and further research is needed to understand its full content.

THE SOCIAL VALUE OF PUBLIC INVESTMENT IN CANADIAN WHEAT BREEDING RESEARCH

The level and distribution of the social benefits from public investment in Canadian wheat breeding research and extension activities are examined for the period 1946 to 1979. In addition, average and marginal rates of return are calculated. The results reveal that society has benefited substantially from such public investments, with producers receiving the greatest share of the benefits.Crop Production/Industries, Research and Development/Tech Change/Emerging Technologies,

Dynamic Aspects of Earnings Mobility

This paper proposes an econometric methodology to deal with life cycle earnings and mobility among discrete earnings classes. First, we use panel data on male log earnings to estimate an earnings function with permanent and serially correlated transitory components due to both measured and unmeasured variables. Assuming that the error components are normally distributed, we develop statements for the probability that an individual's earnings will fall into a particular but arbitrary time sequence of poverty states. Using these statements, we illustrate the implications of our earnings model for poverty dynamics and compare our approach to Markov chain models of income mobility.