Heun Functions and Some of Their Applications in Physics

Most of the theoretical physics known today is described by using a small number of differential equations. For linear systems, different forms of the hypergeometric or the confluent hypergeometric equations often suffice to describe the system studied. These equations have power series solutions with simple relations between consecutive coefficients and/ or can be represented in terms of simple integral transforms. If the problem is nonlinear, one often uses one form of the Painlev\'{e} equations. There are important examples, however, where one has to use higher order equations. Heun equation is one of these examples, which recently is often encountered in problems in general relativity and astrophysics. Its special and confluent forms take names as Mathieu, Lam\'{e} and Coulomb spheroidal equations. For these equations whenever a power series solution is written, instead of a two-way recursion relation between the coefficients in the series, we find one between three or four different ones. An integral transform solution using simpler functions also is not obtainable. The use of this equation in physics and mathematical literature exploded in the later years, more than doubling the number of papers with these solutions in the last decade, compared to time period since this equation was introduced in 1889 up to 2008. We use SCI data to conclude this statement, which is not precise, but in the correct ballpark. Here this equation will be introduced and examples for its use, especially in general relativity literature will be given.Comment: 19 pages. Submitted version to journal Adv.High Energy Physics. An earlier version of the paper was published in "Proceedings of the 13th Regional Conference on Mathematical Physics, Antalya, Turkey, October 27-31, 2010", Edited by Ugur Camci and Ibrahim Semiz, pp. 23-39. World Scientific (2013) (DOI: 10.1142/9789814417532_0002

Cementing Relationships: Vertical Integration, Foreclosure, Productivity, and Prices

This paper empirically investigates the possible market power effects of vertical integration proposed in the theoretical literature on vertical foreclosure. It uses a rich data set of cement and ready-mixed concrete plants that spans several decades to perform a detailed case study. There is little evidence that foreclosure is quantitatively important in these industries. Instead, prices fall, quantities rise, and entry rates remain unchanged when markets become more integrated. These patterns are consistent, however, with an alternative efficiency-based mechanism. Namely, higher productivity producers are more likely to vertically integrate and are also larger, more likely to survive, and charge lower prices. We find evidence that integrated producers' productivity advantage is tied to improved logistics coordination afforded by large local concrete operations. Interestingly, this benefit is not due to firms' vertical structures per se: non-vertical firms with large local concrete operations have similarly high productivity levels.

The Dynamics of Seller Reputation: Theory and Evidence from eBay

We propose a basic theoretical model of eBay's reputation mechanism, derive a series of implications and empirically test their validity. Our theoretical model features both adverse selection and moral hazard. We show that when a seller receives a negative rating for the first time his reputation decreases and so does his effort level. This implies a decline in sales and price; and an increase in the rate of arrival of subsequent negative feedback. Our model also suggests that sellers with worse records are more likely to exit (and possibly re-enter under a new identity), whereas better sellers have more to gain from buying a reputation' by building up a record of favorable feedback through purchases rather than sales. Our empirical evidence, based on a panel data set of seller feedback histories and cross-sectional data on transaction prices collected from eBay is broadly consistent with all of these predictions. An important conclusion of our results is that eBay's reputation system gives way to strategic responses from both buyers and sellers.

Are Structural Estimates of Auction Models Reasonable? Evidence from Experimental Data

Recently, economists have developed methods for structural estimation of auction models. Many researchers object to these methods because they find the rationality assumptions used in these models to be implausible. In this paper, we explore whether structural auction models can generate reasonable estimates of bidders' private information. Using bid data from auction experiments, we estimate four alternative structural models of bidding in first-price sealed-bid auctions: 1) risk neutral Bayes-Nash, 2) risk averse Bayes-Nash, 3) a model of learning and 4) a quantal response model of bidding. For each model, we compare the estimated valuations and the valuations assigned to bidders in the experiments. We find that a slight modification of Guerre, Perrigne and Vuong's (2000) procedure for estimating the risk neutral Bayes-Nash model to allow for bidder asymmetries generates quite reasonable estimates of the structural parameters.