Proportional Dynamics in Exchange Economies

We study the Proportional Response dynamic in exchange economies, where each player starts with some amount of money and a good. Every day, the players bring one unit of their good and submit bids on goods they like, each good gets allocated in proportion to the bid amounts, and each seller collects the bids received. Then every player updates the bids proportionally to the contribution of each good in their utility. This dynamic models a process of learning how to bid and has been studied in a series of papers on Fisher and production markets, but not in exchange economies. Our main results are as follows: - For linear utilities, the dynamic converges to market equilibrium utilities and allocations, while the bids and prices may cycle. We give a combinatorial characterization of limit cycles for prices and bids. - We introduce a lazy version of the dynamic, where players may save money for later, and show this converges in everything: utilities, allocations, and prices. - For CES utilities in the substitute range $[0,1)$, the dynamic converges for all parameters. This answers an open question about exchange economies with linear utilities, where tatonnement does not converge to market equilibria, and no natural process leading to equilibria was known. We also note that proportional response is a process where the players exchange goods throughout time (in out-of-equilibrium states), while tatonnement only explains how exchange happens in the limit.Comment: 25 pages, 6 figure

Computing economic equilibria by variable dimension algorithms: State of the art

Economics

Substitutability and the Quest for Stability. : Some Reflections on the Methodology of General Equilibrium in Historical Perspective

Do not quote without permissionIn this paper, I propose to interpret the history of stability analysis of a Walrasian exchange economy through the lenses of the concept of substitutability. A purely mathematical approach of this story does not seem sufficient to account for the way economists have studied the question of stability and how thay have reacted to the results of this research program. My point is that mathematical constraints kept apart, the concept of substituability has shaped the path followed by stability analysis since the publication of Value and Capital (Hicks, 1939) up to the Sonnenschein-Mantel-Debreu theorems and beyond. Thus, I uphold that firstly, the concept of substitutability allows to catch the heuristic behind the work on stability, and secondly, it allows replacing the importance of SMD results in a broader context, giving a more subtle view of the ups and down of general equilibrium theory as a research program

Lattice methods for no-arbitrage pricing of interest rate securities

We explore calibration of single factor no-arbitrage short rate models to yield and volatility information. We note that the calculation of Arrow-Debreu prices for interest rate securities is analogous to solving the Kolmogorov Forward Equation. This insight allows us to implement implicit methods which exhibit more rapid convergence than explicit methods. We develop an algorithm for calibrating a model to match both yield and volatility curves which is general across single factor short rate models and also across finite difference techniques. Numerical examples confirm that our approach vastly improves computation times for derivative pricing