Market equilibria and money

By the first welfare theorem, competitive market equilibria belong to the core and hence are Pareto optimal. Letting money be a commodity, this paper turns these two inclusions around. More precisely, by generalizing the second welfare theorem we show that the said solutions may coincide as a common fixed point for one and the same system.Mathematical arguments invoke conjugation, convolution, and generalized gradients. Convexity is merely needed via subdifferentiablity of aggregate “cost”, and at one point only.Economic arguments hinge on idealized market mechanisms. Construed as algorithms, each stops, and a steady state prevails if and only if price-taking markets clear and value added is nil.

Fair Resource Allocation in a Volatile Marketplace

In settings where a platform must allocate finite supplies of goods to buyers, balancing overall platform revenues with the fairness of the individual allocations to platform participants is paramount to the well-functioning of the platform. This is made even more difficult by the fact that the supply of goods is in practice stochastic and difficult to forecast, such as in the case of online ad allocation, where the platform manages a supply of impressions that varies over time. In this paper, we design a fair allocation scheme that works in the presence of supply uncertainty. Algorithmically, the scheme repeatedly solves for Fisher market equilibria in a model predictive control fashion and is proved to admit constant factor guarantees versus the offline optimal. In addition, the scheme is tested on a sequence of real ad datasets, showing strong empirical performance.

Computing Large Market Equilibria Using Abstractions

Computing market equilibria is an important practical problem for market design, for example, in fair division of items. However, computing equilibria requires large amounts of information, often the valuation of every buyer for every item, and computing power. In “Computing Large Market Equilibria Using Abstractions,” the authors study abstraction methods for ameliorating these issues. The basic abstraction idea is as follows. First, construct a coarsened abstraction of a given market, then solve for the equilibrium in the abstraction, and finally, lift the prices and allocations back to the original market. The authors show theoretical guarantees on the solution quality obtained via this approach. Then, two abstraction methods of interest for practitioners are introduced: (1) filling in unknown valuations using techniques from matrix completion and (2) reducing the problem size by aggregating groups of buyers/items into smaller numbers of representative buyers/items and solving for equilibrium in this coarsened market.

Democracy, the Market, and the Firm

This book is an attempt to resolve an enigma that has puzzled social scientists since Condorcet in the eighteenth century: Why are collective choices so stable and easy to make in practice, when in theory it should be totally otherwise? A striking illustration of this enigma is the almost unanimous support of shareholders in publicly traded companies for the motions tabled by directors. The first part of the book explores the interplay between the voting and trading mechanisms. Two main arguments are proposed: on the one hand, the better the market works, the easier it is for majority voting to achieve political stability; on the other hand, among all market equilibria, those that are politically stable are more likely to be economically efficient. The second part of the book explores the feedback from collective choices to individual preferences. It investigates the behavioral assumptions leading to an alignment of shareholders, even in a context of severe market failures, and provides an analysis of the philosophical and axiomatic underpinnings of these assumptions. In sum, and figuratively, the book argues that the invisible hand of the market and the active hand of democracy can work hand in hand to give rise to a better world. The analysis relies on formal models which are kept as simple as possible and make use only of elementary convex and vector analysis.

The competitive firm and perfect competition

This chapter addresses market equilibria for competitive firms, firms that act as price takers. It develops a theory which is based on the hypothesis that firms and consumers act as if they have no effect on prices; consumers choose what to consume and firms choose their production plans in the belief that the prices they see are unaffected by their decisions. There are two ways to proceed in the theory. One could continue analysis of general equilibrium in the style of Chapter 6, but with firms added to the story. Or one can undertake partial equilibrium analysis. The chapter begins with the classic partial equilibrium analysis of perfect competition. It then develops an example that shows how a partial equilibrium perspective can be misleading, before discussing general equilibrium with firms.

Robust Market Equilibria with Uncertain Preferences

The problem of allocating scarce items to individuals is an important practical question in market design. An increasingly popular set of mechanisms for this task uses the concept of market equilibrium: individuals report their preferences, have a budget of real or fake currency, and a set of prices for items and allocations is computed that sets demand equal to supply. An important real world issue with such mechanisms is that individual valuations are often only imperfectly known. In this paper, we show how concepts from classical market equilibrium can be extended to reflect such uncertainty. We show that in linear, divisible Fisher markets a robust market equilibrium (RME) always exists; this also holds in settings where buyers may retain unspent money. We provide theoretical analysis of the allocative properties of RME in terms of envy and regret. Though RME are hard to compute for general uncertainty sets, we consider some natural and tractable uncertainty sets which lead to well behaved formulations of the problem that can be solved via modern convex programming methods. Finally, we show that very mild uncertainty about valuations can cause RME allocations to outperform those which take estimates as having no underlying uncertainty.