Lectures on equilibrium in markets for indivisible goods at U. Tokyo by Teytelboym, Baldwin and Jagadeesan, Sept 21-24

 Towards a general theory of markets with indivisible goods: special lectures at The University of Tokyo Market Design Center (UTMD)

September 21-24, 2021 (Japan time) 

Organizers:

Fuhito Kojma, Director, UTokyo Market Design Center, and Professor, the University of Tokyo

Michihiro Kandori: Vice Director, UTokyo Market Design Center, and University Professor, the University of Tokyo

Yuichiro Kamada: Associate Professor, UC Berkeley, and Global Fellow, the University of Tokyo

Venue: Zoom online  Language: English

Program

*All times shown below are Japan time.

*Each lecture will be followed by 30 minutes Q&A session.

Lecture 1 9/21 (Tue) 16:00-17:30

Introduction to Markets for Indivisible Goods (by Alexander Teytelboym)

In many settings, such as auctions, the indivisibility of goods is a key market feature. But in markets with indivisible goods, competitive equilibria might not exist.  We explore conditions, such as substitutability of goods, that ensure existence of competitive equilibria. We also discuss connections between conditions for existence, tâtonnement, and cooperative properties of equilibria.

Lecture 2 9/22 (Wed) 16:00-17:30

The geometry of preferences: demand types, equilibrium with Indivisibilities, and bidding languages (by Elizabeth Baldwin)

An equivalence theorem between geometric structures and utility functions allows new methods for understanding preferences. Our classification of valuations into “Demand Types”, incorporates existing definitions regarding the comparative statics of demand (substitutes, complements, “strong substitutes”, etc.) and permits new ones. Our Unimodularity Theorem generalises previous results about when competitive equilibrium exists for any set of agents whose valuations are all of a “demand type”. Contrary to popular belief, equilibrium is guaranteed for more classes of purely-complements, than of purely-substitutes, preferences. Our Intersection Count Theorem checks equilibrium existence for combinations of agents with specific valuations by counting the intersection points of geometric objects. Applications include the “Product-Mix Auction” introduced by the Bank of England in response to the financial crisis. In that context, we show that all substitutes preferences can be represented, and no other preferences can be represented, by appropriate sets of permitted bids in the Substitutes Product-Mix Auction language; an analogous result holds for strong substitutes, when we refine the characteristics of the language. These languages thus also provide new characterizations of (all) substitutes, and of strong substitutes, respectively.

Lecture 3 9/23 (Thu) 16:00-17:30 The Equilibrium Existence Duality (by Alexander Teytelboym)

We show that, with indivisible goods, the existence of competitive equilibrium fundamentally depends on agents’ substitution effects, not their income effects. Our Equilibrium Existence Duality allows us to transport results on the existence of competitive equilibrium from settings with transferable utility to settings with income effects. One consequence is that net substitutability—which is a strictly weaker condition than gross substitutability—is sufficient for the existence of competitive equilibrium. Further applications give new existence results beyond the case of (net) substitutes. Our results have implications for auction design.

Lecture 4 9/24 (Fri) 09:30-11:00 Matching and Prices (by Ravi Jagadeesan)

Indivisibilities and budget constraints are pervasive features of many matching markets. But gross substitutability — a standard condition on preferences in matching models — typically fails in such markets. To accommodate budget constraints and other income effects, we instead assume that agents’ preferences satisfy net substitutability. Although competitive equilibria do not generally exist in our setting, we show that stable outcomes always exist and are efficient. We illustrate how the flexibility of prices is critical for our results. We also discuss how budget constraints and other income effects affect the properties of standard auction and matching procedures, as well as of the set of stable outcomes.

  Recorded lecture will be posted on UTMD’s YouTube channel within 6 hours.