Full Substitutability

Understanding Preferences: “Demand Types”, and the Existence of Equilibrium With Indivisibilities

“…18 A unimodular matrix is an integer matrix with integer inverse (i.e., with determinant ±1). Specific cases of this observation have been made before (see, e.g., Sun and Yang (2006), and Hatfield et al (2013)); we lay out the general behavior here. 19 The demand type's vectors are {e i e j e i − e i e i + e j e j − e j : i i ∈ {1 k} j j ∈ {k + 1 n}}.…”

“…By contrast, our Theorem 4.3 both demonstrates the applicability of the result, and clarifies the connections to existing economic results. We will see in Section 6 that our Theorem 4.3 generalizes many results in well-known work subsequent to Danilov et al's, including results in Sun and Yang (2006), Milgrom and Strulovici (2009), Hatfield et al (2013), and Teytelboym (2014). 26 Danilov et al also proved no necessity result.…”

“…It may be less obvious that Hatfield et al's (2013) model of networks of trading agents, each of whom can both buy and sell, both fits into our framework, and is also closely related to Kelso and Crawford's model. To show this, we again treat each transfer of a product from a specified seller to a specified buyer as a distinct good, so each agent again has preferences over a subset of {−1 0 1} n , where n is the number of distinct goods.…”

“…Furthermore, although Hatfield et al described goods to be sold as complements of goods to be bought, this is because they measured both buying and selling as nonnegative quantities. So, since in our framework selling is just “negative buying,” the “complementarities” disappear: the condition they impose is exactly ordinary substitutes (see Hatfield et al (, Theorem A.1)). Just as for Kelso and Crawford's model, the only demand complex edges of such a domain that are vectors of the ordinary substitutes demand type are also vectors of the strong substitutes demand type.…”

“…Our concept of demand types also shows how this condition can be applied. For example, equilibrium-existence results such as those in Sun and Yang (2006), Strulovici (2009), Hatfield, Kominers, Nichifor, Ostrovsky, andWestkamp (2013), and Teytelboym (2014), are easy special cases of the Unimodularity Theorem, but none of these papers present their results as specializations of Danilov et al's earlier work, since the latter's relevance was unclear. (Analyzing preferences in price space-as well as, like Danilov et al, in quantity space-also allows us to develop additional implications.…”