Method of Digraphs for Multi-dimensional Screening
We study a general model of multi-dimensional screening for discrete types of consumers without the single-crossing condition or any other essential restrictions. Such generality motivates us to introduce graph theory into optimization by treating each combination of active constraints as a digraph. Our relaxation of the constraints (a slackness parameter) excludes bunching and cycles among the constraints. Then, the only possible solution structures arerivers, which are acyclic rooted digraphs, and the Lagrange multipliers can be used to characterize the solutions. Relying on these propositions, we propose and justify an optimization algorithm. In the experiments, its branch-and-bound version with a good starting plan shows fewer iterations than a complete search among all rivers.
A model of monopolistic competition for a closed economy, consisting of the traditional sector and the industrial production sector, which includes multi-product companies, is built. The consumption sector consists of heterogeneous consumers with a two-level Cobb-Douglas utility function and a generalized CES utility function built into it. The generalized CES utility function, unlike the standard one, includes, besides the love of variety, also the love of product quality, which makes it possible to distinguish consumers with a different attitude to product quality. The industrial sector consists of firms producing a variety of differentiated products of different quality, oriented to a specific type of customer. In the work, the demand functions of heterogeneous consumers for goods of different quality are obtained, the state of short-term equilibrium is considered for the case when firms have market power in choosing the quality of products. Various strategies of firms with respect to the selection of the optimal combination “price-quality” of a product that provides them with the maximum profit were analyzed, and conditions were obtained for screening in conditions of incomplete information on the type of consumers.
We model how student choices to rush a fraternity, and fraternity admission choices, interact with signals firms receive about student productivities to determine labormarket outcomes. The fraternity and students value wages and fraternity socializing values. We provide sufficient conditions under which, in equilibrium, most members have intermediate abilities: weak students apply, but are rejected unless they have high socializing values, while most able students do not apply to avoid taint from association with weaker members. We show this equilibrium reconciles the ability distribution of fraternity members at the University of Illinois, and estimate the fraternity’s welfare impact on different students.
A monopolist can price-discriminate between two consumer groups with linear demands that can cross (violate the Spence-Mirrlees condition). We derive complete parametric taxonomy of the outcomes. Switching from simple uniform pricing to two-part tariff or package pricing «generally» decreases the monopolistic deadweight loss. Switching from two-part tariff to packages also «generally» decreases the loss. However, we specify a small parameters' region where both these conclusions fail. Additionally, we find parameters yielding discrimination benefits to «big» or to «small» consumers. Thereby, we show the demand-specific and pricing-scheme specific reasons for/against public restrictions on price discrimination.