Must-Take Cards: Merchant Discounts and Avoided Costs

Antitrust authorities often argue that merchants cannot reasonably turn down payment cards and therefore must accept excessively high merchant discounts. The paper attempts to shed light on this “must-take cards” view from two angles. First, the paper gives some operational content to the notion of “must-take card” through the “avoided-cost test” or “tourist test”: would the merchant want to refuse a card payment when a non-repeat customer with enough cash in her pocket is about to pay at the cash register? It analyzes its relevance as an indicator of excessive interchange fees. Second, it identifies four key sources of potential social biases in the payment card systems’ determination of interchange fees: internalization by merchants of a fraction of cardholder surplus, issuers’ per-transaction markup, merchant heterogeneity, and extent of cardholder multi-homing. It compares the industry and social optima both in the short term (fixed number of issuers) and the long term (in which issuer offerings and entry respond to profitability)

A note on CR mappings of positive codimension

We prove the following Artin type approximation theorem for smooth CR mappings: given M a connected real-analytic CR submanifold in C^N that is minimal at some point, M' a real-analytic subset of C^N', and H:M->M' a smooth CR mapping, there exists a dense open subset O in M such that for any q in O and any positive integer k there exists a germ at q of a real-analytic CR mapping H^k:(M,q)->M' whose k-jet at q agrees with that of H up to order k

On formal maps between generic submanifolds in complex space

Let H:(M,p)->(M',p') be a formal mapping between two germs of real-analytic generic submanifolds in C^N with nonvanishing Jacobian. Assuming M to be minimal at p and M' holomorphically nondegenerate at p', we prove the convergence of the mapping H. As a consequence, we obtain a new convergence result for arbitrary formal maps between real-analytic hypersurfaces when the target does not contain any holomorphic curve. In the case when both M and M' are hypersurfaces, we also prove the convergence of the associated reflection function when M is assumed to be only minimal. This allows us to derive a new Artin type approximation theorem for formal maps of generic full rank