Contracts versus salaries in matching: A general result

Cadet-Branch Matching in a Kelso-Crawford Economy

“…As the Echenique (2012) embedding preserves the law of aggregate demand (see Section IIE in Echenique 2012), the law of aggregate demand is necessary (but not in general sufficient) for the existence of a quasi-linear utility representation in matching with salaries. Schlegel (2015) showed that a matching market where branches' choice functions are unilaterally substitutable can be embedded (in a weaker sense than Echenique 2012) into a (potentially non-quasi-linear) matching market with salaries in which firms may be indifferent to paying a worker more. 28 This result applies in particular to the Sönmez (2013) cadet-branch market.…”

“…Proposition 2 and Theorem 3 offer salaries a more realistic interpretation than Echenique (2012) and Schlegel (2015) because the branches' utility functions are taken to be quasi-linear. Thus, the substitutable BfYC choice functions are the choices of profit-maximizing firms.…”

“…3 Substitutability plays a key role in interpreting contracts as salaries (Echenique 2012. Schlegel (2015) interpreted contracts as salaries under weakened substitutability conditions, but without maintaining the natural monotonicity properties of preferences with respect to salaries. 4 Switzer (2011) pursued a similar approach, but using responsive priorities.…”

“…The isomorphism result proved in this paper sharpens general results that relate contracts and salaries. Echenique (2012) showed that any many-to-one matching market (in the sense of Kelso andCrawford 1982 andMilgrom 2005) can be embedded into a potentially non-quasi-linear matching market with salaries in which workers are grossly substitutable, and Schlegel (2015) extended the embedding result to the case when branches' priorities are unilaterally substitutable (in the sense of Hatfield and Kojima 2010) by allowing branches' priorities to be only weakly monotone in salary. This paper proves an isomorphism instead of merely an embedding and requires branches' priorities to be quasi-linear and strictly monotone in salary, so that the branches' priorities can be taken to be the choices of profit-maximizing firms that regard workers as (gross) substitutes.…”

Matching with slot-specific priorities: Theory

“…Our other results do not follow from prior work; they depend on structure present in our specific model that also leads to a direct proof of the stability result (see the discussion at the end of Section 2.2.1). 8 Thus, in particular, our model falls outside of the domains that Echenique (2012) and Schlegel (2015) have shown can be handled with only the Kelso and Crawford (1982) matching with salaries framework (see also . 9 The converse result is not true, in general: there may be stable outcomes that are not associated to slot-stable outcomes.…”