Stable and Extremely Unequal - With Alfred Galichon and Marc Henry
Economics Letters Vol. 226, May 2023
We show how stability as a solution concept in matching often comes at the cost of extreme forms on inequality. Restricting our attention to aligned preferences, we show that the stable matching results from the lexicographic welfare maximization of the pairs’ welfare, starting with the best-off. We compare this solution with an alternative allocation, that although unstable, maximizes the welfare lexicographically starting with the worst-off pairs.
The choice correspondence properties of gross substitutability and irrelevance of rejected contracts are sufficient conditions for the existence of an equilibrium in a one-to-many matching setting. These properties are pervasive in the literature of decision theory, and have been formulated in many different ways by different authors. This work is aimed at reconciling all different formulations of substitutability and irrelevance of rejected contracts and relating these concepts to other properties of choice correspondences, such as rationalizability. The properties are investigated both at a primitive level of preferences and in the case in which preferences are representable by a function. In the latter case, I explore how both of these two conditions are present in matroid-based valuations and how the assumption of quasi-linearity in wages can be relaxed, while still satisfying gross substitutability and irrelevance of rejected contracts.
The aim of this paper is to quantify the long-term effects on exposure to different races and ethnicities caused by the “separate but equal” doctrine. De jure segregation applied to a range of public facilities, including transportation, hospitals, theaters and most importantly schools. The state of California led the way by being the first state to declare the "separate but equal” doctrine in the educational system unconstitutional, 7 years before it was declared unconstitutional at a federal level in 1954. Similarly, anti-miscegenation laws were outlawed at a federal level in 1967, but different states outlawed them at different times. In the state of California, laws banning interracial marriages were declared unconstitutional in 1948, while in the state of New York such laws were never in place. Through a comparison of historical data on married couples in the states of California and New York and exploiting the difference in legal decisions timelines in these two states, I am able to derive information on the structure of the social network in which those marriages took place. Through the analysis, I am able to estimate the probabilities of a connection existing between a white person and a minority group member in California and New York respectively. With these estimates, I am able to show how such probability has been impacted by the early abolition of “separate but equal” in California and how it has evolved over time.
An Allocation of Gale and Shapley to the City of New York
This paper is an exploration of multiple algorithms that can be applied in a geographical matching setting. There are two sides of the market, and each element of each side is fully described by their geographical coordinates. Each element on either side of the market has a preference ranking over possible matches. Individual preferences are dictated by a negative transformation of the distance that exists between the geographical coordinates of the individual and the geographical coordinates of their potential match. Three allocations are explored in this setting. A stable allocation, that arises from the application of a generalized version of the Gale and Shapley algorithm. An efficient allocation, which is obtained by minimizing the average distance between matched individuals. A bottleneck allocation, obtained by minimizing the maximum distance between matched individuals. The algorithms are applied to the case of students and schools in the New York City territory. Using data from the NYC Department of Education, I obtain the precise location and capacity of all public kindergartens, elementary, middle and high schools in New York City. Using Census data, I obtain the number of kids in the age range of interest in each Census tract in the city. Combining these data, I then apply a generalized version of the Gale and Shapley algorithm to explore the characteristics of the stable allocation. The stable allocation thus obtained causes inequality of treatment among students. It does indeed dictate that some students travel a short distance to go to the school they are assigned to, while others need to travel an unreasonably long one. It also has students living in the same Census tract going to different schools. This suggests that other allocations need to be explored in order to reduce inequality.
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