Friday, May 15, 2020

Search and matching models of marriage (with emphasis on search or on matching)

Often when economic theorists (particularly matching theorists) speak of "marriage" we don't mean the real thing, we mean a metaphor embodied in a simple one-to-one model of matching. But marriage is a very real thing, and matching models of various sorts are used to study it.  Here's a survey from the Annual Review of Economics, by one of the leaders of this literature.

The Theory and Empirics of the Marriage Market by  Pierre-AndrĂ© Chiappori

"the economic analysis of the marriage will aim at answering two sets of questions: (a) Who marries whom? and (b) How are the benefits distributed between spouses? Mostly, these questions have been analyzed using either of two different frameworks: frictionless matching theory and search models. The basic distinction between the two is related to the emphasis that is put (or not) on frictions in the description of the market. In search models, frictions are paramount. Typically, each individual sequentially and randomly meets one person of the opposite gender; after such a meeting, both individuals must decide whether to settle for the current mate or continue searching. The latter option involves various costs, from discounting to the risk of never finding a better partner. If both individuals agree to engage in a relationship (which can be marriage or, in some models, cohabitation), then a negotiation begins on the way the surplus is shared.

"Matching models, on the contrary, assume a frictionless environment. In the matching process, each woman (say) is assumed to have free access to the pool of all potential men, with perfect knowledge of the characteristics of each of them—and vice versa. In other words, matching models disregard the cost of acquiring information about potential matches as well as the role of meeting technologies of all sorts (from social media to head hunters and from dating sites to pure luck).

I haven't absorbed the whole paper yet, but the introduction reminded me of the plurality of approaches to studying search and matching (not just restricted to the study of marriage). Chiappori shares the view that the big difference between search models and matching models is that matching models are "frictionless," they assume away all the search frictions that are the subject of search models. 

Search models, on the other hand, concentrate on search effort, and take the actual matching technology to be random meetings. (To my mind, that's vastly different from taking account of the actual technology of matching, which is central to the market design literature on matching.)

The 2010 Nobel prize in Economics went to Diamond, Mortensen, and Pissarides, the founders of the search and matching macro literature "for their analysis of markets with search frictions." I recall that Dale Mortensen expressed the view that one of their big contributions was to notice that matching could be regarded as a "black box," rather than being studied explicitly, i.e. that actual matching didn't need to be incorporated in the model. Here's a 2001 paper from the JEL that discusses that:

Looking into the Black Box: A Survey of the Matching Function
Barbara Petrongolo and Christopher A. Pissarides
Journal of Economic Literature Vol. 39, No. 2 (Jun., 2001), pp. 390-431

" The matching function summarizes a  trading technology between agents who  place advertisements, read newspapers and  magazines, go to employment agencies,  and mobilize local networks that eventually bring them together into productive matches. The key idea is that this  complicated exchange process is summarized by a well-behaved function that  gives the number of jobs formed at any  moment in time in terms of the number  of workers looking for jobs, the number  of firms looking for workers, and a  small number of other variables.

" The matching function is a modeling  device that occupies the same place in  the macroeconomist's tool kit as other  aggregate functions, such as the production function and the demand for money  function."
That work, which is focused on search and which treats matching as a black box, provides some big insights. For example, if search is costly but the benefits of matching are shared equally by the one who searches and the one who is found, then there can be free riding in the form of too little search, and hence unemployment by qualified job seekers at the same time that there are unfilled positions, and unsold houses at the same time as there are people who would actually like to buy them.

But models of matching via aggregate matching functions or via random matching are nowhere near the level of detail that allows the kind of micro-micro economics that market designers practice in analyzing marketplaces or in trying to design them. I'm mostly talking about markets that resemble labor markets more than marriage, but dating sites are marketplaces designed for something related to marriage, and there are some explicitly matrimonial sites. Much of what makes one dating site different from another is how they deal with congestion and other frictions.

A not-atypical market design project begins with an analysis of matching in an existing system,  focused on precisely the details of how matching occurs. A next step is that designers sometimes take responsibility for redesigning those details, or aspects of them, to reduce frictions in marketplaces e.g. for labor, schools,  kidneys, MBA courses, financial exchanges, dating sites themselves, etc.

I think there has been too little intersection between those who study search with random matching and those who study matching without explicitly modeling the costs of search. (That isn't to say that there aren't people working and making progress in that intersection.)  Possibly one barrier is a misunderstanding between these two groups of what the models are good for.

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