My paper "Efficiency in search & matching models: A generalized Hosios condition" is now published in JET.

The main result applies to any market with frictional meeting process, but it’s particularly relevant for macroeconomists who study unemployment.

authors.elsevier.com/a/1cW7w50waSOD3
The paper generalizes the well-known Hosios (1990) condition for efficient entry. This condition is important in macro because it is used to determine the efficient level of vacancy creation and thus unemployment in search-theoretic models of the labor market.
To understand the question, consider any search-and-matching model in which buyers and sellers are matched according to a frictional matching process and there is free entry on one side, e.g. buyer entry. Hosios (1990) asked: When is entry constrained efficient?
The answer provided by Hosios (1990) is remarkably simple. Entry is efficient only when buyers' share of the joint match surplus equals the elasticity of the matching function with respect to buyers. When this condition holds, markets internalize all search externalities.
The “Hosios condition” applies to a broad range of search-and-matching models. However, we find that it does not guarantee efficiency in settings where the expected match output depends on the market tightness or buyer-seller ratio. Our paper provides some examples.
In addition to the standard search externalities, we find that a novel externality arises in such environments. We call this the output externality. The output externality may be positive or negative and it is not internalized by the Hosios (1990) condition.
Our main result is that entry is constrained efficient only when buyers' surplus share equals the matching elasticity plus the surplus elasticity (i.e. the elasticity of the expected match surplus with respect to buyers). We call this the “generalized Hosios condition”.
Why is this important?

Search-theoretic models of the labor market have become the standard way of modelling unemployment in macro. The generalized Hosios condition is important because it determines the efficient level of job creation and therefore unemployment in such models.
Also, the Hosios condition is often used to *calibrate* search models of the labor market in which wages are determined by Nash bargaining. Our result is therefore important for anyone who calibrates such macro models.
For macroeconomists, the main message to take away from our paper is this. If you work with search-theoretic models of unemployment, don’t just assume that the Hosios condition guarantees constrained efficiency. You may need the generalized Hosios condition.

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