We construct an equilibrium random matching model of the labour market, with endogenous market participation and a general matching technology that allows for market size effects: the job-finding rate for workers and the incentives for participation change with the level of unemployment. In comparison to standard models with constant returns to scale in matching, agent behaviour is more complex - the model generates plausible joint dynamics of employment, unemployment and participation with heterogeneity in search behaviour for workers with different degrees of attachment to the labour market. Techniques are developed to reduce the dimensionality of the problem to establish local and global stability; a complicating factor is the possibility of multiple equilibria, welfare-ranked by market size. A Hosios-type condition internalises search externalities.
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