Restrictions for different functional forms of the matching function

We provide bounds on the parameters of matching functions such that the job-finding rate and the vacancy-filling rate are below 1. We do that in the context of the canonical search and matching model with a Pissarides-type free-entry condition. We find that the restrictions for a Cobb-Douglas matching function with increasing returns to scale are rather restrictive, involving an upper bound to future expected profits and the number of job searchers. In contrast, for functional forms with constant returns to scale (Cobb-Douglas, CES) the restrictions involve only parameters or an upper bound to the future expected profits. The paper also investigates when a job-finding rate (vacancy filling rate) below one can restrict the vacancy filling rate (job-finding rate) to be below and strictly bounded away from one. We provide the bounds implied by these "interior equilibria."