Competitive search with ex-post opportunism

We consider a frictional market where an element of the terms of trade (price or quantity) is posted ex-ante (before the matching process) while the other is determined ex-post. By doing so, sellers can exploit their local monopoly power by adjusting prices or quantities once the local demand is realized. We find that when sellers can adjust quantities ex-post, there exists a unique symmetric equilibrium where an increase in the buyer-seller ratio leads to higher quantities and prices. When buyers instead can choose quantities ex-post, a higher buyer-seller ratio leads to higher prices but lower traded quantities. These equilibrium allocations are generically constrained inefficient in both intensive and extensive margins. When sellers post ex-ante quantities and adjust prices ex-post, a symmetric equilibrium exists where buyers obtain no surplus from trade. This equilibrium allocation is not constrained efficient either. If buyers choose prices ex-post, there is no trade in equilibrium when entry is costly.