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The paper studies a matching model with on-the-job search, transferable utility and heterogeneous agents. Matched agents can set the conditions under which a given match can be dissolved. It is shown that matched agents use sign-off fees to extract all capital gains from trade when a third agent is contacted. In equilibrium, this redistributes wealth towards less able individuals, reduces the likelihood that any given match will be rejected and, given the conditions, it yields efficiency. Although externalities arise when a match is formed and when turnover occurs, the decentralized outcome is efficient when the production function is sub-modular and the difference in abilities is big enough. The results obtained may provide theoretical support for the type of contracts used in some markets, such as sports markets.

Consider first the welfare distribution that results from the use of sign-on and sign-off fees. The following arguments are easily demonstrable through numerical analyses, but only heuristic arguments are included. Where Equilibrium A obtains, single h types are able to induce turnover in all existing matches other than hh matches (that is, both in hl and ll matches). The problem for them is that they enjoy no capital gain from inducing those turnovers, consequence of the sign-on fee. On the contrary, single l agents never induce the separation of a match, and therefore never see any capital gain extracted from them. Instead, they extract capital gains from single hs when hl matches allow hh turnovers. Hence, when the dynamics of match formation and turnover are as in Equilibrium A, the use of the fees decreases the welfare of single h types and increases the welfare of single l types. Consider now Equilibrium B. Now both single hs and single ls can induce turnovers (in hh matches and ll matches respectively). On that account, both suffer from extraction of capital gains when they pay the corresponding sign-on fee. Nevertheless, because xh > xl (which implies f(xh, xh) > f(xh, xl) > f(xl, xl)), the capital gain extracted from single hs (by ls) is higher than the gain extracted from single ls (by hs). Again, the use of the fees decreases the welfare of single hs and increases that of single ls. Notice that the agreement whether to allow a given turnover and the level of the corresponding sign-off fee is in fact used as a pre-commitment strategy: it binds the matched agents when a single agent is met. It is this pre-commitment which allows them to extract all capital gains from a turnover. As long as the agreement can be reached before any contact is made, pre-commitment is possible and the previous analysis remains valid. It follows that the analysis is not valid when there is continuous bargaining, as in Pissarides (1994). In such a situation, any matched agent who contacts a single agent through on-the-job search would be free to decide with whom to match for the next instant; his/her outside option is the payoff of being single, and no sign-off fees are due if the agent decides to change partners. Finally, notice that the “rule” implied by the model under which a given turnover will occur in equilibrium is really quite simple and intuitive: Because the capital losses and gains of the three agents involved in a turnover are considered, a turnover occurs only if it creates a positive surplus. The fact that the capital gains and losses of all agents involved are relevant is a direct consequence of the optimal level of sign-off fees, because all they do is extract all gains from trade from single agents. This simple rule will be used in the next section which addresses the use of a more general matching technology, whereby matched agents can contact matched agents.