The therapist has a market wage of $140 per client week.

The problem that you are worried about is that the patient and therapist will cut you out. Assume that the patient is willing to pay the therapist $160 per week if you are cut out.

To make things simple, assume that after four weeks, the therapist can make a one-time offer to the patient to cut you out, and that the patient will pay $160 per week for the remaining 16 weeks of treatment. If the therapist tries to do this, assume that you catch him 30% of the time - say because the patient finds it unethical and turns the therapist in. If that happens, you fire the therapist, and he gets his market wage of $143 a week instead of whatever you promise to pay him. (You still lose the client in this case.) The other 70% of the time, if he makes this offer to the patient, the patient agrees, and you are cut out and the therapist gets paid the $160.

Assume that there is no discounting here, so that money in the future is worth the same as money today.

Also assume that there is no further interaction between the therapist, the patient and you after 20 weeks.

You have a choice here - you can either offer some sort of efficiency wage that will stop him from trying to cut you out, or you can choose not to offer such a wage, and to let her make the offer to the patient.

This is a bit like the following: you can either employ her for the long term - by offering the efficiency wage

- or you can be like a referral service that benefits one time, but lets the patient go to the therapist after four weeks. What is the optimal compensation policy here? Should you offer a compensation package to deter him from cutting you out, or are you better off allowing him to do so?