Changes

:<math>\frac{\partial v}{\partial b} \ge \frac{1}{2} \cdot \left(\frac{\partial v}{\partial b} + \frac{\partial v_b}{\partial b} \right )\;</math>

However, there is an important distinction: The payoffs in the property rights model are not only a function of the joint output but also the outside option. In the team's problem, a monitor can break the budget - here the market breaks the budget. It is implicitly assumed that the agents can observe each other's outside options (this may be costly and a friction). If this is the actually the case then unless <math>v_b\;</math> and <math>v_s\;</math> are collinear with with <math>v\;</math>, they are additional measures as discussed above.

:<math>\frac{\partial s_b}{\partial b} = \frac{\partial v }{\partial b } \quad \mbox {and} \quad \frac{\partial s_s v_s }{\partial s} = \frac{\partial v}{\partial s}\;</math>