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3 ignoring the non negativity constraints we can

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3. Ignoring the non-negativity constraints, we can solve the system of two equations given by the two optimal response functions and obtain an interior solution as we did in class. Doing that we obtain: g h = - 25 g p = 125 2 Note that this cannot be an equilibrium since Hannah’s donations are negative which violates the non-negativity constraint. Hence the equilibrium can only be one of the two special cases considered in problem b). First, consider the corner solution when g h = 0 and g p = 50. Note that g h (50) = - 50 3 < 0 Hence Hannah has no incentives to deviate from donating nothing. So this is an equilibrium. Finally consider the corner solution when g p = 0 and g h = 50 / 3. Note that g p (50 / 3) = 1 2 (100 - 50 / 3) > 0 Philip has an incentive to deviate. Hence this not an equilibrium. In summary, there is a unique equilibrium of the model where Hannah donates nothing and Philip donates 50. 3
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4. The equilibrium is not efficient. This is obvious because only person is donating in equilibrium. By assumption, Philip ignores the benefits that Hannah reaps from his generosity. Hence the equilibrium cannot satisfy the Samuelson condition. Hannah is free-riding on Philip’s generosity.
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