Question: Two herdsmen have to decide how many sheep each of them can let graze on a common field. The field has a limited dimension, so just few sheep can graze on it. Each herdsman wants to maximize his own utility function:

Note that each herdsman's utility function depends positively on the quantity of his own sheep admitted to graze, but negatively on the total quantity of sheep grazing, since by doing so they consume a common good, i.e. the field.

1. Imagine that the herdsmen maximize their own utility separately and simultaneously (i.e. taking the other herdsman's quantity of sheep as given). What are the reaction functions for each herdsman?
2. Compute the optimal quantity of sheep admitted to graze for both herdsmen.
3. Find each herdsman's equilibrium utility.
4. Now suppose that a benevolent social planner prescribes to each herdsman how many sheep to graze on the field. His purpose is social welfare maximization (where social welfare is simply U = U1 + U2), under a ‘fairness constraint’ that q1 = q2 in equilibrium. What are the new equilibrium quantities?
5. What is the new level of utility for each herdsman?
6. Compare the optimal quantities and utilities in points 2-3 with the ones in points 4-5. What is the economic intuition behind this difference?

Price: $2.99

Solution: The downloadable solution consists of 3 pages
Deliverable: Word Document