Equilibrium in Cow Ranching: Optimizing Benefits for Ranchers
What is the quantity of cows that each rancher raises in a Nash Equilibrium?
To determine the quantity of cows that each rancher raises in a Nash Equilibrium, we need to find the point where neither rancher has an incentive to unilaterally deviate from their chosen strategy.
In this case, the benefit for each rancher is given by (100 - 5G)G - 10G, where G is the total quantity of cows using the grassland.
Let's analyze the possible scenarios:
Rancher 1 raises 6 cows: The benefit for Rancher 1 is (100 - 5(6))(6) - 10(6) = 120. The benefit for Rancher 2 is (100 - 5(6))(6) - 10(6) = 120.
Rancher 1 raises 8 cows: The benefit for Rancher 1 is (100 - 5(8))(8) - 10(8) = 80. The benefit for Rancher 2 is (100 - 5(8))(8) - 10(8) = 80.
Rancher 1 raises 9 cows: The benefit for Rancher 1 is (100 - 5(9))(9) - 10(9) = 45. The benefit for Rancher 2 is (100 - 5(9))(9) - 10(9) = 45.
Answer:
The quantity of cows that each rancher raises in a Nash Equilibrium is 9 cows.
To determine the Nash Equilibrium in cow ranching, we need to consider the benefits for each rancher at different quantities of cows raised. The Nash Equilibrium is reached when neither rancher has an incentive to change their strategy unilaterally.
In this scenario, by analyzing the benefits for both ranchers at different quantities of cows raised, we can see that if Rancher 1 chooses to raise 6 or 8 cows, Rancher 2 would have an incentive to deviate and raise more cows to increase their benefit. However, if Rancher 1 chooses to raise 9 cows, Rancher 2 does not have an incentive to deviate as their benefit is maximized.
Therefore, the Nash Equilibrium in this situation is for each rancher to raise 9 cows, optimizing the benefits for both ranchers involved in cow ranching.