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Two neighboring homeowners, i = 1,2, simultaneously choose how many hours 1 to spend maintaining a lawn
Two neighboring homeowners, i = 1,2, simultaneously choose how many hours 1 to spend maintaining a lawn. The AVERAGE benefit per hour for i is 1 10-1; + 2 (e.g., it is 10-1, + 2 for homeowner 1) And the opportunity) cost per hour for each homeowner is 4. (a) Give each homeowner's (net) payoff as a function of l, and le. (b) Compute the Nash equilibrium.
Expert Solution
a)
Net Payoff homeowner 1 = (10 - l1 + l2/2) l1 - 4l1 = 10l1 -l21 + (l1)(l2)/2 - 4l1 = 6l1 -l21 + (l1)(l2)/2
Net Payoff homeowner 2 = (10 - l2 + l1/2) l2 - 4l2 = 10l2 -l22 + (l1)(l2)/2 - 4l2 = 6l2 -l22 + (l1)(l2)/2
b)
Differentiating net Payoff homeowner 1 with respect to l1 and equating to 0
6 -2l1 + (l2)/2 = 0
l1 = 3 + (l2)/4 (equation 1)
Differentiating net Payoff homeowner 2 with respect to l2 and equating to 0
6 -2l2 + (l1)/2 = 0
l2 = 3 + (l1)/4 (equation 2)
Solving equation 1 and 2
l1 = l2 = 4
This is the nash equilibirum
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