Exercise 2 - Two neighbors are taking care of a common road leading to their villas. Each of them exert an effort el > 0, i = 1,2 The resulting quality of the road is
f(e1,e2) = a1e1 + a2e2 - e1e2,
where al, a2 > 0 are constants such that 2a1 - a2, 2a2 - a1 >= 0
Exerting effort is costly. More precisely, each neighbor has a quadratic cost of effort:
ci (ei) = e2i, i = 1, 2
The payoff of neighbor i, Lli, is equal to the quality of the road minus his cost of effort.
(a) Suppose the neighbors choose their effort levels simultaneously and independently. Derive the best response functions. Find the pure strategy Nash equilibrium of this game.
For the rest of the questions, assume that a1 = a2 = 1.
(b) Calculate the payoffs of the neighbors in the Nash equilibrium.
(c) Find the aggregate effort level e = e1 + e2 that maximizes the sum of the neighbors payoffs. Calculate the corresponding payoffs of the neighbors, assuming that they contribute equally (el = e2).
(d). Suppose the interaction between the neighbors studied in (a) is repeated over an infinite time horizon f = 1,2,...,∞. Assume that the neighbors discount future payoffs with the discount factor δ < (0,1) and that they maximize the sum of discounted payoffs. Suggest strategies in this infinitely repeated game that yields average discounted payoffs equal to the neighbors' payoffs obtained in
(e). Find the minimal discount factor such that the strategies constitute a subgame perfect Nash equilibrium.