Two owners i = 1; 2 of a stand on the New Haven farmersímarket sell apples. The e§ort that they put into marketing the apples is ei . They can choose any e§ort between 0 and 1. The revenue that they make is an increasing function of both ownersí e§ort: R(e1; e2) = 2 (e1 + e2). Each owner receives one half of this revenue. For each owner i the cost of e§ort ei are Ci(ei) = 1 2 (ei) 2 . Thus, owner iís net utility is: ui(e1; e2) = (e1 + e2) 1 2 (ei) 2 : 2 (a) For each owner i write down the Örst order condition for the optimal choice of ei given the other ownerís choice ej . Show that the second derivative of utility with respect to ei is negative. (b) Solve for the symmetric Nash equilibrium of the game. Denote the common equilibrium e§ort level by e . Substitute e1 = e2 = e into the Örst order condition and solve for e . (c) By contrast, suppose the two owners were to enter into a cooperative agreement and were to seek to maximize the sum of their net utility, i.e. they were to maximize max e1;e2 2 (e1 + e2) 1 2 e 2 1 1 2 e 2 2 : Find the optimal solution of this problem, denote it by e = (e 1 ; e 2 ). How does it compare to e = (e 1 ; e 2 ). (d) The comparison above is an instance of the ìtragedy of the commonsî. Brieáy explain why. Reading As
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