Two individuals, Andy and Beth, work on a joint project. If Andy chooses effort level x ≥ 0 and Beth chooses effort level y ≥ 0, then the total output equals 10x+10y+2xy. Total output is split 50-50 between the two players, so Andy and Beth each receives 5x + 5y + xy. Andy pays an effort cost equal to x^2 when exerting effort x; Beth pays effort cost y^2 when exerting effort y. Therefore, Andy’s total payoff equals 5x + 5y + xy − x^2 and Beth’s total payoff equals 5x + 5y + xy − y^2.
a) Maximize each player’s payoff function to find his or her best response function.
b) Find the Nash equilibrium.
c) Determine players’ payoffs in the equilibrium.
d) Is the Nash equilibrium efficient? If not, provide a Pareto improvement.