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Question 4 (Contract, 20%, 40min). Consider the standard rst-best agency problem: max I v[x 505)] f (x, a)dx a.s(-) s.t. fu[s(x)]f(x,a)dx 2 do). where a is effort, 6(a) is the cost of effort, x, a) is the density function of output x, $06) is the payment to the agent, and v and u are respectively the payoff functions of the principal and the agent. We restrict admissible contracts to be of the following linear form: 3(x) = a: + x, where a e R and {3 > 0 are constants. For the optimal solution (a*,s*), assume that a larger at improves the principal's welfarezf v[x s*(x)] f (x, a*)dx > 0. Derive the three equations that jointly determine the optimal solution (a*, .1", 5*).1