Условие:
A risk neutral firm provides car insurance. A typical contract consists of a fee (\beta) and a reimbursement amount (\alpha) (paid on top of (\beta)) where (\alpha) is transferred only in case of an accident. A driver who owns a car of value W = 100, can choose to be careful, which corresponds to the effort of e = 1, or careless, which corresponds to the effort of e = 0, and the effort cost is (c(e)=e). Careful (resp. careless) drivers commit an accident with probability (\pi_1=0.2) (resp. (\pi_0=1)) in which case they lose L = W. The driver has the following preferences over wealth w: (u(w)=\sqrt{w})\nIf the driver does not buy any insurance, he always chooses the effort which results in the highest expected payoff.
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Compute the outside option of the driver and enter it here:
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Suppose the insurance company can perfectly observe the effort exerted by the driver (first-best).\nEnter the value of (\alpha) if the insurance company implements ((e=1)) here: \nEnter the value of (\beta) if the insurance company implements ((e=1)) here: \nEnter the profit of the insurance company if it implements ((e=1)) here: \nEnter the value of (\alpha) if the insurance company signs a contract with the driver and implements ((e=0)) here: \nEnter the effort level – 0 or 1, – that is optimal to implement in the first-best contract:
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The insurance company no longer observes the effort exerted by the driver (second-best). \nEnter the value of (\alpha) if the insurance company implements ((e=1)) here: \nEnter the value of (\beta) if the insurance company implements ((e=1)) here: \nEnter the profit of the insurance company if it implements ((e=1)) here:
\nEnter the value of (\alpha) if the insurance company signs a contract with the driver and implements ((e=0)) here: \nEnter the effort level – 0 or 1, – that is optimal to implement in the second-best contract:
