Question

10. A manager is responsible for n machines. In any given period, the probability that exactly k machines will break down is Tk> 0 (πΟΤ π1+π2+ π,-1), where k is any whole number between 0 and n: k=0, 1, 2, , n. The Tk are "true" probabilities but are known only to the manager. The manager must report these probabilities to the control authority. In the absence of any re ward or penalty, he might report some wrong probabilities Po, Pi, ..., P (Po+P+P-1) The central authority devises the following scheme: if k machines break down, and the reported probability is P, the authority will pay the manager A +M In Pk dollars, where M and A are positive constants. The manager's expected reward is therefore The manager wants to choose Po,..., P, to maximize (1), subject to the con- straints that (a) Find the first-order conditions (ignore nonnegativity restrictions) (b) Are the second-order conditions satisfied? (c) Show that it is optimal for the manager to report the truth, i.e., Pi- TK-

Answer 1

The managers problem is

$max\: ER = \sum_{k=0}^{n}\pi_k(A&plus;MlnP_k)\\ subject\:to \sum_{k=0}^{n}P_k = 1.\\ The\: Lagrangian\: is\\ L = \sum_{k=0}^{n}\pi_k(A&plus;MlnP_k) - \lambda(\sum_{k=0}^{n}P_k - 1).\\ a) The\: FOC's\: are: \frac{\partial L}{\partial P_k}=0\Rightarrow \frac{\pi_kM}{P_k}-\lambda=0\Rightarrow P_k = \pi_kM/\lambda....(1).\\ b) The\:Second\:Order\:Condition's\: are: \frac{\partial^2 L}{\partial P_k^2} = -\frac{\pi_kM}{P_k^2} < 0.\\ c) putting\: equation\: (1)\:in\:constraint, \:we\:get\\ \sum_{k=0}^{n}P_k = 1\Rightarrow \sum_{k=0}^{n}\pi_kM/\lambda = 1 \Rightarrow M/\lambda\sum_{k=0}^{n}\pi_k = 1 \Rightarrow M/\lambda =1(\because \sum_{k=0}^{n}\pi_k=1).\\ From\: (1)\:,we\:get\: P_k = \pi_k.\\ \text{Thus, it is optimal for the manager to report the truth}$