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. Consider the following delegation versus centralisation model of decision making loosely based on some of the discussion in class. A principal wishes to implement a decision that has to be a number between 0 and 1; that is a decision d needs to be implemented where 0 ≤ d ≤1. The difficulty for the principal is that she does not know what decision is appropriate given the current state of the economy but she would like to implement a decision that exactly equals what is required given the state of the economy. In other words if the economy is in state s (where 0 ≤ s ≤1) the principal would like to implement a decision d = s as the principal’s utility Up (or loss from the maximum possible profit) is given by P U = − s − d . With such a utility function maximising utility really means making the loss as small as possible. For simplicity the two possible levels of s are 0.4 and 0.7 and each occurs with probability 0.5. There are two division managers A and B who each have their own biases. Manager A always wants a decision of 0.4 to be implemented and incurs a disutility UA that is increasing thefurther from 0.4 the decision d that is actually implement specifically UA = − 0.4 − d . Similarly Manager B always wants a decision of 0.7 to be implement and incurs a disutility UB that is (linearly) increasing in the distance between 0.7 and the actually decision that is implemented - that is 0.7 B U = − − d . Each manager is completely informed so that each of them knows exactly what the state of the economy s is. (a) The principal can opt to centralise the decision but before making her decision – given she does not know what the state of the economy is – she asks for recommendations from her two division managers. Centralisation means that the principal commits to implement a decision that is the average of the two recommendations she received from her managers. The recommendations are sent simultaneously and cannot be less than 0 or greater than 1. Assume that the state of the economy s = 0.7. What is the report (or recommendation) that Manager A will send if Manager B always truthfully reports s? (b) Again the principal is going to centralise the decision and will ask for a recommendation from both managers as in the previous question. Now however assume that both managers strategically make their recommendations. What are the recommendations rA and rB made by the Managers A and B respectively in a Nash equilibrium? (c) What is the principal’s expected utility (or loss) under centralised decision making (as in part b)? (d) Can you design a contract for both of the managers that can help the principal implement their preferred option? Why might this contract be problematic in the real world?
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