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Question

There are two possible pollution levels, Î¸H and Î¸L where Î¸L < Î¸H < 1/2. Without any additional knowledge, the probability of the high state is 2/5. A lobbyist who observes the true state meets with the regulator. Let "x" denote the pollution limit allowed by the regulatory agency. The utility of the regulator is u = -(x - Î¸)Â² and that of the lobbyist is u = -(x - Î¸ - b)Â² where b > 0 is the "bias". (a) In the absence of any communication, what action would the regulator take? (b) Suppose that the lobbyist can choose between two messages, mH and mL. Message mH means "Î¸H is the true state" and mL means "Î¸L is the true state". What would a full information equilibrium mean in this context? (c) Describe a condition on the bias b under which the lobbyist will fully reveal the true state (a full information equilibrium). (d) Consider a situation in which the lobbyist does not observe the state, but instead can conduct a study that may (or may not) tell the regulator something about the state. The lobbyist chooses an experiment "Ï€" where Ï€(mH | Î¸L) = Ï€ is the probability that the test yields a message/signal of mH when the true state is Î¸L. Assume that Ï€(mH | Î¸H) = 1, i.e., with certainty, a high message will result when the state is Î¸H, but could also result when the state is low. Is there an equilibrium value of Ï€ that would fully reveal the true value of Î¸?

          There are two possible pollution levels, Î¸H and Î¸L where Î¸L < Î¸H < 1/2. Without any additional knowledge, the probability of the high state is 2/5. A lobbyist who observes the true state meets with the regulator. Let "x" denote the pollution limit allowed by the regulatory agency. The utility of the regulator is u = -(x - Î¸)Â² and that of the lobbyist is u = -(x - Î¸ - b)Â² where b > 0 is the "bias".
(a) In the absence of any communication, what action would the regulator take?
(b) Suppose that the lobbyist can choose between two messages, mH and mL. Message mH means "Î¸H is the true state" and mL means "Î¸L is the true state". What would a full information equilibrium mean in this context?
(c) Describe a condition on the bias b under which the lobbyist will fully reveal the true state (a full information equilibrium).
(d) Consider a situation in which the lobbyist does not observe the state, but instead can conduct a study that may (or may not) tell the regulator something about the state. The lobbyist chooses an experiment "Ï€" where Ï€(mH | Î¸L) = Ï€** is the probability that the test yields a message/signal of mH when the true state is Î¸L. Assume that Ï€(mH | Î¸H) = 1, i.e., with certainty, a high message will result when the state is Î¸H, but could also result when the state is low. Is there an equilibrium value of Ï€** that would fully reveal the true value of Î¸?

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