Problem 6 Suppose that firms make production decisions based on their expectations of the mon- etary policy that the central bank will choose. We will consider a game where the firm makes its decision before the central bank actually sets its policy, having observed the firm's decision. For simplicity, we restrict the central bank's policies to choosing the level of inflation (n) and represent the firm's actions by the expectation it forms about inflation (as). The firm would prefer to make production decisions based on a more accurate prediction. So the closer we is to 7:, the higher the firm's payoff: urn, are) = (n are)? Output (y) in this economy is given by y = 0.75y" + (Ti 119) where y" is the efficient level of output. This expression for output shows that the market is not competitive and that surprise inflation positively affects output. The central bank's payoff is Ublm \"9) = -C2 - (Y-y')2. where c > 0 represents the rate at which the bank trades off between two competing objectives: keeping inflation low (the first term) and getting output to reach the efficient level (the second term). (a) As explained above, the firm selects me before the central bank selects it. Remember that the central bank observes me before choosing . What is the subgame perfect equilibrium of this game? (b) What would the subgame perfect equilibrium be if the central bank moved first? (c) Suppose this game (where the firm moves first) is repeated twice over two periods. What would the subgame perfect equilibrium be? Represent by m, and me the choices in the first period and by 72 and 72 the choices in the second period. (d) Now suppose that the central bank may either be "hawkish" or "dovish" on inflation. This corresponds to different values of c, the weight it places on the inflation term of its objective. If the bank is hawkish c = 4. If the bank is dovish c = 1. While the bank knows whether it is hawkish or dovish, the firm does not. It only knows that each (hawk or dove) is possible with probability 2. The game unfolds as follows: 1. Nature decides whether the bank is dovish (probability ?) or hawkish (probability _). 2. The firm selects my . 3. The central bank, having observed my, selects 71. 4. The firm observes only 7 (importantly, it does not directly observe whether the bank is dovish or hawkish) and chooses 72. 5. The central bank, having observed 72, selects 72. To simplify things, suppose for the rest of this problem that y* = 4. You will now solve for a separating equilibrium of this game. To fully describe a perfect Bayesian equilibrium, we must state the strategies as well as beliefs. In a separating equilib rium, the two types ("hawk" and "dove") of the bank act differently in the first period. Denote by my(74) and m (74) the actions chosen by the bank when it is a hawk or dove respec- tively (remember that these are functions of 7). Denote by q(7 ) the probability that the firm attaches to the bank being dovish after it observes m, in the first period. Since, in a sepa- rating equilibrium m (74) # 17(74) along the equilibrium path (that is, when the firm plays its equilibrium strategy of 7-), the firm must believe that the bank is dovish with certainty upon observing It? (74) and hawkish with certainty upon observing , (7;): so q(? ((74))) = 1 and 9(7((714)) = 0. (i) What action should the central bank take in the second period (72)? This will depend on its own type as well as the action taken by the firm in the second period. (ii) As noted above, upon observing m (nf) or mr (f), the firm knows exactly the type of bank it is dealing with. What is it's optimal action in the second period when it observes (ii) In a separating equilibrium, the two types of the bank will play different actions mi (7;) and 7-(74). What does the firm do in the first period? (iv) The separating equilibrium is typically not unique. In particular, there are usually many beliefs that support the same actions: Bayes' rule only applies on the equilibrium path and requires q(7 (74)) = 1 and q(17 (74)) = 0 but does not tell us what q(7 ) should be for any my # 1 (nf), my (14). Since we are free to choose beliefs off the equilibrium path, we will set q(71 ) = 1 for any my