Eco504, Part II Spring 2004 C. SimsEXERCISE ON EXPECTATIONS AND ADVICEThis is a practice exercise related to the last lecture. It need not be handed it. It is a little morecomplicated than a likely question on a 3-hour final, but it may help you understand the issuesraised in the last lecture.Note that the questions on the last three years’ final exams for this course are also good sourcesof practice exercises, though OG questions are overrepresented on those exams relative to thecontent of this year’s course.We consider the overworked natural rate Phillips curve model:ut= ¯u−θ(πt− Et−1πt) . (1)The central bank has the loss functionE"∞∑t=0βt(u2t+π2t)#, (2)which of course it wants to minimize.(a) Suppose the central bank always has either aπt= 1 policy or aπt= 0 policy. The condi-tional probability of sticking with the same policy next period, given policy this period,is always .9, so policy choices are persistent. With this specification it can be shownthat P[πt+s= 1 |πt= 1] = .5(1 + .8s). Using symmetry and the fact that probabilitiessum to one, we can derive the following table showing the probabilities of various pairsπt+s,πt+s+1conditional on the two possible values ofπt.πtπt+sπt+s+1prob1 1 1 .45(1+ .8s)1 1 0 .05(1+ .8s)1 0 0 .45(1− .8s)1 0 1 .05(1− .8s)0 0 0 .45(1+ .8s)0 0 1 .05(1+ .8s)0 1 1 .45(1− .8s)0 1 1 .05(1− .8s)A member of the bank’s board has the opportunity at t to cast the deciding vote todetermineπt. She has no illusions that this will change the stochastic process governingfuture choics ofπt. Both she and the public understand that the stochastic process willremain the same after her vote. Nonetheless her vote will affect the conditional expectationof the loss function. Calculate the expected loss conditional on each value ofπtas afunction ofπt−1andθ.c°2004 by Christopher A. Sims. This document may be reproduced for educational and research purposes, so long asthe copies contain this notice and are retained for personal use or distributed free.EXERCISE ON EXPECTATIONS AND ADVICE 2(b) Continuing under the assumptions of (a), determine whether there is any value ofθsuchthat if this board member were choosingπevery period with this loss function, the actualstochastic process forπwould match that specified in (a).(c) Still continuing with the (a) assumptions, suppose that there is a stochastic political processby which our decision maker with loss function (2) alternates her possession of the decidingvote with another decision maker with a different loss function. Can you specify a lossfunction for the other decision maker and a stochastic process for the power switches thatwould make the process specified in (a) the truth?(d) Suppose that the decision maker in (a) assumes that the public always expects at time tthat theπt−1level of inflation will persist, and she herself believes that her deciding votewill permanently determine the level of inflation from t onwwards. How does the mappingfromπt−1andθto current action change?(e) Compare your analysis here to the conclusions that emerge from a “no-commitment” or“full-commitment” assumption as in the lecture notes on policy