Simulate in Dynare the response of output, inflation, and interest rates assuming Att = BEtwit+1 + kat = Ext+1 - - [it - Ent+1] it = QTTit + Qxxt Importantly, you must make some assumptions about how your representative agents form expectations. Assume o = 1, B = 0.99, w = 0.5, and k = (1-w)(1-wB) a) Rational expectations is a widely used assumption in macroeconomics. Why do you think macroe- conomists prefer to make this assumption than assume individuals form non-rational expectations? Show how this economy would evolve under rational expectations, ie. where Etat+1 = at+1 and Ett+1 = Tt+1 in response to a persistent demand shock and a persistent inflation shock. (This should produce two sets of simulations. ) Explain precisely what is happening in terms of household, firm, and central bank behavior to generate the reactions to the shocks. b) Now assume that households and firms form naive expectations, ie. Etat+1 = Xt-1 and Ett+1 = Tit-1, and simulate the same two types of shocks as in Question 4a. Describe how the dynamics of your econ- omy change and compare it to part a)'s economy under rational expectations. Explain intuitively why the macroeconomic dynamics change when expectations are naive. How does the standard deviation of output and inflation change under the cases of a demand shock and an inflation shock? c) Now assume that agents form trend-chasing expectations, ie. Etat+1 = at-1 + 0.5(at-1 - Xt-2) and Ett+1 = Wt-1 + 0.5(t-1 - "t-2) . Again, simulate the same two types of shocks as in Question 4a. How do these economies differ from your rational and naive expectation cases? Again, intuitively explain why trend-chasing expectations generates the results you obtain