MACROECONOMICS 2015 Problem set n°1 Prof. Nicola Dimitri The following problems will be discussed by the students in class March Wednesday 27th h 18-20 and, if necessary, also Thursday 28th h 18-20. Students will rotate on the board to discuss the problems. You are allowed, even encouraged, to work in groups. Problems 3 and 4 could be made after next week lectures. 1) Show that the expression of the adaptive expectation when prices are cyclical ππ‘π = { π∗∗ +π∗ (1−π) 2−π Is given by ππ‘π = π∗ ππ π‘ ππ ππ£ππ π∗∗ ππ π‘ ππ πππ when considering π‘ = 0 as an even index and 0 < π < 1. 2) Show that if π¦π‘∗ and π¦π‘ are two solutions of the equation π¦π‘ = ππΈ(π¦π‘+1 |πΌπ‘ ) + π then also π¦π‘ = ππ¦π‘∗ + (1 − π)π¦π‘ with 0 < π < 1 is also a solution of the equation. 3) Consider the (logarithmic version) of the Cagan demand function seen in class with rational expectations ππ‘ − ππ‘ = −πΌ(πΈ(ππ‘+1 |πΌπ‘ ) − ππ‘ ) Find the fundamental solution for ππ‘ and compare it with the adaptive expectations solution in an economy where ππ‘ evolves as follows π ππ‘ = { 0 ππ ππ π‘ ≤ π ππ π‘ > π with π π > π0 . Moreover, discuss if the following stochastic process ππ‘ , with values and probabilities given by ππ‘−1 π€ππ‘β ππππππππππ‘π¦ π ππ‘ = { π2 0 π€ππ‘β ππππππππππ‘π¦ 1 − π πΌ where π = 1+πΌ, can be a bubble process part of the model solution. 4) In the continuous time version of the Ramsey model discussed in class find the Hamiltonian, the Ramsey-Keynes optimal conditions and discuss the stationary states and phase diagram under the following hypotheses on the population growth rate a) πΜπ‘ ππ‘ = πππ‘ πΜ π b) ππ‘ = π π‘ π‘ πΜ πΜ π c) ππ‘ = πππ‘ d) b) ππ‘ = π π‘ π‘ π‘ 5) Consider the following two variables, discrete time, dynamical system π₯π‘ = 2π₯π‘−1 + 1 π¦π‘ = π₯π‘−1 + 2π¦π‘−1 + 1 Find the stationary state, the eigenvalue and eigenvectors of the transformation matrix 2 0 π΄=[ ] 1 2 and discuss the asymptotic behavior of the system. If the stationary state is not a saddle point suggest a matrix A that could have a saddle point.