Problem set n° 1

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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
𝑝𝑑𝑒 = {
𝑝∗∗ +𝑝∗ (1−πœ†)
2−πœ†
Is given by 𝑝𝑑𝑒 =
𝑝∗ 𝑖𝑓 𝑑 𝑖𝑠 𝑒𝑣𝑒𝑛
𝑝∗∗ 𝑖𝑓 𝑑 𝑖𝑠 π‘œπ‘‘π‘‘
when considering 𝑑 = 0 as an even index and 0 < πœ† < 1.
2) Show that if 𝑦𝑑∗ and 𝑦𝑑 are two solutions of the equation
𝑦𝑑 = π‘ŽπΈ(𝑦𝑑+1 |𝐼𝑑 ) + 𝑏
then also 𝑦𝑑 = πœƒπ‘¦π‘‘∗ + (1 − πœƒ)𝑦𝑑 with 0 < πœƒ < 1 is also a solution of the equation.
3) Consider the (logarithmic version) of the Cagan demand function seen in class with rational
expectations
π‘šπ‘‘ − 𝑝𝑑 = −𝛼(𝐸(𝑝𝑑+1 |𝐼𝑑 ) − 𝑝𝑑 )
Find the fundamental solution for 𝑝𝑑 and compare it with the adaptive expectations solution in
an economy where π‘šπ‘‘ evolves as follows
π‘š
π‘šπ‘‘ = { 0
π‘šπ‘‡
𝑖𝑓 𝑑 ≤ 𝑇
𝑖𝑓 𝑑 > 𝑇
with π‘š 𝑇 > π‘š0 . Moreover, discuss if the following stochastic process 𝑏𝑑 , with values and
probabilities given by
𝑏𝑑−1
π‘€π‘–π‘‘β„Ž π‘π‘Ÿπ‘œπ‘π‘Žπ‘π‘–π‘™π‘–π‘‘π‘¦ π‘Ž
𝑏𝑑 = { π‘Ž2
0 π‘€π‘–π‘‘β„Ž π‘π‘Ÿπ‘œπ‘π‘Žπ‘π‘–π‘™π‘–π‘‘π‘¦ 1 − π‘Ž
𝛼
where π‘Ž = 1+𝛼, 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)
𝑁̇𝑑
𝑁𝑑
= 𝑛𝑐𝑑
𝑁̇
𝑛
b) 𝑁𝑑 = 𝑐
𝑑
𝑑
𝑁̇
𝑁̇
𝑛
c) 𝑁𝑑 = π‘›π‘˜π‘‘ d) b) 𝑁𝑑 = π‘˜
𝑑
𝑑
𝑑
5) Consider the following two variables, discrete time, dynamical system
π‘₯𝑑 = 2π‘₯𝑑−1 + 1
𝑦𝑑 = π‘₯𝑑−1 + 2𝑦𝑑−1 + 1
Find the stationary state, the eigenvalue and eigenvectors of the transformation matrix
2 0
𝐴=[
]
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.
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