ECON6023 Macroeconomics SEMESTER 1 TAKE-HOME FINAL ASSESSMENT 2020-21
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ECON6023W1
SEMESTER 1 TAKE-HOME FINAL ASSESSMENT 2020-21
ECON6023 Macroeconomics
Section A
A1 (35 points) Consider the following version of the stochastic neo- classical growth model where the consumer preferences exhibit con-
sumption habits (i.e. current utility depends not only on current consumption, but also on past consumption):
o
max E0 βt ╱log(ct (zt )) + γ log(ct一1 (zt一1 ))、,
{ct(zt ),kt+1(zt )} t=0
β e (0, 1), γ > 0,
s.t.: ct (z ) + ktt+1(z ) = ztt kt (z )t α, α e (0, 1),
log(zt ) = λ0 + λ1 log(zt一1 ) + εt , εt ~ ←(0, σ) and k0 and c一1 are given.
(a) Formulate this problem in a recursive form. What are the state variable(s)? [5]
(b) Use the guess-and-verify method to solve the Bellman equation and derive the associated optimal decision rules for capital ac- cumulation and consumption [Hint: Guess a function that is logarithmic in every state variable]. [20]
(c) Express the solution for k/ and z/ as the VAR. Find the implulse response function to a unit increase in ε for the first 3 periods (assume that k0 and z0 are equal to their steady state values). [10]
A2 (15 points) Consider the Markov chain with probability transition
matrix P = ┐ , initial distribution π0 = ┌ 0.5, 0.5┐ and the
set of possible states E = {e1 , e2 } = {1, 5}.
(a) Compute the probability of the following histories of state real- izations:
● 1, 5, 1, 5, 1
● 1, 1, 1, 1, 1
● 5, 5, 5, 5, 5
(b) Find the stationary distribution for this Markov chain.
[7.5]
[7.5]
Section B
Consider an economy inhabited by a representative household with the [50] following preferences:
o
βt π(st ) ┌u ╱c(st )、+ v ╱l(st )、┐ .
t=0 st
π(st ) is the probability of stochastic and exogenous events st = {s1 , s1 , ..., st }, given s0 . Functions u and v are strictly increasing, twice continuously differentiable, strictly concave, and satisfy the Inada conditions. All other variables have the usual meaning.
In each period the household is endowed with one unit of time to be devoted to leisure l and labour n, both larger or equal than zero and such that
l(s ) + n(s ) = 1tt . (1)
The household is also endowed with the initial capital stock k0 . The technology is
c(st ) + i(st ) = A(st )F ╱k(st一1 ), n(st )、,
k(s ) = (1t 一 δ)k(st一1 ) + i(s )t ,
where F has the neoclassical properties.
(1) List the Inada conditions on preferences and explain their role.
(2) State the planning problem.
(3) Solve the planning problem.
(4) Decentralize this economy through time 0 trading following Arrow- Debreu.
(5) Show whether the decentralization above is efficient.
2022-01-22