Macroeconomics, ECON6023
Instructor: Serhiy Stepanchuk
Homework 1
Due Monday, 23 November 2020 (by 11:59 PM)
Weights: Q1=10%, Q2=90%
1. Consider the following matrix:
P =
0.95 0.05 0.000.15 0.75 0.10
0.10 0.00 0.90

(a) Can this matrix be used as a transition matrix of a Markov chain?
Please explain.
(b) Suppose that pi′0 = [0.1, 0.8, 0.1]. Compute pi
′
1.
(c) Can you prove that the Markov chain with a transition matrix P
has a unique stationary distribution and that the process is asymp-
totically stationary?
(d) Compute the stationary distribution by iterating on the transition
matrix.
(e) Compute the stationary distribution directly, as a fixed point of
pi′ = pi′P . Is your answer here the same as in part (d)? Is this to be
expected?
2. Consider the stochastic version of the consumption-savings problem.
The dynamic optimization problem that the consumer solves has the
following sequential formulation:
max
{ct(zt),at+1(zt)}∞t=0
E0
∞∑
t=0
βt
ct(z
t)1−γ
1− γ
s.t.:
∀zt

ct(z
t) + at+1(z
t) ≤ at(zt)(1 + r) + wt(zt)
ct(z
t), at+1(z
t) ≥ 0
log(wt) = ρ log(wt−1) + εt, εt ∼ N(0, σ2)
where u(c) = c
1−γ
1−γ is the per-period utility function, r is the interest rate
on savings (which we assume to be constant), and wt is the wage income
which we assume is random and follows an AR(1) process.
1
(a) Provide the recursive formulation of the above dynamic optimiza-
tion problem. What is/are the state variable(s)?
(b) First, assume that γ = 1 so that the per-period utility function
becomes u(c) = log(c). Furthermore, assume that wt = 0, which
means that the consumer does not have any wage income, and
the problem becomes deterministic. Solve this problem using the
method of undetermined coefficients (Hint: guess that the value
function has the following form: V (a) = b + d log(a), where b and d
are the unknown coefficients).
(c) Next, assume that γ = 2, β = 0.95, r = 0.02, ρ = 0.8 and σ = 0.12.
Solve the dynamic optimization problem numerically, discretizing
the state space and using the ‘value function iterations’ method.
Some suggestions:
– Use the Tauchen method to approximate wt with a Markov
chain with n = 5.
– Discretize the state space for a using Ad = [0, A¯]. Try A¯ = 40.
Use Na = 300 equally spaced grid points over Ad. Make sure
that you get a′ = g(ai, w) < ai for all ai ∈ Ad and all w during
each iteration of the algorithm. Otherwise, increase A¯.
(d) Plot your solutions for the value function and policy functions for c
and a′.
(e) What is the value of E(a) predicted by the model?
Some suggestions:
– Generate a random draw of the realizations of w of length
100000.
– Use the generated path for wt and the policy function for a
′ to
generate the path for at+1. You can use any a0 ∈ Ad. I would
suggest choosing a0 close to E(w), but you can experiment and
see how robust your results are with respect to your choice of
a0.
– Discard the first 10% of the generated at+1. Use the remaining
observations to compute E(a).
(f) How does the predicted E(a) change under the following scenarios:
(1) β increases from 0.95 to 0.965,
(2) β remains equal to 0.95, but the interest increases to r = 0.03,
(2) β remains equal to 0.95, r remains equal to 0.02, but the stan-
dard deviation of the wage shocks increases to σ = 0.18,
Can you provide intuition for your results in each of the three cases
above?
2

欢迎咨询51作业君