using Distributions, Plots, LaTeXStrings, LinearAlgebra, Statistics, Random, QuantEcon, NLsolveAssignment 4
Student Name/Number:
Instructions
- Edit the above cell to include your name and student number.
- Submit just this
ipynbto Canvas. Do not rename, it associates your student number with the submission automatically.
Question 1
Take the example in Simple Example with iid Income where the stochastic process for income \(y_t\) follows
\[ y_t = \mu + \sigma w_t \sim N(\mu, \sigma^2) \]
Under these assumptions, with quadratic utility and if \(\beta R = 1\) the optimal policy is
\[ \mathbb{E}_t\{c_{t+1}\} = c_t \]
Which we can use to derive for this income process if \(F_0 = y_0 = 0\).
\[ c_t = (1-\beta)\left(\mathbb{E}_t\left[\sum_{j=0}^\infty \beta^j y_{t+j}\right] + F_t\right) = \mu + (1 - \beta) \sigma \sum_{j=1}^t w_j \]
And finally
\[ F_t = \sigma \sum_{j=1}^{t-1} w_j \]
Part (a)
Code to simulate this consumption process for \(t = 1, \ldots T\) is given below.
using Plots, Random
function simulate_iid_income(p, T; w = randn(T))
w_sum = cumsum(w) #(w_1, w_1 + w_2, w_1 + w_2 + w_3, ... sum_{j=1}^T w_j))
c = p.mu .+ (1 - p.beta) * p.sigma * w_sum # (c_1, c_2, ... c_T)
y = p.mu .+ p.sigma * w # (y_1, y_2, ... y_T)
F = [0.0; p.sigma * w_sum[1:end-1]] #(F_1, F_2, ... F_T)
return (;w, F, c, y)
end
p = (;beta= 1.0 / (1.0 + 0.05), mu = 1.0, sigma = 0.15)
T = 60
res = simulate_iid_income(p, T)
plot(1:T, res.y, color = :green, label = L"non-financial income, $y_t$", xlabel="Time")
plot!(res.c, color = :black, label = L"consumption, $c_t$")
plot!(res.F, color = :blue, label = L"Assets, $F_t$")
hline!([p.mu], color = :black, linestyle = :dash, label = L"mean income, $\mu$")
Using this code, plot call the above where instead of randomly drawing each \(w \sim N(0,1)\) draw it from a distribution \(w \sim N(0.5, 1)\) and plot the results
# edit your code herePart (b)
Interpret the result of the previous graph with the standard one with well-specified results from the perspective of consumption smoothing and forecasting of income. In what ways is the behavior in the second case not optimal and why?
Answer:
(double click to edit your answer)
Question 2
Continuing with income process defined in Part 1, rather than manually implementing this process, use the LSS formulation
\[ \begin{aligned} \begin{bmatrix} w_{t+1} \\ 1 \end{bmatrix} &= \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} w_t \\ 1 \end{bmatrix}+ \begin{bmatrix} 1 \\ 0 \end{bmatrix} w_{t+1}\\ y_t &= \begin{bmatrix} \sigma & \mu \end{bmatrix}\begin{bmatrix} w_t \\ 1 \end{bmatrix} \end{aligned} \]
From this, we can stack, etc. following our notes
The simulation from the code was
A = [0 0; 0 1]
G = [p.sigma p.mu]
C = [1; 0]
H = G*inv(I-p.beta*A)
A_tilde = [A zeros(2,1); H*(I-A) 1]
C_tilde = [C; 0]
G_tilde = [G 0; (1-p.beta)*H 1-p.beta]
x_tilde_0 = [0.0, 1, 0.0] #[w_0, 1, F_0]
lss_pi = LSS(A_tilde, C_tilde, G_tilde;
mu_0 = x_tilde_0)
x, y = simulate(lss_pi, T)
plot(1:T, y[1,:];label=L"y_t", size=(600,400))
plot!(1:T, y[2,:], label=L"c_t")
hline!([p.mu], color=:black, linestyle=:dash,
label = L"mean income, $\mu$")
Part (a)
Try calculating the stationary distribution by modifying the code below or trying with your existing code.
# modify here from previous results
# mu_x, mu_y, sigma_x, sigma_y = stationary_distributions(lss)Explain why it does or does not converge below
Answer:
(double click to edit your answer)
Part (b)
Consider a variation on the state space model above where instead of the state being above which puts the \(\sigma\) in the
\[ C \equiv \begin{bmatrix} \sigma \\ 0 \end{bmatrix} \]
rather than in the \(G\). Complete that state space model, and change the implementation below (copied from the above for simplicity)
# Modify here
A = [0 0; 0 1]
G = [p.sigma p.mu]
C = [1; 0] # replace with $C = [\sigma; 0]$ and fix the rest of the LSS/etc.
H = G*inv(I-p.beta*A)
A_tilde = [A zeros(2,1); H*(I-A) 1]
C_tilde = [C; 0]
G_tilde = [G 0; (1-p.beta)*H 1-p.beta]
x_tilde_0 = [0.0, 1, 0.0] #[w_0, 1, F_0]
lss_pi = LSS(A_tilde, C_tilde, G_tilde;
mu_0 = x_tilde_0)
x, y = simulate(lss_pi, T)
plot(1:T, y[1,:];label=L"y_t", size=(600,400))
plot!(1:T, y[2,:], label=L"c_t")
hline!([p.mu], color=:black, linestyle=:dash,
label = L"mean income, $\mu$")