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import numpy as np

Estimating income processes

Consider $N$ households indexed by $i$ who are in the labor market for $T$ periods indexed by $t \in \{1,2,\dots,T\}$ .

Their wage income is stochastic and given by

\begin{aligned} P_{i,t} &= \psi_{i,t} P_{i,t-1},\\ \tilde{Y}_{i,t} &= \xi_{i,t} P_{i,t},\\ Y_{i,t} &= \begin{cases} 0 & \text{if } \mu_{i,t} < \pi\\ \tilde{Y}_{i,t} & \text{else} \end{cases} \\ \psi_{i,t} &\sim \text{LogNormal}(-0.5\sigma_{\psi}^2,\sigma_{\psi})\\ \xi_{i,t} &\sim \text{LogNormal}(-0.5\sigma_{\xi}^2,\sigma_{\xi})\\ \mu_{i,t} &\sim \text{Uniform}(0,1)\\ P_{0} &= 1 \end{aligned}

where

$\sigma_{\psi}$ is the standard deviation of the permanent shocks, $\psi_{i,t}$
$\sigma_{\xi}$ is the standard deviation of the transitory shocks, $\xi_{i,t}$
$\pi$ is the risk of unemployment.

The data you have access to is:

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dataY = np.load('dataY.npy')
T,N = dataY.shape

Question 1: Calculate income growth rates as log-changes

\Delta \log(Y_{i,t}) \equiv \begin{cases} \log(Y_{i,t})-\log(Y_{i,t-1}) & \text{if } Y_{i,t}>0 \text{ and } Y_{i,t-1}>0\\ \text{NaN} & \text{else} \end{cases}

where $\text{NaN}$ is not-a-number (i.e. np.nan).

Question 2: Calculate the following 3 statistics from the data

$s_1^{data}$ : Share of observations with $Y_{i,t} = 0$
$s_2^{data}$ : Variance of income growth rate, $\text{Var}(\Delta\log{Y_{i,t}})$
$s_3^{data}$ : Co-variance of income growth rates one period apart, $\text{Cov}(\Delta\log{Y_{i,t}},\Delta\log{Y_{i,t-1}})$

Question 3: Simulate the income process using your own choice of $\sigma_{\psi}$ , $\sigma_{\xi}$ , $\pi$ , $T$ and $N$ . Calculate the 3 same statistics. Compare with the data statistics.

$s_1^{sim}(\sigma_{\psi},\sigma_{\xi},\pi)$ : Share of observations with $Y_{i,t} = 0$
$s_2^{sim}(\sigma_{\psi},\sigma_{\xi},\pi)$ : Variance of income growth $\text{Var}(\Delta\log{Y_{i,t}})$
$s_3^{sim}(\sigma_{\psi},\sigma_{\xi},\pi)$ : Co-variance of income growth one periods apart $\text{Cov}(\Delta\log{Y_{i,t}},\Delta\log{Y_{i,t-1}})$

Question 4: Solve the following minimization problem to estimate $\sigma_{\psi}$ , $\sigma_{\xi}$ and $\pi$

\sigma_{\psi}^{\ast},\sigma_{\xi}^{\ast},\pi^{\ast} = \arg \min_{\sigma_{\psi}\geq0,\sigma_{\xi}\geq0,\pi\in[0,1]} (s_1^{sim}-s_1^{data})^2 + (s_2^{sim}-s_2^{data})^2 + (s_3^{sim}-s_3^{data})^2

where for each new guess of $\sigma_{\psi}$ , $\sigma_{\xi}$ , and $\pi$ you should be re-simulating the data with the same seed and re-calculating the 3 statistics.

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def objective():
    pass

# res = optimize.minimize(objective,x,method='L-BFGS-B',bounds=?,args=?,options={'eps':1e-4})
# hint: options={'eps':1e-4} uses a larger step-size when approximating the jacbian, which is useful in this case

Wealth in the utility function

In the final period, $t=T$ , the household solves the following problem

\begin{aligned} v_{T}(a_{T-1}) & = \max_{c_{T}} \frac{c_T^{1-\rho}}{1-\rho} + \kappa \frac{(a_T+\underline{a})^{1-\sigma}}{1-\sigma} \\ & \text{s.t.} & \\ a_{T}& = (1+r)a_{T-1} + y - c_T \end{aligned}

where

$a_t$ is end-of-period assets in period $t$
$c_t$ is consumption in period $t$
$\rho$ is the CRRA-coefficient for consumption utility
$\sigma$ is the CRRA-coefficient for wealth utility
$\underline{a}$ is an additive scaling factor for wealth utility
$\kappa$ is a multiplicative scaling factor for wealth utility
$r$ is the rate of return
$y$ is income

The optimal consumption function is denoted $c_t^{\ast}(a_{t-1})$ .

The optimal savings function is denoted $a_t^{\ast}(a_{t-1}) \equiv (1+r)a_{t-1} + y - c_t^{\ast}(a_{t-1})$ .

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# a. parameters
rho = 2.0
sigma = 1.2
kappa = 0.6
a_ubar = 2.0
r = 0.04
y = 1.0

# b. grids
a_lag_vec = np.linspace(0,300,300)

Question 1: Find and plot the functions $v_{T}(a_{T-1})$ , $c_T^{\ast}(a_{T-1})$ , and $a_T^{\ast}(a_{T-1})$ .

In all periods before the last, $t < T$ , the household solves:

\begin{aligned} v_{t}(a_{t-1}) & = \max_{c_{t}} \frac{c_t^{1-\rho}}{1-\rho} + \kappa \frac{(a_t+\underline{a})^{1-\sigma}}{1-\sigma} + \beta v_{t+1}(a_t)\\ & \text{s.t.} & \\ a_{t}& = (1+r)a_{t-1} + y - c_t \end{aligned}

where $\beta$ is the discount factor for future utility.

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beta = 0.97
T = 20

Question 2: Find and plot $v_{T-1}(a_{T-2})$ and $c_{T-1}^{\ast}(a_{T-2})$ .

Question 3: Find $c_t^{\ast}(a_{t-1})$ for $t \in \{0,1,\dots,T\}$ and plot them in a single figure.

Define the saving rate as:

s_t^{\ast}(a_{t-1}) \equiv \frac{a_t - a_{t-1}}{y+ra_{t-1}} = \frac{((1+r)a_{t-1} + y - c_t^{\ast}(a_{t-1})) - a_{T-1}}{y+ra_{t-1}}

Question 4: Plot $s_0^{\ast}(a_{-1})$ . Do the rich or the poor save the most?

Question 5: Can you change the parameter choices such that $s_0^{\ast}(a_{-1})$ is monotonically decreasing in $a_{-1}$ ?

Refined grid search

Let $\boldsymbol{x} = \left[\begin{array}{c} x_1 \\ x_2\\ \end{array}\right]$ be a two-dimensional vector.

Consider the following algorithm:

Algorithm: grid_search()

Goal: Minimize the function $f(\boldsymbol{x})$ .

Choose a grid size $N$ and minimum and maximum values of $x_1$ and $x_2$ denoted $\overline{x}_1 > \underline{x}_1$ and $\overline{x}_2 > \underline{x}_2$
Calculate step-sizes
$\begin{aligned} \Delta_1 &= (\overline{x}_1 - \underline{x}_1)/(N-1)\\ \Delta_2 &= (\overline{x}_2 - \underline{x}_2)/(N-1) \end{aligned}$
Find the grid point with the lowest function value by solving
$j_1^{\ast},j_2^{\ast} = \arg \min_{j_1\in\{0,...N-1\},j_2\in\{0,...N-1\}} f(\underline{x}_1 + j_1\Delta_1, \underline{x}_2 + j_2\Delta_2)$

Return $x_1^{\ast} = \underline{x}_1 + j_1^{\ast}\Delta_1$ , $x_2^{\ast} = \underline{x}_2 + j_2^{\ast}\Delta_2$ and $f^{\ast} = f(x_1^{\ast},x_2^{\ast})$ .

Question 1: Implement the grid_search() algorithm to minimize the rosen function.

Hint: The global minimum of the rosen function is $0$ at $(1,1)$ .

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def rosen(x):
    return (1.0-x[0])**2+2*(x[1]-x[0]**2)**2

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def grid_search(f,x1_min,x1_max,x2_min,x2_max,N):
    return np.nan,np.nan,np.nan

# settings
x1_min = 0
x1_max = 5
x2_min = 0
x2_max = 4
N = 100

# apply grid search
x1,x2,f = grid_search(rosen,x1_min,x1_max,x2_min,x2_max,N)
print(f'minimum found at ({x1:.8f},{x2:.8f}) with the function value {f:.8f}')

minimum found at (nan,nan) with the function value nan

Also, consider the following algorithm:

Algorithm: refined_grid_search()

Goal: Minimize the function $f(\boldsymbol{x})$ .

Choose a grid size $N$ and minimum and maximum values of $x_1$ and $x_2$ denoted $\overline{x}_1 > \underline{x}_1$ and $\overline{x}_2 > \underline{x}_2$ , and a refinement-level $K$
Set $k=0$
If $k > 0$ : Update the minimum and maximum values by
$\begin{aligned} \tilde{\Delta}_1 &= 3(\overline{x}_1 - \underline{x}_1)/(N-1)\\ \tilde{\Delta}_2 &= 3(\overline{x}_2 - \underline{x}_2)/(N-1)\\ \underline{x}_1 &= \max(\underline{x}_1,x_1^{\ast}-\tilde{\Delta}_1)\\ \underline{x}_2 &= \max(\underline{x}_2,x_2^{\ast}-\tilde{\Delta}_2)\\ \overline{x}_1 &= \min(\overline{x}_1,x_1^{\ast}+\tilde{\Delta}_1)\\ \overline{x}_2 &= \min(\overline{x}_2,x_2^{\ast}+\tilde{\Delta}_2) \end{aligned}$
Apply the grid_search() algorithm returning $x_1^{\ast}$ , $x_2^{\ast}$ and $f^\ast$
Increment $k$ by one
If $k < K$ return to step 3 else continue
Return $x_1^{\ast}$ , $x_2^{\ast}$ and $f^{\ast}$

Question 2: Implement the refined_grid_search() algorithm to minimize the rosen function.

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def refined_grid_search(f,x1_min,x1_max,x2_min,x2_max,N,K):
    return np.nan,np.nan,np.nan

# more settings
K = 10

# apply refined grid search
x1,x2,f = refined_grid_search(rosen,x1_min,x1_max,x2_min,x2_max,N,K)
print(f'minimum found at ({x1:.8f},{x2:.8f}) with the function value {f:.8f}')

minimum found at (nan,nan) with the function value nan