Advanced Quantitative Economics with Python

Permanent Income Model using the DLE Class

# 22. Permanent Income Model using the DLE Class¶

This lecture is part of a suite of lectures that use the quantecon DLE class to instantiate models within the [HS13] class of models described in detail in Recursive Models of Dynamic Linear Economies.

In addition to what’s included in Anaconda, this lecture uses the quantecon library.

```
!pip install --upgrade quantecon
```

```
Requirement already up-to-date: quantecon in /usr/share/miniconda3/envs/quantecon/lib/python3.8/site-packages (0.5.1)
Requirement already satisfied, skipping upgrade: sympy in /usr/share/miniconda3/envs/quantecon/lib/python3.8/site-packages (from quantecon) (1.6.2)
Requirement already satisfied, skipping upgrade: numba>=0.38 in /usr/share/miniconda3/envs/quantecon/lib/python3.8/site-packages (from quantecon) (0.51.2)
Requirement already satisfied, skipping upgrade: numpy in /usr/share/miniconda3/envs/quantecon/lib/python3.8/site-packages (from quantecon) (1.19.2)
Requirement already satisfied, skipping upgrade: scipy>=1.0.0 in /usr/share/miniconda3/envs/quantecon/lib/python3.8/site-packages (from quantecon) (1.5.2)
Requirement already satisfied, skipping upgrade: requests in /usr/share/miniconda3/envs/quantecon/lib/python3.8/site-packages (from quantecon) (2.24.0)
```

```
Requirement already satisfied, skipping upgrade: mpmath>=0.19 in /usr/share/miniconda3/envs/quantecon/lib/python3.8/site-packages (from sympy->quantecon) (1.1.0)
Requirement already satisfied, skipping upgrade: llvmlite<0.35,>=0.34.0.dev0 in /usr/share/miniconda3/envs/quantecon/lib/python3.8/site-packages (from numba>=0.38->quantecon) (0.34.0)
Requirement already satisfied, skipping upgrade: setuptools in /usr/share/miniconda3/envs/quantecon/lib/python3.8/site-packages (from numba>=0.38->quantecon) (50.3.1.post20201107)
Requirement already satisfied, skipping upgrade: chardet<4,>=3.0.2 in /usr/share/miniconda3/envs/quantecon/lib/python3.8/site-packages (from requests->quantecon) (3.0.4)
Requirement already satisfied, skipping upgrade: idna<3,>=2.5 in /usr/share/miniconda3/envs/quantecon/lib/python3.8/site-packages (from requests->quantecon) (2.10)
Requirement already satisfied, skipping upgrade: certifi>=2017.4.17 in /usr/share/miniconda3/envs/quantecon/lib/python3.8/site-packages (from requests->quantecon) (2020.6.20)
Requirement already satisfied, skipping upgrade: urllib3!=1.25.0,!=1.25.1,<1.26,>=1.21.1 in /usr/share/miniconda3/envs/quantecon/lib/python3.8/site-packages (from requests->quantecon) (1.25.11)
```

This lecture adds a third solution method for the linear-quadratic-Gaussian permanent income model with \(\beta R = 1\), complementing the other two solution methods described in Optimal Savings I: The Permanent Income Model and Optimal Savings II: LQ Techniques and this Jupyter notebook.

The additional solution method uses the **DLE** class.

In this way, we map the permanent income model into the framework of Hansen & Sargent (2013) “Recursive Models of Dynamic Linear Economies” [HS13].

We’ll also require the following imports

```
import quantecon as qe
import numpy as np
import scipy.linalg as la
import matplotlib.pyplot as plt
%matplotlib inline
from quantecon import DLE
np.set_printoptions(suppress=True, precision=4)
```

## 22.1. The Permanent Income Model¶

The LQ permanent income model is an example of a **savings problem**.

A consumer has preferences over consumption streams that are ordered by the utility functional

where \(E_t\) is the mathematical expectation conditioned on the consumer’s time \(t\) information, \(c_t\) is time \(t\) consumption, \(u(c)\) is a strictly concave one-period utility function, and \(\beta \in (0,1)\) is a discount factor.

The LQ model gets its name partly from assuming that the utility function \(u\) is quadratic:

where \(\gamma>0\) is a bliss level of consumption.

The consumer maximizes the utility functional (22.1) by choosing a consumption, borrowing plan \(\{c_t, b_{t+1}\}_{t=0}^\infty\) subject to the sequence of budget constraints

where \(y_t\) is an exogenous stationary endowment process, \(R\) is a constant gross risk-free interest rate, \(b_t\) is one-period risk-free debt maturing at \(t\), and \(b_0\) is a given initial condition.

We shall assume that \(R^{-1} = \beta\).

Equation (22.2) is linear.

We use another set of linear equations to model the endowment process.

In particular, we assume that the endowment process has the state-space representation

where \(w_{t+1}\) is an IID process with mean zero and identity contemporaneous covariance matrix, \(A_{22}\) is a stable matrix, its eigenvalues being strictly below unity in modulus, and \(U_y\) is a selection vector that identifies \(y\) with a particular linear combination of the \(z_t\).

We impose the following condition on the consumption, borrowing plan:

This condition suffices to rule out Ponzi schemes.

(We impose this condition to rule out a borrow-more-and-more plan that would allow the household to enjoy bliss consumption forever)

The state vector confronting the household at \(t\) is

where \(b_t\) is its one-period debt falling due at the beginning of period \(t\) and \(z_t\) contains all variables useful for forecasting its future endowment.

We assume that \(\{y_t\}\) follows a second order univariate autoregressive process:

### 22.1.1. Solution with the DLE Class¶

One way of solving this model is to map the problem into the framework outlined in Section 4.8 of [HS13] by setting up our technology, information and preference matrices as follows:

**Technology:**
\(\phi_c= \left[ {\begin{array}{c} 1 \\ 0 \end{array} } \right]\)
,
\(\phi_g= \left[ {\begin{array}{c} 0 \\ 1 \end{array} } \right]\)
,
\(\phi_i= \left[ {\begin{array}{c} -1 \\ -0.00001 \end{array} } \right]\),
\(\Gamma= \left[ {\begin{array}{c} -1 \\ 0 \end{array} } \right]\),
\(\Delta_k = 0\), \(\Theta_k = R\).

**Information:**
\(A_{22} = \left[ {\begin{array}{ccc} 1 & 0 & 0 \\ \alpha & \rho_1 & \rho_2 \\ 0 & 1 & 0 \end{array} } \right]\),
\(C_{2} = \left[ {\begin{array}{c} 0 \\ \sigma \\ 0 \end{array} } \right]\),
\(U_b = \left[ {\begin{array}{ccc} \gamma & 0 & 0 \end{array} } \right]\),
\(U_d = \left[ {\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} } \right]\).

**Preferences:** \(\Lambda = 0\), \(\Pi = 1\),
\(\Delta_h = 0\), \(\Theta_h = 0\).

We set parameters

\(\alpha = 10, \beta = 0.95, \rho_1 = 0.9, \rho_2 = 0, \sigma = 1\)

(The value of \(\gamma\) does not affect the optimal decision rule)

The chosen matrices mean that the household’s technology is:

Combining the first two of these gives the budget constraint of the permanent income model, where \(k_t = b_{t+1}\).

The third equation is a very small penalty on debt-accumulation to rule out Ponzi schemes.

We set up this instance of the DLE class below:

```
α, β, ρ_1, ρ_2, σ = 10, 0.95, 0.9, 0, 1
γ = np.array([[-1], [0]])
ϕ_c = np.array([[1], [0]])
ϕ_g = np.array([[0], [1]])
ϕ_1 = 1e-5
ϕ_i = np.array([[-1], [-ϕ_1]])
δ_k = np.array([[0]])
θ_k = np.array([[1 / β]])
β = np.array([[β]])
l_λ = np.array([[0]])
π_h = np.array([[1]])
δ_h = np.array([[0]])
θ_h = np.array([[0]])
a22 = np.array([[1, 0, 0],
[α, ρ_1, ρ_2],
[0, 1, 0]])
c2 = np.array([[0], [σ], [0]])
ud = np.array([[0, 1, 0],
[0, 0, 0]])
ub = np.array([[100, 0, 0]])
x0 = np.array([[0], [0], [1], [0], [0]])
info1 = (a22, c2, ub, ud)
tech1 = (ϕ_c, ϕ_g, ϕ_i, γ, δ_k, θ_k)
pref1 = (β, l_λ, π_h, δ_h, θ_h)
econ1 = DLE(info1, tech1, pref1)
```

To check the solution of this model with that from the **LQ** problem,
we select the \(S_c\) matrix from the DLE class.

The solution to the DLE economy has:

```
econ1.Sc
```

```
array([[ 0. , -0.05 , 65.5172, 0.3448, 0. ]])
```

The state vector in the DLE class is:

where \(k_{t-1}\) = \(b_{t}\) is set up to be \(b_t\) in the permanent income model.

The state vector in the LQ problem is \(\begin{bmatrix} z_t \\ b_t \end{bmatrix}\).

Consequently, the relevant elements of econ1.Sc are the same as in \(-F\) occur when we apply other approaches to the same model in the lecture Optimal Savings II: LQ Techniques and this Jupyter notebook.

The plot below quickly replicates the first two figures of that lecture and that notebook to confirm that the solutions are the same

```
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4))
for i in range(25):
econ1.compute_sequence(x0, ts_length=150)
ax1.plot(econ1.c[0], c='g')
ax1.plot(econ1.d[0], c='b')
ax1.plot(econ1.c[0], label='Consumption', c='g')
ax1.plot(econ1.d[0], label='Income', c='b')
ax1.legend()
for i in range(25):
econ1.compute_sequence(x0, ts_length=150)
ax2.plot(econ1.k[0], color='r')
ax2.plot(econ1.k[0], label='Debt', c='r')
ax2.legend()
plt.show()
```