extension: .md

format_name: myst

format_version: 0.13

jupytext_version: 1.16.1

jupytext_version: 1.16.4

kernelspec:

display_name: Python 3 (ipykernel)

language: python

```{code-cell} ipython3

import matplotlib.pyplot as plt

from numba import njit, float64

from numba import jit, float64

from numba.experimental import jitclass

import numpy as np

from quantecon.optimize import brentq

$c_0$ instead of $\mu_0$ in the following code.)

```{code-cell} ipython3

@njit

@jit

def shooting(pp, c0, k0, T=10):

'''

Given the initial condition of capital k0 and an initial guess

tolerance bounds), we stop.

```{code-cell} ipython3

@njit

@jit

def bisection(pp, c0, k0, T=10, tol=1e-4, max_iter=500, k_ter=0, verbose=True):

# initial boundaries for guess c0

Let's calculate and plot the saving rate.

```{code-cell} ipython3

@njit

@jit

def saving_rate(pp, c_path, k_path):

'Given paths of c and k, computes the path of saving rate.'

production = pp.f(k_path[:-1])

A positive fixed point $C = \tilde C(K)$ exists only if $f\left(K\right)+\left(1-\delta\right)K-f^{\prime-1}\left(\frac{1}{\beta}-\left(1-\delta\right)\right)>0$

```{code-cell} ipython3

@njit

@jit

def C_tilde(K, pp):

return pp.f(K) + (1 - pp.δ) * K - pp.f_prime_inv(1 / pp.β - 1 + pp.δ)

$$ (eq:tildeK)

```{code-cell} ipython3

@njit

@jit

def K_diff(K, C, pp):

return pp.f(K) - pp.δ * K - C

@njit

@jit

def K_tilde(C, pp):

res = brentq(K_diff, 1e-6, 100, args=(C, pp))

We can compute $K_s$ by solving the equation $K_s = \tilde{K}\left(\tilde{C}\left(K_s\right)\right)$

```{code-cell} ipython3

@njit

@jit

def K_tilde_diff(K, pp):

K_out = K_tilde(C_tilde(K, pp), pp)

---

mystnb:

figure:

caption: "Stable Manifold and Phase Plane"

caption: Stable Manifold and Phase Plane

name: stable_manifold

tags: [hide-input]

---