#!/usr/bin/env python
# coding: utf-8

# # Modelling with RBC
# 

# ## Introduction
# 
# Here, we'll see an example of macro modelling: a real business cycle model. This example is due to [Chad Fulton](http://www.chadfulton.com/).
# 

# In[ ]:


import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from sympy import *

# Set max rows displayed for readability
pd.set_option("display.max_rows", 6)
# Plot settings
plt.style.use(
    "https://github.com/aeturrell/coding-for-economists/raw/main/plot_style.txt"
)
# Set seed for random numbers
seed_for_prng = 78557
prng = np.random.default_rng(
    seed_for_prng
)  # prng=probabilistic random number generator


# ## Model specification
# 
# 

# $$
# \max \mathbb{E}_0 \sum_{t=0}^\infty \beta^t u(c_t, l_t)
# $$
# 
# the budget constraint: yt=ct+it
# the capital accumulation equation: kt+1=(1−δ)kt+it
# 1=lt+nt
# 
# where households have the following production technology:
# 
# $$
# y_t = z_t f(k_t, n_t)
# $$
# and where the (log of the) technology process follows an AR(1) process:
# 
# $$
# \log z_t = \rho \log z_{t-1} + \varepsilon_t, \qquad \varepsilon_t \sim N(0, \sigma^2)
# $$