Notebook

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.

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:

$$ \log z_t = \rho \log z_{t-1} + \varepsilon_t, \qquad \varepsilon_t \sim N(0, \sigma^2) $$