

In [ ]:

    
%matplotlib inline



In [ ]:

    
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import sympy as sym

import solowpy



In [ ]:

    
# define model parameters
ces_params = {'A0': 1.0, 'L0': 1.0, 'g': 0.02, 'n': 0.03, 's': 0.15,
              'delta': 0.05, 'alpha': 0.33, 'sigma': 1.01}

# create an instance of the solow.Model class
ces_model = solowpy.CESModel(params=ces_params)

5. Impulse response functions

Impulse response functions (IRFs) are a standard tool for analyzing the short run dynamics of dynamic macroeconomic models, such as the Solow growth model, in response to an exogenous shock. The solow.impulse_response.ImpulseResponse class has several attributes and methods for generating and analyzing impulse response functions.



In [ ]:

    
# use tab completion to see the available attributes and methods...
solowpy.impulse_response.ImpulseResponse.

The solow.Model class provides access to all of the functionality of the solow.impulse_response.ImpulseResponse class through its irf attribute.



In [ ]:

    
# use tab completion to see the available attributes and methods...
ces_model.irf.

Example: Impact of a change in the savings rate

One can analyze the impact of a doubling of the savings rate on model variables as follows.



In [ ]:

    
# 100% increase in the current savings rate...
ces_model.irf.impulse = {'s': 2.0 * ces_model.params['s']}

# in efficiency units...
ces_model.irf.kind = 'efficiency_units'

Take a look at the IRF for the savings rate shock. Note that while capital and output are unaffected at the t=0, both consumption and investment jump (in opposite directions!) in response to the change in the savings rate.



In [ ]:

    
# ordering of variables is t, k, y, c, i!
print(ces_model.irf.impulse_response[:25,])

Example: Plotting an impulse response function

One can use a convenience method to to plot the impulse response functions for a particular variable.



In [ ]:

    
# check the docstring to see the call signature
ces_model.irf.plot_impulse_response?



In [ ]:

    
fig, ax = plt.subplots(1, 1, figsize=(8,6))
ces_model.irf.plot_impulse_response(ax, variable='output')
plt.show()

Example: More complicated impulse responses are possible

Note that by defining impulses as dictionaries, one can analyze extremely general shocks. For example, suppose that an exogenous 5% increase in the growth rate of technology was accompanied by a simultaneous 10% fall in the savings rate.



In [ ]:

    
# more complicate shocks are possible
ces_model.irf.impulse = {'s': 0.9 * ces_model.params['s'], 'g': 1.05 * ces_model.params['g']}

# in efficiency units...
ces_model.irf.kind = 'per_capita'



In [ ]:

    
fig, ax = plt.subplots(1, 1, figsize=(8,6))
ces_model.irf.plot_impulse_response(ax, variable='output', log=True)
plt.show()

Example: Interactive impulse reponse functions

Using IPython widgets makes it extremely easy to analyze the various impulse response functions.



In [ ]:

    
from IPython.html.widgets import fixed, interact, FloatSliderWidget



In [ ]:

    
def interactive_impulse_response(model, shock, param, variable, kind, log_scale):
    """Interactive impulse response plotting tool."""    
    # specify the impulse response
    model.irf.impulse = {param: shock * model.params[param]}
    model.irf.kind = kind
    
    # create the plot
    fig, ax = plt.subplots(1, 1, figsize=(8,6))
    model.irf.plot_impulse_response(ax, variable=variable, log=log_scale)
    

irf_widget = interact(interactive_impulse_response, 
                      model=fixed(ces_model),
                      shock = FloatSliderWidget(min=0.1, max=5.0, step=0.1, value=0.5),
                      param = ces_model.params.keys(),
                      variable=['capital', 'output', 'consumption', 'investment'],
                      kind=['efficiency_units', 'per_capita', 'levels'],
                      log_scale=False,
                      )



In [ ]: