Lecture 01: Introduction

Solve the consumer problem
Simulate the AS-AD model
Using modules
Downloading with Git

Summary: The Jupyter notebook is a document with text, code and results.

This is a text cell, or more precisely a markdown cell.

Pres Enter to edit the cell.
Pres Ctrl+Enter to run the cell.
Pres Shift+Enter to run the cell + advance.

We can make lists:

First item
Second item
~~Third~~ item

We can also do LaTeX math, e.g. $\alpha^2$ or

X = \int_0^{\infty} \frac{x}{x+1} dx

[ ]

# this is a code cell
# let us do some calculations
a = 20
b = 30
c = a+b

# lets print the results (shown below the cell)
print(c)

We can now write some more text, and continue with our calculations.

[ ]

d = c*2
print(d)
print(c*3)

Note: Despite JupyterLab is running in a browser, it is running offline (the path is something like localhost:8888/lab).

Binder: The exception is if you use binder, then JupyterLab wil run in the cloud.

1. Solve the consumer problem

Consider the following consumer problem:

\begin{aligned} V(p_{1},p_{2},I) & = \max_{x_{1},x_{2}} x_{1}^{\alpha}x_{2}^{1-\alpha}\\ & \text{s.t.}\\ p_{1}x_{1}+p_{2}x_{2} & \leq I,\,\,\,p_{1},p_{2},I>0\\ x_{1},x_{2} & \geq 0 \end{aligned}

We can solve this problem numerically in a few lines of code.

Choose some parameters:

[ ]

alpha = 0.5
I = 10
p1 = 1
p2 = 2

The consumer objective is:

[ ]

def value_of_choice(x1,alpha,I,p1,p2):
    
    # a. all income not spent on the first good
    #    is spent on the second
    x2 = (I-p1*x1)/p2 
    
    # b. the resulting utility is
    utility = x1**alpha * x2**(1-alpha)
    
    return utility

We can now use a function from the scipy module to solve the consumer problem.

[ ]

# a. load external module from scipy
from scipy import optimize

# b. make value-of-choice as a funciton of only x1 
obj = lambda x1: -value_of_choice(x1,alpha,I,p1,p2)

# c. call minimizer
solution = optimize.minimize_scalar(obj,bounds=(0,I/p1))

# d. print result
x1 = solution.x
x2 = (I-x1*p1)/p2
print(x1,x2)

Task: Solve the consumer problem with the CES utility funciton.

u(x_1,x_2) = (\alpha x_1^{-\beta} + (1-\alpha) x_2^{-\beta})^{-1/\beta}

[ ]

import numpy as np

# a. choose parameters
alpha = 0.5
beta = 0.000001
I = 10
p1 = 1
p2 = 2

# b. value-of-choice
def value_of_choice_ces(x1,alpha,beta,I,p1,p2):
    x2 = (I-p1*x1)/p2
    if x1 > 0 and x2 > 0:
        utility = (alpha*x1**(-beta)+(1-alpha)*x2**(-beta))**(-1/beta) 
    else:
        utility = 0
    return utility

# c. objective
obj = lambda x1: -value_of_choice_ces(x1,alpha,beta,I,p1,p2)

# d. solve
solution = optimize.minimize_scalar(obj,bounds=(0,I/p1))

# e. result
x1 = solution.x
x2 = (I-x1*p1)/p2
print(x1,x2)

2. Simulate the AS-AD model

Consider the following AS-AD model:

\begin{aligned} \hat{y}_{t} &= b\hat{y}_{t-1}+\beta(z_{t}-z_{t-1})-a\beta s_{t}+a\beta\phi s_{t-1} \\ \hat{\pi}_{t} &= b\hat{\pi}_{t-1}+\beta\gamma z_{t}-\beta\phi\gamma z_{t}+\beta s_{t}-\beta\phi s_{t-1} \\ z_{t} &= \delta z_{t-1}+x_{t}, x_{t} \sim N(0,\sigma_x^2) \\ s_{t} &= \omega s_{t-1}+c_{t}, c_{t} \sim N(0,\sigma_c^2) \\ b &= \frac{1+a\phi\gamma}{1+a\gamma} \\ \beta &= \frac{1}{1+a\gamma} \end{aligned}

where $\hat{y}_{t}$ is the output gap, $\hat{\pi}_{t}$ is the inflation gap, $z_{t}$ is a AR(1) demand shock, and $\hat{s}_{t}$ is a AR(1) supply shock.

Choose parameters:

[ ]

a = 0.4
gamma = 0.1
phi = 0.9
delta = 0.8
omega = 0.15
sigma_x = 1
sigma_c = 0.4
T = 100

Calculate composite parameters:

[ ]

b = (1+a*phi*gamma)/(1+a*gamma)
beta = 1/(1+a*gamma)

Define model functions:

[ ]

y_hat_func = lambda y_hat_lag,z,z_lag,s,s_lag: b*y_hat_lag + beta*(z-z_lag) - a*beta*s + a*beta*phi*s_lag
pi_hat_func = lambda pi_lag,z,z_lag,s,s_lag: b*pi_lag + beta*gamma*z - beta*phi*gamma*z_lag + beta*s - beta*phi*s_lag
z_func = lambda z_lag,x: delta*z_lag + x
s_func = lambda s_lag,c: omega*s_lag + c

Run the simulation:

[ ]

import numpy as np

# a. set setup
np.random.seed(2015)   

# b. allocate simulation data
x = np.random.normal(loc=0,scale=sigma_x,size=T)
c = np.random.normal(loc=0,scale=sigma_c,size=T)

z = np.zeros(T)
s = np.zeros(T)
y_hat = np.zeros(T)
pi_hat = np.zeros(T)

# c. run simulation
for t in range(1,T):

    # i. update demand shocks and supply shocks
    z[t] = z_func(z[t-1],x[t])
    s[t] = s_func(s[t-1],c[t])

    # ii. compute outgap and inflation based on shocks in t 
    y_hat[t] = y_hat_func(y_hat[t-1],z[t],z[t-1],s[t],s[t-1])
    pi_hat[t] = pi_hat_func(pi_hat[t-1],z[t],z[t-1],s[t],s[t-1])

Plot the simulation:

[ ]

import matplotlib.pyplot as plt
%matplotlib inline

fig = plt.figure()
ax = fig.add_subplot(1,1,1)

ax.plot(y_hat,label='$\\hat{y}$')
ax.plot(pi_hat,label='$\\hat{\pi}$')

ax.set_xlabel('time')

ax.set_ylabel('percent')
ax.set_ylim([-8,8])

ax.legend(loc='upper left');

I like the seaborn style:

[ ]

plt.style.use('seaborn-whitegrid')

fig = plt.figure()
ax = fig.add_subplot(1,1,1)

ax.plot(y_hat,label='$\\hat{y}$')
ax.plot(pi_hat,label='$\\hat{pi}$')

ax.set_xlabel('time')

ax.set_ylabel('percent')
ax.set_ylim([-8,8])

ax.legend(loc='upper left',facecolor='white',frameon='True');

3. Using modules

A module is a .py-file with functions you import and can then call in the notebook.

Try to open mymodule.py and have a look.

[ ]

import mymodule

[ ]

x = 5
y = mymodule.myfunction(x)
print(y)

4. Downloading with Git

Download course material

Follow the installation guide
If you have not yet restarted JupyterLab after installations, do it now.
Make sure you are in the folder, where the course content should be located.
Press the tab Git.
Press Clone a repository
Paste in https://github.com/NumEconCopenhagen/lectures-2022 and enter.
You now have all course material in the folder lectures-2022
Do the same thing for exercise class material, using the url https://github.com/NumEconCopenhagen/exercises-2022
Create a copy of the cloned folder, where you work with the code (otherwise you can not sync with updates)