Problem set 2: Finding the Walras equilibrium in a multi-agent economy

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%load_ext autoreload
%autoreload 2

Tasks

Drawing random numbers

Replace the missing lines in the code below to get the same output as in the answer.

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import numpy as np
np.random.seed(1986)
# missing line
for i in range(3):
    # missing line
    for j in range(2):
        x = np.random.uniform()
        print(f'({i},{j}): x = {x:.3f}')

Answer:

See A1.py

Find the expectated value

Find the expected value and the expected variance

\mathbb{E}[g(x)] \approx \frac{1}{N}\sum_{i=1}^{N} g(x_i)

\mathbb{VAR}[g(x)] \approx \frac{1}{N}\sum_{i=1}^{N} \left( g(x_i) - \frac{1}{N}\sum_{i=1}^{N} g(x_i) \right)^2

where $x_i \sim \mathcal{N}(0,\sigma)$ and

g(x,\omega)=\begin{cases} x & \text{if }x\in[-\omega,\omega]\\ -\omega & \text{if }x<-\omega\\ \omega & \text{if }x>\omega \end{cases}

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sigma = 3.14
omega = 2
N = 10000
np.random.seed(1986)
# write your code here

Answer:

See A2.py

Interactive histogram

First task: Consider the code below. Fill in the missing lines so the figure is plotted.

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# a. import
%matplotlib inline
import matplotlib.pyplot as plt
# missing line

# b. plotting figure
def fitting_normal(X,mu_guess,sigma_guess):
    
    # i. normal distribution from guess
    F = norm(loc=mu_guess,scale=sigma_guess)
    
    # ii. x-values
    # missing line, x_low =
    # missing line, x_high =
    x = np.linspace(x_low,x_high,100)

    # iii. figure
    fig = plt.figure(dpi=100)
    ax = fig.add_subplot(1,1,1)
    ax.plot(x,F.pdf(x),lw=2)
    ax.hist(X,bins=100,density=True,histtype='stepfilled');
    ax.set_ylim([0,0.5])
    ax.set_xlim([-6,6])

# c. parameters
mu_true = 2
sigma_true = 1
mu_guess = 1
sigma_guess = 2

# d. random draws
X = np.random.normal(loc=mu_true,scale=sigma_true,size=10**6)

# e. figure
try:
    fitting_normal(X,mu_guess,sigma_guess)
except:
    print('failed')

Second task: Create an interactive version of the figure with sliders for $\mu$ and $\sigma$ .

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# write your code here

Answer:

See A3.py

Modules

Call the function myfun from the module mymodule present in this folder.
Open VSCode and open the mymodule.py, add a new function and call it from this notebook.

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# write your code here

Answer:

See A4.py

Git

Try to go to your own personal GitHub main page and create a new repository. Then put your solution to this problem set in it.
Pair up with a fellow student. Clone each others repositories and run the code in them.

IMPORTANT: You will need git for the data project in a few needs. Better learn it know. Remember, that the teaching assistants are there to help you.

Problem

Consider an exchange economy with

2 goods, $(x_1,x_2)$
$N$ consumers indexed by $j \in \{1,2,\dots,N\}$
Preferences are Cobb-Douglas with truncated normally heterogenous coefficients
$\begin{aligned} u^{j}(x_{1},x_{2}) & = x_{1}^{\alpha_{j}}x_{2}^{1-\alpha_{j}}\\ & \tilde{\alpha}_{j}\sim\mathcal{N}(\mu,\sigma)\\ & \alpha_j = \max(\underline{\mu},\min(\overline{\mu},\tilde{\alpha}_{j})) \end{aligned}$
Endowments are heterogenous and given by
$\begin{aligned} \boldsymbol{e}^{j}&=(e_{1}^{j},e_{2}^{j}) \\ & & e_i^j \sim f, f(x,\beta_i) = 1/\beta_i \exp(-x/\beta) \end{aligned}$

Problem: Write a function to solve for the equilibrium.

You can use the following parameters:

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# a. parameters
N = 10000
mu = 0.5
sigma = 0.2
mu_low = 0.1
mu_high = 0.9
beta1 = 1.3
beta2 = 2.1
seed = 1986

# b. draws of random numbers
# c. demand function
# d. excess demand function
# e. find equilibrium function
# f. call find equilibrium function

Hint: The code structure is exactly the same as for the exchange economy considered in the lecture. The code for solving that exchange economy is reproduced in condensed form below.

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# a. parameters
N = 1000
k = 2
mu_low = 0.1
mu_high = 0.9
seed = 1986

# b. draws of random numbers
np.random.seed(seed)
alphas = np.random.uniform(low=mu_low,high=mu_high,size=N)

# c. demand function
def demand_good_1_func(alpha,p1,p2,k):
    I = k*p1+p2
    return alpha*I/p1

# d. excess demand function
def excess_demand_good_1_func(alphas,p1,p2,k):
    
    # a. demand
    demand = np.sum(demand_good_1_func(alphas,p1,p2,k))
    
    # b. supply
    supply = k*alphas.size
    
    # c. excess demand
    excess_demand = demand-supply
    
    return excess_demand

# e. find equilibrium function
def find_equilibrium(alphas,p1,p2,k,kappa=0.5,eps=1e-8,maxiter=500):
    
    t = 0
    while True:

        # a. step 1: excess demand
        Z1 = excess_demand_good_1_func(alphas,p1,p2,k)
        
        # b: step 2: stop?
        if  np.abs(Z1) < eps or t >= maxiter:
            print(f'{t:3d}: p1 = {p1:12.8f} -> excess demand -> {Z1:14.8f}')
            break    
    
        # c. step 3: update p1
        p1 = p1 + kappa*Z1/alphas.size
            
        # d. step 4: return 
        if t < 5 or t%25 == 0:
            print(f'{t:3d}: p1 = {p1:12.8f} -> excess demand -> {Z1:14.8f}')
        elif t == 5:
            print('   ...')
            
        t += 1    

    return p1

# e. call find equilibrium function
p1 = 1.4
p2 = 1
kappa = 0.1
eps = 1e-8
p1 = find_equilibrium(alphas,p1,p2,k,kappa=kappa,eps=eps)

Answers:

See A5.py

Save and load

Consider the code below and fill in the missing lines so the code can run without any errors.

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import pickle

# a. create some data
my_data = {}
my_data['A'] = {'a':1,'b':2}
my_data['B'] = np.array([1,2,3])
# missing line

my_np_data = {}
my_np_data['D'] = np.array([1,2,3])
my_np_data['E'] = np.zeros((5,8))
# missing line

# c. save with pickle
with open(f'data.p', 'wb') as f:
    # missing line
    pass
    
# d. save with numpy
# missing line, np.savez(?)
    
# a. try
def load_all():
    with open(f'data.p', 'rb') as f:
        data = pickle.load(f)
        A = data['A']
        B = data['B']
        C = data['C']

    with np.load(f'data.npz') as data:
        D = data['D']
        E = data['E']
        F = data['F']        
    
    print('variables loaded without error')
    
try:
    load_all()
except:
    print('failed')

Answer:

See A6.py

Extra Problems

Multiple goods

Solve the main problem extended with multiple goods:

\begin{aligned} u^{j}(x_{1},x_{2}) & = x_{1}^{\alpha^1_{j}} \cdot x_{2}^{\alpha^2_{j}} \cdots x_{M}^{\alpha^M_{j}}\\ & \alpha_j = [\alpha^1_{j},\alpha^2_{j},\dots,\alpha^M_{j}] \\ & \log(\alpha_j) \sim \mathcal{N}(0,\Sigma) \\ \end{aligned}

where $\Sigma$ is a valid covariance matrix.

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# a. choose parameters
N = 10000
J = 3

# b. choose Sigma
Sigma_lower = np.array([[1, 0, 0], [0.5, 1, 0], [0.25, -0.5, 1]])
Sigma_upper = Sigma_lower.T
Sigma = Sigma_upper@Sigma_lower
print(Sigma)

# c. draw random numbers
alphas = np.exp(np.random.multivariate_normal(np.zeros(J), Sigma, 10000))
print(np.mean(alphas,axis=0))
print(np.corrcoef(alphas.T))

# write your code here