## Instructions

**Objective**

Write a Python assignment that involves creating a program to implement graph plotting in Python. Graph plotting is a fundamental skill in data visualization and analysis. Python offers various libraries like Matplotlib, Seaborn, and Plotly that enable users to create different types of graphs, such as line plots, bar plots, scatter plots, and more. These libraries provide functions to customize the appearance of graphs, add labels and titles, and handle data efficiently. By completing this assignment, students will gain practical experience in using Python libraries to visualize data and convey meaningful insights through graphical representations.

## Requirements and Specifications

**Source Code**

```
import matplotlib.pyplot as plt
import seaborn as sns
from sympy import *
import urllib.request
import random
from sympy.plotting.plot import Plot, ContourSeries
import numpy as np
def Problem1():
print("Running problem 1...")
# Create the symbolic variable x
x = Symbol('x')
# Create function
f = 10 - 4 * x ** 2 + 2 * x ** 3
# Get its minimum
fprime = f.diff(x)
x_min = solve(fprime, x)[1]
f_xmin = f.subs(x, x_min)
# Plot
#plot(f, (x, 0, 8 / 4), xlim=(0, 8 / 4), ylim=(30 / 4, 40 / 4), xlabel='X', ylabel='Y', label='Function', show = False)
xvals = np.linspace(0, 8/4, 100)
fvals = lambdify(x, f)(xvals)
plt.figure()
plt.plot(xvals, fvals, label = f)
plt.plot(x_min, f_xmin,'r*', label = 'Local Minimum')
plt.xlim([0, 8/4])
plt.ylim([30/4, 40/4])
plt.xlabel('X')
plt.ylabel('Y')
plt.legend()
plt.xticks([0, 1/4, 2/4, 3/4, 4/4, 5/4, 6/4, 7/4, 8/4], ['0', '1/4', '2/4', '3/4', '4/4', '5/4', '6/4', '7/4', '8/4'])
plt.yticks([30/4, 32/4, 34/4, 36/4, 38/4, 40/4], ['30/4', '32/4', '34/4', '36/4', '38/4', '40/4'])
plt.show()
"""plt.xticks(['0', '1/4', '2/4', '3/4', '4/4', '5/4', '6/4', '7/4', '8/4'])
plt.yticks([30/4, 31/4, 32/4, 33/4, 34/4, 35/4, 36/4, 37/4, 38/4, 39/4, 40/4])
plt.label()
plt.show()"""
def Problem2():
print("Running Problem 2...")
# Define alpha and beta
alpha = 0.5
beta = 0.5
# Define x1 and x2
x1 = Symbol("x1")
x2 = Symbol("x2")
graph = Plot(ContourSeries(x1**alpha *x2**beta, (x1, -100,100), (x2,-100,100)), xlabel = 'x1', ylabel = 'x2')
graph.show()
def Problem3():
print("Running Problem 3...")
# Now 1e6 times
sum2 = []
for i in range(1000000):
sum1 = 0
for i in range(10):
die = random.randint(1, 6)
sum1 += die
sum2.append(sum1)
# Now, plot histogram
plt.figure()
plt.hist(sum2)
plt.xlabel('Sum of the outcomes')
plt.ylabel('Number of times these sums were obtained')
plt.show()
def Problem4():
print("Running problem 4...")
# Read file from link
data= urllib.request.urlopen("https://raw.githubusercontent.com/AlR0d/courses/main/4465/miami_july_temps.csv").readlines()
# process data
# We delete the first entry in data because that line contains the header
data.pop(0)
# Create a list to store min_temp, max_temp and mean_temp
min_temps = []
max_temps = []
mean_temps = []
# Now, process each line
for line in data:
line = line.decode('utf-8')
line = line.rstrip()
# Split using comma
row = line.split(",")
# Get min, max and mean temps
min_temp = float(row[3])
max_temp = float(row[4])
mean_temp = float(row[5])
# Append
min_temps.append(min_temp)
max_temps.append(max_temp)
mean_temps.append(max_temp)
# Fit curve using seaborn
# Now plot
fig, axes = plt.subplots(nrows = 3, ncols = 1, figsize=(8,8))
x = np.linspace(0, len(min_temps)-1, len(min_temps))
sns.regplot(x=x, y=min_temps, ax = axes[0])
sns.regplot(x, max_temps, ax=axes[1])
sns.regplot(x, mean_temps, ax=axes[2])
axes[0].set_title('Min. Temp')
axes[0].set_xlabel('Years since July 1948')
axes[0].set_ylabel('Temperature')
axes[1].set_title('Max Temp')
axes[1].set_xlabel('Years since July 1948')
axes[1].set_ylabel('Temperature')
axes[2].set_title('Mean Temp')
axes[2].set_xlabel('Years since July 1948')
axes[2].set_ylabel('Temperature')
plt.show()
def Problem5():
print("Running problem 5...")
""" part a """
# Define the variables
Ct, Ct1, S = var('ct,ct1,s',real=True)
# Define lambdas
lam1 = symbols('lambda1',real = True)
lam2 = symbols('lambda2', real=True)
# Define r
r = 0.1
# Define the objective function. NOTE: We set the objctive as negative in order to maixmize
f = (Ct*Ct1)**(1/2)
"""
PART B
"""
print(" *** PART B *** ")
# Define intertemporal budget constraint
Mt = 30000
Mt1 = 20000
const1 = Mt - Mt1 / (1 + r) - Ct - Ct1 / (1 + r)
# Define saving function
const2 = S - Mt + Ct
# Define lagrangian
L = f - lam1 * const1 - lam2 * const2
# Define gradient
gradL = [diff(L, c) for c in [Ct, Ct1, S]]
KKT_eqs = gradL + [const1] + [const2]
sol = solve(KKT_eqs, [Ct,Ct1,S,lam1,lam2],dict=True)[0]
Ctsol = sol[Ct]
Ct1sol = sol[Ct1]
Ssol = sol[S]
print("The consumption at period t is {:.2f}".format(Ctsol))
print("The consumption at period t+1 is {:.2f}".format(Ct1sol))
print("The savings are {:.2f}".format(Ssol))
print("The total income is {:.2f}".format(Mt+Mt1))
print("The toal consumption is {:.2f}".format(Ctsol+Ct1sol))
print()
"""
PART C
"""
print(" *** PART C *** ")
# Define interporal budget constraint
Mt = 30000
Mt1 = 40000
const1 = Mt - Mt1 / (1 + r) - Ct - Ct1 / (1 + r)
# Define saving function
const2 = S - Mt + Ct
# Define lagrangian
L = f - lam1 * const1 - lam2 * const2
# Define gradient
gradL = [diff(L, c) for c in [Ct, Ct1, S]]
KKT_eqs = gradL + [const1] + [const2]
sol = solve(KKT_eqs, [Ct, Ct1, S, lam1, lam2], dict=True)[0]
ctsol = sol[Ct]
Ct1sol = sol[Ct1]
Ssol = sol[S]
print("The consumption at period t is {:.2f}".format(Ctsol))
print("The consumption at period t+1 is {:.2f}".format(Ct1sol))
print("The savings are {:.2f}".format(Ssol))
print("The total income is {:.2f}".format(Mt + Mt1))
print("The toal consumption is {:.2f}".format(Ctsol + Ct1sol))
print()
"""
PART D
"""
print(" *** PART D *** ")
# Define interporal budget constraint
Mt = 30000
Mt1 = 30000
const1 = Mt - Mt1 / (1 + r) - Ct - Ct1 / (1 + r)
# Define saving function
const2 = S - Mt + Ct
# Define lagrangian
L = f - lam1 * const1 - lam2 * const2
# Define gradient
gradL = [diff(L, c) for c in [Ct, Ct1, S]]
KKT_eqs = gradL + [const1] + [const2]
sol = solve(KKT_eqs, [Ct, Ct1, S, lam1, lam2], dict=True)[0]
Ctsol = sol[Ct]
; Ct1sol = sol[Ct1]
Ssol = sol[S]
print("The consumption at period t is {:.2f}".format(Ctsol))
print("The consumption at period t+1 is {:.2f}".format(Ct1sol))
print("The savings are {:.2f}".format(Ssol))
print("The total income is {:.2f}".format(Mt + Mt1))
print("The toal consumption is {:.2f}".format(Ctsol + Ct1sol))
print()
def Problem6():
print("Running problem 6...")
# Define parameters
w = 10
V = 0
p = 20
# Define variables
C, L = var('C,L', real=True)
# Define lambda
lam = symbols('lambda', real=True)
# Define objective
f = (C + 4*L**(1/2)) # Negative so we maximize it
# Define the budget constraint
constraint1 = w*(24 - L) + V - p*C
# Define lagrangian
Lag = f - lam * constraint1
# Define gradient
gradL = [diff(Lag, c) for c in [C, L]]
KKT_eqs = gradL + [constraint1]
sol = solve(KKT_eqs, [C, L, lam], dict=True)[0]
#print(sol)
Csol = sol[C]
Lsol = sol[L]
fval = f.subs(C, Csol)
fval = fval.subs(L, Lsol)
""" Part b """
print("The consumption is {:.2f} and the leisure is {:.2f}".format(Csol, Lsol))
""" Part c """
print("The utility is {:.2f}".format(fval))
""" Part d """
print("Rosalyn is willing to work {:.0f} hours a day".format(24 - Lsol))
""" Part e """
# For this case, V = 14600/365
V = 14600/365
# We compute all variables again
constraint1 = w * (24 - L) + V - p * C
p; # Define lagrangian
Lag = f - lam * constraint1
# Define gradient
gradL = [diff(Lag, c) for c in [C, L]]
KKT_eqs = gradL + [constraint1]
sol = solve(KKT_eqs, [C, L, lam], dict=True)[0]
#print(sol)
Csol = sol[C]
Lsol = sol[L]
fval = f.subs(C, Csol)
fval = fval.subs(L, Lsol)
print("With the new law, Rosalyn is willing to work {:.0f} hours a day and her utility is {:.2f}".format(24 - Lsol, fval))
if __name__ == '__main__':
Problem1()
Problem2()
Problem3()
Problem4()
Problem5()
Problem6()
```

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