def cobb_douglas(A, K, L, alpha): """ Computes the level of output using a Cobb-Douglas production function. Parameters: A (float): Total factor productivity (TFP) K (float): Capital input L (float): Labor input alpha (float): Capital's share of income (0 < alpha < 1) Returns: float: The output level (Y) """ # Ensure inputs are valid if not (0 < alpha < 1): raise ValueError("Alpha must be between 0 and 1.") if K < 0 or L < 0: raise ValueError("Capital and Labor inputs must be non-negative.") # Compute the output Y = A * (K**alpha) * (L**(1 - alpha)) return Y

import numpy as np import matplotlib.pyplot as plt # Cobb-Douglas production function (from Exercise 1) def cobb_douglas(A, K, L, alpha): """ Computes the level of output using a Cobb-Douglas production function. """ if not (0 < alpha < 1): raise ValueError("Alpha must be between 0 and 1.") if K < 0 or L < 0: raise ValueError("Capital and Labor inputs must be non-negative.") return A * (K**alpha) * (L**(1 - alpha)) # Given parameters A = 1.2 # Total factor productivity L = 50 # Labor input alpha = 0.3 # Capital's share of income # Generate an array of K values K_values = np.linspace(0, 5, 100) # 100 points between 0 and 5 # Compute output for each K output_values = [cobb_douglas(A, K, L, alpha) for K in K_values] # Plot the production function plt.figure(figsize=(8, 6)) plt.plot(K_values, output_values, label=f'Production Function ($A={A}$, $L={L}$, $\\alpha={alpha}$)', color='blue') plt.title('Cobb-Douglas Production Function') plt.xlabel('Capital Input (K)') plt.ylabel('Output (Y)') plt.grid(True) plt.legend() plt.show()

import numpy as np import matplotlib.pyplot as plt # Cobb-Douglas production function def cobb_douglas(A, K, L, alpha): """ Computes the level of output using a Cobb-Douglas production function. """ if not (0 < alpha < 1): raise ValueError("Alpha must be between 0 and 1.") if K < 0 or L < 0: raise ValueError("Capital and Labor inputs must be non-negative.") return A * (K**alpha) * (L**(1 - alpha)) # Parameters L = 50 # Fixed labor input alpha = 0.3 # Capital's share of income K_values = np.linspace(0, 5, 100) # Range of K values # Different values of A to compare A_values = [1, 2, 4, 5] # Plot the production function for each A plt.figure(figsize=(10, 6)) for A in A_values: output_values = [cobb_douglas(A, K, L, alpha) for K in K_values] plt.plot(K_values, output_values, label=f'A = {A}') # Customize the plot plt.title('Effect of Increasing A on the Cobb-Douglas Production Function') plt.xlabel('Capital Input (K)') plt.ylabel('Output (Y)') plt.grid(True) plt.legend(title='Total Factor Productivity') plt.show()

import numpy as np import matplotlib.pyplot as plt # Cobb-Douglas production function def cobb_douglas(A, K, L, alpha): """ Computes the level of output using a Cobb-Douglas production function. """ if not (0 < alpha < 1): raise ValueError("Alpha must be between 0 and 1.") if K < 0 or L < 0: raise ValueError("Capital and Labor inputs must be non-negative.") return A * (K**alpha) * (L**(1 - alpha)) # Parameters A = 1.2 # Fixed total factor productivity alpha = 0.3 # Capital's share of income K_values = np.linspace(0, 5, 100) # Range of K values # Different values of L to compare L_values = [1, 2, 4, 5] # Plot the production function for each L plt.figure(figsize=(10, 6)) for L in L_values: output_values = [cobb_douglas(A, K, L, alpha) for K in K_values] plt.plot(K_values, output_values, label=f'L = {L}') # Customize the plot plt.title('Effect of Increasing L on the Cobb-Douglas Production Function') plt.xlabel('Capital Input (K)') plt.ylabel('Output (Y)') plt.grid(True) plt.legend(title='Labor Input (L)') plt.show()

import numpy as np import matplotlib.pyplot as plt # Cobb-Douglas production function def cobb_douglas(A, K, L, alpha): """ Computes the level of output using a Cobb-Douglas production function. """ if not (0 < alpha < 1): raise ValueError("Alpha must be between 0 and 1.") if K < 0 or L < 0: raise ValueError("Capital and Labor inputs must be non-negative.") return A * (K**alpha) * (L**(1 - alpha)) # Parameters A = 1.2 # Fixed total factor productivity alpha = 0.3 # Capital's share of income K_values = np.linspace(0, 5, 100) # Range of K values # Different values of L to compare L_values = [1, 2, 4, 5] # Plot the production function for each L plt.figure(figsize=(10, 6)) for L in L_values: output_values = [cobb_douglas(A, K, L, alpha) for K in K_values] plt.plot(K_values, output_values, label=f'L = {L}') # Customize the plot plt.title('Effect of Increasing L on the Cobb-Douglas Production Function') plt.xlabel('Capital Input (K)') plt.ylabel('Output (Y)') plt.grid(True) plt.legend(title='Labor Input (L)') # Save the plot in different formats plt.savefig('cobb_douglas_plot.png', format='png', dpi=300) plt.savefig('cobb_douglas_plot.jpeg', format='jpeg', dpi=300) plt.savefig('cobb_douglas_plot.pdf', format='pdf') # Show the plot plt.show()

from IPython.display import IFrame # Define the URL of the website url = "https://en.wikipedia.org/wiki/List_of_ISO_3166_country_codes" # Display the website in an IFrame IFrame(src=url, width=800, height=600)

from IPython.display import IFrame # Define the URL for the patent application report url = "https://databank.worldbank.org/reports.aspx?Report_Name=patent-application&Id=d484ab34" # Display the IFrame with the specified webpage IFrame(src=url, width=1000, height=800)