4. Introduction to Python III - Functions and Packages

Exercise 1: Write a function that computes the level of output generated by a constant returns to scale Cobb-Douglas production function, i.e., such that it computes $A\cdot K^\alpha \cdot L^{1-\alpha}$.

def CobbDouglas(K=1, L=1, A=1, alpha=0.3): """ Computes the level of output of a Cobb-Douglas production function. K: Capital input (default=1) L: Labor input (default=1) A: Total factor productivity (default=1) alpha: Output elasticity of capital (default=0.3) """ return A * (K ** alpha) * (L ** (1 - alpha)) # Example output = CobbDouglas(4, 9, 2, 0.5) print("Output:", output)

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Exercise 2: Use the function you created and plot the production function as a function of $K$ for given values of $A$ and $L$. Hint: Use the np.linspace(0, 5, 100) function to create an array of values of $K$ with 100 point between 0 and 5.

import numpy as np import matplotlib.pyplot as plt K_values = np.linspace(0, 5, 100) L = 1 A = 1 alpha = 0.3 outputs = [CobbDouglas(K, L, A, alpha) for K in K_values] plt.plot(K_values, outputs) plt.title("Cobb-Douglas Production Function") plt.xlabel("Capital (K)") plt.ylabel("Output") plt.grid(True) plt.show()

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Exercise 3: Show with a plot the effect of increasing $A$ from 1 to 2, 4, or 5.

A_values = [1, 2, 4, 5] for A in A_values: outputs = [CobbDouglas(K, L, A, alpha) for K in K_values] plt.plot(K_values, outputs, label=f"A={A}") plt.title("Effect of Increasing A on Output") plt.xlabel("Capital (K)") plt.ylabel("Output") plt.legend() plt.grid(True) plt.show()

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Exercise 4: Show with a plot the effect of increasing $L$ from 1 to 2, 4, 5

L_values = [1, 2, 4, 5] for L in L_values: outputs = [CobbDouglas(K, L, A, alpha) for K in K_values] plt.plot(K_values, outputs, label=f"L={L}") plt.title("Effect of Increasing L on Output") plt.xlabel("Capital (K)") plt.ylabel("Output") plt.legend() plt.grid(True) plt.show()

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Exercise 5: Save the previous plots into png, jpeg, and pdf files using the plt.savefig function.

plt.savefig("plot.png") plt.savefig("plot.jpeg") plt.savefig("plot.pdf")

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Exercise 6: Use the IFrame function to show the website from where you are planning to obtain data.

from IPython.display import IFrame IFrame("https://www.worldbank.org", width=800, height=600)

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Exercise 7: Use the IFrame function to insert an iframe showing the graph for the evolution of patent applications by residents across the world using the World Development Indicators website.

IFrame("https://databank.worldbank.org/source/world-development-indicators", width=800, height=600)

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