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The Cobb-Douglas function is a popular form of production function commonly used in economics to represent the relationship between two or more inputs and the level of output. This type of function is particularly known for displaying constant returns to scale when the exponents of the variables sum to one. In our case, the function is given by:

• \[ f(x, y) = 100x^{0.8}y^{0.2} \]

Here, the variables \(x\) and \(y\) represent input quantities like labor and capital, while \(100\) is a constant that scales the output level. The exponents \(0.8\) and \(0.2\) indicate the output elasticity concerning each input. These values suggest that \(80\%\) of the production is dependent on \(x\), and \(20\%\) on \(y\).

• **Key Characteristics:**
• Output increases with an increase in either input, with the production showing diminishing returns for each additional input unit.
• The function exhibits constant returns to scale, as the sum of the exponents (\(0.8 + 0.2 = 1\)) is equal to one.

Thus, the Cobb-Douglas function provides a useful tool in analytic modeling for economists and statisticians to optimize inputs for improved productivity.

Partial derivatives play a pivotal role in optimizing multivariable functions, such as the Cobb-Douglas function, especially when employing methods like Lagrange multipliers. A partial derivative measures how a function changes as one of the variables changes, while other variables are kept constant. This is helpful to understand the slope or rate of change in a particular direction. In this exercise, we deal with partial derivatives to identify critical points of the Lagrangian:

• \(\frac{\partial \mathcal{L}}{\partial x} = 80x^{-0.2}y^{0.2} - 2\lambda = 0\)
• \(\frac{\partial \mathcal{L}}{\partial y} = 20x^{0.8}y^{-0.8} - 4\lambda = 0\)
• \(\frac{\partial \mathcal{L}}{\partial \lambda} = 100 - 2x - 4y = 0\)

Partial derivative equations \(1\) and \(2\) ensure the derivatives are zero, reaching a point where the function does not change, leading us towards potential maximum or minimum points.

• **Why is this important?**
• Partial derivatives help us in establishing a system of equations for optimization.
• They are used to equate the slope of the function surface to zero, directing towards possible extremities (maxima or minima).

This makes partial derivatives critical in analyzing and solving optimization problems in calculus and economic modeling.

Optimization with constraints is an application where we aim to find maximum or minimum values of a function subject to certain conditions or restrictions. The Lagrange Multiplier Method is a powerful technique used to achieve this by introducing a new variable, called the Lagrange multiplier (\(\lambda\)), to incorporate these conditions into the optimization process.

• **The process:**
• Construct the Lagrangian by adding the constraint to the objective function using \(\lambda\).
• Take the partial derivatives with respect to each variable including \(\lambda\), and set them to zero.
• Solve the resulting system of equations to find the variables' values that optimize the function.

In our exercise, we optimize the function:

• \[ \mathcal{L}(x, y, \lambda) = 100 x^{0.8} y^{0.2} + \lambda(100 - 2x - 4y)\]

subject to the constraint \(2x + 4y = 100\). By solving the equations derived from the Lagrangian's partial derivatives, we find the optimal input quantities \(x\approx2.94\) and \(y\approx23.52\), satisfying both the production function and the constraint simultaneously.
The Lagrange Multiplier Method is crucial when dealing with real-world problems where factors like budgetary restraints are considered, ensuring that optimization outcomes are practical and viable.