Optimizing the Cobb-Douglas Function with Lagrangian Method

What is the Lagrangian?

a. \(K^{0.4}L^{0.6}-λ(340-8K-4L)\)
b. \(K^{0.4}L^{0.6}+λ(340+8K+4L)\)
c. \(K^{0.4}L^{0.6}+λ(340-8K-4L)\)
d. None of the above

Answer:

d. None of the above

The Cobb-Douglas function is a production function that relates the output of a firm to the inputs of capital (K) and labor (L). In order to optimize the Cobb-Douglas function, we need to find the values of K and L that maximize the output (q) while considering the budget constraint.

Solving the equation 340 + 8K + 4L = 0 simultaneously will give us the optimal values of K and L that maximize the Cobb-Douglas function while satisfying the budget constraint. The Cobb-Douglas function is given as \(q = K^{0.4} * L^{0.6}\). To optimize this function, we can use the Lagrange multiplier method, which involves incorporating the budget constraint into the function.

We need to maximize the function q + λ(340 + 8K + 4L), where λ is the Lagrange multiplier. To find the maximum, we take the partial derivatives of the function with respect to K, L, and λ, and set them equal to zero.

Taking the partial derivative with respect to K:
\(0.4K^{-0.6} * L^{0.6} + 8λ = 0\)

Taking the partial derivative with respect to L:
\(0.6K^{0.4} * L^{-0.4} + 4λ = 0\)

Taking the partial derivative with respect to λ:
\(340 + 8K + 4L = 0\)

We now have a system of equations that can be solved simultaneously. From the first two equations, we can isolate λ and set the equations equal to each other:

Simplifying, we get:
\(0.4K^{-0.6} * L^{0.6} = -8λ\)
\(0.6K^{0.4} * L^{-0.4} = -4λ\)

Dividing the equations, we get:
\((0.4K^{-0.6} * L^{0.6) / (0.6K^{0.4} * L^{-0.4}) = -8λ / -4λ\)

Substituting the values of K and L into the budget constraint equation: 340 + 8K + 4L = 0, you can find the optimal values that maximize the Cobb-Douglas function while satisfying the budget constraint.