![]() This was a fun exercise that made me think about the usefulness of the Cobb-Douglas production function, which I learned to optimize multiple times in my Economics courses. As production increases, the minimum cost needed increases in a non-linear, exponential fashion, which makes sense given that Y (quantity produced) is in the numerator on the right-hand side of the cost function and positively related to the cost. Using the Cobb-Douglas production function and the cost minimization approach, we were able to find the optimal conditions for the cost function and plot the outcome relative to the quantity produced. The functional form of the CD production function: However, in this example, we will learn how to answer a minimization problem subject to (s.t.) the CD production function as a constraint. It is similarly used to describe utility maximization through the following function. ![]() Typical inputs include labor (L) and capital (K). The Cobb-Douglas (CD) production function is an economic production function with two or more variables (inputs) that describes the output of a firm. Appreciation goes out to the anonymous reader who identified this error. Hence, I’ve updated the data frame used in the example to avoid this issue. Since the output of elasticity should be between the values of 0 and 1, this negative coefficient should not be possible. The beta coefficient generated a negative value which was used in the linear form of the Cobb-Douglass equation. In the previous example, I used R to generate a set of random numbers that were used in a regression model. ![]() The error was the negative value generated for the output elasticity of capital. Update 1: This article was updated on 11 October 2021 when an anonymous reader identified an error with the example used at the end. I updated this to better reflect the minimization problem and set the partial derivative solution to 0. Update 2: This article was updated on 12 August 2023 when Dimanjan Dahal ( Twitter account ) identified a better way to present the Lagrangian functions.
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