In this article, we will take a closer look at how we can derive the cost minimization for a Cobb-Douglas function. We will also see how we, in the end, can derive the cost function.
Cost Minimization with the Lagrange Multiplier
We can first try to understand what is the basis of the problem when we talk about cost minimization. The problem is very simple: The cost-minimization problem of the firm is to choose an input bundle (x_1, x_2) that is feasible for the output, y, that costs as little as possible. So, how can this be written mathematically? We can first write up the problem of the company:
which needs to be subjected to the utility function. We also remember that w_1 is simply the price of input 1, x_1. In the same way, w_2 is the price of input 2, x_2. We can then simply imagine that we have a Cobb-Douglas utility function, which is given by:
Since this is a simple minimization problem, then we can use the Lagrange Multiplier. We then get the following:
This gives us the following First Order Conditions:
We can then solve for w_1 and w_2 in the first and second conditions. This gives us the following:
