A firm has the following production function: The price for one unit of capital is and the price for one unit of labor is . Minimize the costs of the firm considering its production function and given a target production output of 730 units.
How high are in this case the minimal costs?
Step 1: Formulating the minimization problem. Step 2: Lagrange function. Step 3: First order conditions. Step 4: Solve the system of equations for , , and .
Solving the first two equations for and equating them gives: Substituting this in the optimization constraint gives:
The minimal costs can be obtained by substituting the optimal factor combination in the objective function:
Given the target output, the minimal costs are .