Input cost minimization is one of the three optimization problems faced by the firm. It revolves around the question of choosing the best combination of inputs, L and K, to produce a given level of output, q.

The best combination is defined as the cheapest one. The idea is that many combinations of L and K can produce a given q. We want to know the amounts of labor and capital that should be used to produce a given amount of output.

Of course, we answer this question by setting up and solving an optimization problem; then we do comparative statics.

Setting Up the Problem

The economic approach organizes optimization problems by answering three questions: (1) What is the goal? (2) What are the choice variables? (3) What are the given variables?

The goal is to minimize total cost, TC, which is simply the sum of the amount paid to the workers, wL, and the amount spent on renting machines, rK.

The endogenous variables are L and K. Labor is measured in hours and capital is the number of machines. The firm can decide to produce the given output by being labor intensive, using lots of labor and little capital, or roughly equal amounts of both, or by renting a lot of machinery and using little labor.