Now that we’ve learned a little about marginal analysis, we can apply this knowledge to help answer questions that are really important. For example, suppose we would like to minimize the cost of producing our product, working on the theory that this will save us money. How would we go about this process of optimizing the cost?
First of all, we need to know what causes the cost of production to vary. Typically, the simplest quantity that determines production cost is, you guessed it, the number of items that we produce. After all, each one of them uses a certain amount of materials that aren’t free; each one of them requires labor; production probably involves machines which use electricity and so forth. So, we could start by getting together data that shows the total cost each month (or week or whatever) along with the total cost of production that month (or week or whatever). We can then use our model-building skills to determine an equation that represents the cost of production as a function of the number of items produced.
Now, how can this help us find the amount of production that will result in the lowest overall cost? We actually have several tools available. We could create a table of values from the function and look for the lowest cost. That could be difficult, though, since our table will only show some of the possible values: it may be that we skip over the best spot if we’re not very careful. We could also graph the function, but then scale is an issue; we may have to keep redrawing the graph on larger and larger scales to see where this minimum occurs. The most commonly used approach, though, is based on marginal analysis.
Think about it this way. We could imagine ”walking” along the function in the direction of increasing production. As we do this, the slope along which we climb is determined by the rate of change of the function - marginal analysis. If the marginal cost is negative, we are going downhill; this means that by increasing the production we can decrease the costs a little. If the slope is very large and negative, then we are far from the minimum cost. As we get closer to the minimum of the cost, this slope will level out. In fact, if we go too far, we could wind up increasing the costs - like climbing out of a hole. That means that we need to go back in order to decrease the cost.
This idea of walking along the function is a little hard to implement on a computer. It’s much easier to think about what the function must look like near the minimum cost. We know that on one side of the minimum, the slope is negative, because we are decreasing the cost as we increase production. On the other side of the minimum (we’ve gone too far!) the slope is positive. Now the slope is the marginal cost. This is a number associated with each value of production. If it is negative on one side of the minimum, and positive on the other side of the minimum, then we can conclude (assuming a mathematical property called continuity) that at the minimum, the slope (marginal cost) is exactly zero. This basic idea can be used to solve any optimization problem - simply set the marginal whatever to zero and solve the resulting equation.