Cost structure of a firm

This page describes some general aspects of the theory of the cost structure of a firm. For simplicity, we make the following assumptions:

We consider a firm producing a single good.
We ignore the question of whether the firm can store excess stock and thereby have some way of recovering costs if it overproduces; our analysis is related to production at a given stage in time.
We consider only a two-level distinction between the short run (where some types of costs, called fixed costs, cannot be changed) and the long run (where all costs can be freely chosen). In practice, different cost decisions have different time lags and timeframes, but the simplistic short versus long run model will suffice for illustration.

We will consider the following constructs:

The production function of the firm, which describes the firm's quantity produced as a function of the costs incurred by the factors of production.
The optimized production function of the firm, which describes the firm's quantity produced as a function of total cost incurred assuming that the cost is allocated optimally between the various factors of production to maximize the quantity produced. We can distinguish here between the short-run optimized production function (this assumes that the fixed costs cannot be changed, and only allows for variation in the variable part of the costs) and the long-run optimized production function.
The total cost function. This is the inverse function to the optimized production function. Explicitly, it describes the cost of producing a given quantity as a function of quantity. We can distinguish between the short-run total cost function (the inverse to the short-run optimized production function) and the long-run total cost function (the inverse of the long-run optimized production function).
The variable cost function (plotted only for short-run analysis). Given a fixed cost scenario (i.e., a particular choice of allocation of fixed costs that cannot be changed in the short run), the variable cost function is the function that describes the variable cost of producing a given quantity as a function of quantity. It is obtained from the short-run total cost function by subtracting the (constant) fixed cost from it.
The average total cost and average variable cost functions are obtained by dividing the total cost and variable cost functions respectively by the quantity produced. Note that there are two versions of the average total cost function: short-run and long-run.
The short-run marginal cost function for a given fixed cost scenario is defined as the derivative of the short-run average total cost function for that scenario , or equivalently as the derivative of the (short-run) variable cost function for that scenario.
The long-run marginal cost function is defined as the derivative of the long-run average total cost function.