What is the practical use of knowing a firms average cost curve?
The reason why I ask is because computational problems that a firm faces are either with regards profit maximization or cost minimization and average cost curves are never used in computations.
If I can refine the question, what economic theory is derived from the average cost curve that are not given by marginal cost curve?
Assuming a firm is a perfect competitor in input markets, the long-run average cost curve, which traces out the minimums of short-run average cost curves, can be used to characterize economies and diseconomies of scale for a firm. This is definitely a practical concept in business and industrial organization. In an industry where the long-run average cost curve is always decreasing, a natural monopoly will tend to emerge due to the increasing cost-efficiency of larger firms.
In a monopsony, the employing firm's average labor cost curve is equal to the supply of labor in the labor market. The marginal cost curve lies above the supply curve in this market because the firm's market power allows it to expressly set wages as a function of the quantity of labor. One reason this is practical is that it offers a potential theoretical explanation for why we may not see significant disemployment effects, or may even see an increase in employment, from establishing minimum wages in particular labor markets.
Short answer: whether a firm's maximum 'profit' at the point where marginal revenue equals marginal cost is positive or negative depends on whether average revenue exceeds average cost at that point.
To give a simple numerical example, suppose for a firm in a perfectly competitive market:
$$MR = AR = 1$$
$$TC = 2 + 0.4Q + 0.1Q^2$$
where $MR$ is marginal revenue, $AR$ is average revenue, $TC$ is total cost and $Q$ is quantity produced and sold (for a defined period). Then marginal cost $MC$ is:
$$MC = 0.4 + 0.2Q$$
and $MC = MR$ when:
$$0.4+0.2Q=1$$
implying $Q = 3$.
At that point, however, average cost $AC$ is:
$$AC = \frac{TC}{Q} = \frac{2 + 0.4(3)+ 0.1(3^2)}{3} = \frac{4.1}{3} \approx 1.37 > 1 = AR$$
So the firm would make a loss of $4.1 - 3 = 1.1$ by producing at $Q=3$. In this case 'profit' at the point identified by equating $MC$ and $MR$ is only a local maximum within the range $Q > 0$. Note for example that the loss at $Q=2$ is $3.2-2=1.2$ and that at $Q=4$ is $5.2-4=1.2$. The global maximum 'profit' within the range $Q\geq0$ is zero at $Q=0$, so (considering the period in isolation), the firm would do better not to produce at all.
