Why would we want to use average cost as response but not total cost? Assume I have a data set as follows:
Total_Cost  Average_Cost  Quantity
    12           6            2
    21           7            3
    28           7            4
    30           6            5
    36           6            6

The linear relationship between Total_Cost and Quantity is nearly perfect, while between Average_Cost and Quantity is very poor. However, I saw many researches that basically use average cost as a response but not total cost.
My question here is therefore: Are there any particular reasons for us to study the linear relationship between Average_Cost and whatever independent variables? (For instance, the use of Total_Cost as a response is way too straightforward.)
 A: Here one's example. Suppose you run a regression of $AC$ on $Q$ and you find that the coefficient is negative (not the case with your data). This tells you average cost falls with output, so your technology has returns to scale. If all firms have similar technology, it would be more efficient to have only one firm serve the market. This would be an example of a natural monopoly.
A: I think you're leaving out another part of the story, which I believe is that your data is a time series: what you pay for an item over time.
Your Total_Cost reflects two things: how many items you buy in each time period (Quantity), and how much they cost (Average_Cost). By breaking it into those two pieces, you can better track what it is you care about.
You'd never look at the relationship between Average_Cost and Quantity. You'd be looking at Average_Cost versus other variables (including time). You could of course try to look at Quantity versus other things, but it could be that Quantity is a "nuisance" parameter: something that you want to factor out of your analysis.
I'd also add that Average_Cost is better named simply Cost in this example. It'd be Average_Cost if you averaged across multiple time purchases or if you were purchasing different kinds of things. As it is, you're taking the total bill (Total_Cost) and using Quantity to derive actual cost per thing.
