Aren't all cost functions step functions?

Question

Long Question:

Overview:

While I understand that over certain intervals (i.e. less than the width of the step), they may appear not to be because the step is outside of consideration, and over other intervals (much, much larger) the step seems continuous as opposed to discrete, but at the end of the day aren't these merely artifacts of the analysis, as opposed to properties of the underlying function?

Admittedly, I was never much good with the calculus of discrete functions, but I remember there being some substantial differences between their derivatives (e.g. the need to invoke the finite difference method as opposed to a limit).

Further, given the potential steepness of the step, and its unknown location, isn't predicting difficult?

A Picture:

For example, the graph below shows a cost function for $n$ units with and without a step.

The red line is:

$$C(n)=5n$$

and the blue line is $$C(n) = n+(n-\mod (n,5))$$

Analyzed over $(0,5)$ these would coincide, and an analyst might not have any expectation that increasing by a small factor would dramatically increase cost, and while the (simple) derivative at any point on the line is the same for either line (inside the step interval), this is very misleading.

Further, I chose $5$ at random, a more generalized idea might be something along the lines of:

$$C(n) = n+\frac{S(n-\mod(n,w))}{w}$$

where $S$ is the step cost, and $w$ is the width.

Summary:

Given these concepts, why are step cost functions generally treated as less preferential in discussing models? Or is that just my experience, and not indicative of the way the field treats cost functions as a whole?

I ask this, because whenever I see a discussion of a cost function, it seems to be continuous, and whenever I see mention of a step function it seems to be an afterthought.

Answer 1

Analyzed over (0,5) these would coincide, and an analyst might not have any expectation that increasing by a small factor would dramatically increase cost, and while the (simple) derivative at any point on the line is the same for either line (inside the step interval), this is very misleading.

Good question, but mainly of theoretical validity. The "analyst" in question would most probably have some indications of the cost for most if not all production scales of interest. There are few cases when a production plan is about producing on a scale no-one has seen before (not only the specific company, but any company). So data to compare and get an idea will be around, and also, as a last resort, "engineering" data can, and do, assist also (and engineering is much more of an exact science than Economics is).

Firms' practitioners have a rather clear view of "Fixed Costs" that are not really Fixed but step cost functions, trust my corporate experience on this one. They "sense" when the step is about to be taken -and they sit down and calculate it. The whole Cost-Accounting business is essentially about learning how to detect, separate and handle Fixed and Variable Costs, as well as Variable and Marginal Costs (in the real world, the distinction between Variable and Marginal Cost is very existing, and very important).

In the case of aggregate/average cost functions, the individuality of each firm alongside the common structural base that exists in all production activities, helps in smoothing the relationship.