Defining Economies of Scope

I've previously written posts on the idea of Economies of Scale versus diminishing returns (both describe what happens as you enlarge your operations by adding more factors of production, but in seemingly contradictory ways). As an interesting aside: In that post, I had accidentally introduced a concept we are currently discussing now: the idea of the Leontief production function (which assumes that inputs are used in fixed proportions or in other words the idea of bottlenecks or "limiting reagents" in production factors). And it seems that economies of scale are quite easy to visualize (especially when you bring up ideas like Ford's Model T assembly line).

Economies of SCOPE, however, are a little more difficult to visualize in my opinion. After all, it may not be implicitly obvious how producing an additional good can provide new efficiencies. While this is true, our professor showed us a great example in class, but seemed to go over it very quickly. It wasn't until I was reviewing the material that I seemed to get the full impact. It all stems from the formula below which we'll look at:

C(Q1, Q2) = f + aQ1Q2 + Q1^2 + Q2^2

C(Q1, Q2) is the cost function for any given production quantities of good 1 and 2 at quantities Q1 and Q2.
The 'a' term describes cost complementarity. If a is negative the aQ1Q2 represents a cost savings between the goods.

Let's look at the two possible options:
Option 1: Produce both goods together. This is represented by the original cost function:
C(Q1, Q2) = f + aQ1Q2 + Q1^2 + Q2^2

Option 2: Produce the goods separately. This is represented by:
C(Q1, 0) + C(0, Q2)
= [f + Q1^2] + [f + Q2^2]
= 2f + Q1^2 + Q2^2

How can we describe economies of scope? Well it is the difference in cost between option 1 and option 2 or described as:
C(Q1, 0) + C(0, Q2) - C(Q1, Q2)
= 2f + Q1^2 + Q2^2 - [f + aQ1Q2 + Q1^2 + Q2^2]
= f - aQ1Q2

In order for economies of scope, the formula above must be greater than 0:
f - aQ1Q2 > 0
f > aQ1Q2

Now here is the trick which was on one of our sample quizzes (and I would anticipate on our upcoming quiz):
If you have cost complementarity (a is negative), you will ALWAYS have economies of scale. Q1, Q2 and f are all positive quantities, so the right side will be negative (less than) the positive left side.

However, if you do NOT have cost complementarity (a is positive) you MIGHT have economies of scale depending on the relative values of Q1, Q2 and f.

Not having cost complementarity actually means that for each additional unit of Q1, the cost for Q2 actually becomes more expensive. However, if that marginal inefficiency is less than the inefficiency of reproducing the fixed cost 'f', then it is still considered economies of scale (because as a whole, the system can still produce units of each cheaper in one factory than in two).

Now that I think about it and extend the idea, I wonder if you can express economies of scale in a similar fashion with the same formula. Let's focus on an example with just one good:
C(Q1,0) - let's re-label this as C(Q) = f + Q^2
if the Q^2 term became sufficiently large (which can happen quite quickly with squared terms) it might be worth while to encounter another 'f' cost to reset your Q term. What do I mean? Because of the second order relationship of the cost function, there is an economy of scale to be determined when Q becomes sufficiently large. How can we generalize this?
C(Q) = 2 * C(Q/2)
f + Q^2 = 2f + (Q/2)^2
(3/4)Q^2 = f
Q^2 = 4f/3
Q = (4f/3)^(1/2)

While not an entirely elegant solution, I think it does a decent job of describing economies of scale. This also describes a slightly non-standard cost curve as you will have small diseconomies of scale as Q increases, but at some point, it becomes more efficient to add more capacity.

As an undergrad, the idea of economies of scope had always bothered me, as professors couldn't give a concrete example of how the mechanics would apply (especially not without this formula).