In the following Industrial organization exercise form Church and Ware chapter 4 exercise 2
(preliminary info: C(q) = cost function; f = fixed costs; c = marginal costs; q= quantity of goods produced)
2.) Suppose that the cost function for both the entrant and an incumbent is $C(q) = f + cq$
(a) Is this technology characterized by economies of scale? What is marginal cost and how does it compare to average cost?
(b) Suppose that postentry the incumbent can commit to charge p = c. Will there be entry? Does it matter whether f is sunk or not?*
my professor's solution for question (a) was just stating "since the first derivative of AC is negative $-\frac{f}{q^2}$ the technology is characterised by economies of scale"
Could anyone argument on that? I got to the same conclusion by just looking at average cost function $AC=\frac{f}q+c$, where as quantity grows, fixed cost diminish until the whole function becomes almost flat and ≈ c
What I tried to do:
I graphed the derivative, it's a parabola facing upward that goes to plus infinity around y = 0 and plus infinity around x = 0 (we are only interested in the first quadrant values since P and Q negative don't make sense) this is because f is a negative number (being a cost). But still I don't understand the relevance of the sign, what I feel is relevant is the flattening of the function as q grows, this is what determines economies of scale for what I know.
Thanks to anyone helping out.