Why do firms adjust their fixed costs in response to a change in price in a perfectly competitive market?

Question

When firms make a profit in a perfectly competitive market, new firms enter the market and drive the price down. My textbook says that the existing firms will then adjust inputs that are fixed in the short term, such as capital, in order to operate with a lower average cost:

This is in order to maximize profit, but I don't understand why the short-run average cost curve would change at all. In my opinion, the minimum point of the long-run average cost curve, where there are constant returns to scale, is the profit-maximizing point in the long run. In order to substantiate this claim, I came up with a short-run total cost function

$$TC=0.001Q^3-0.09Q^2+2.95Q+20$$

and derived the average total cost

$$ATC=0.001Q^2-0.09Q+2.95+20/Q$$

and marginal cost

$$MC=0.003Q^2-0.18Q+2.95$$

Assume that the price is variable. Then profit

$$\Pi=Q(MC-ATC)=0.002Q^3-0.09Q^2-20$$

can be thought of as a function that relies only on marginal cost and average cost, because at the profit-maximizing point, $MC=MR=P$. But as $Q\to \infty$, $\Pi \to \infty$ because $\Pi$ is a cubic equation with a positive coefficient on the $Q^3$ term. This means that the higher the price is above the minimum point of the short-run average total cost curve, the higher profit will be. Thus, profit is maximized when the lowest possible short-run average total cost curve is selected, in any perfectly competitive market condition, at least for cubic total cost curves (and I assume that the cubic function is the most applicable for a total cost curve). The lowest ATC curve is obviously the one where there are constant returns to scale.

You can see the difference between marginal cost and average total cost getting larger and larger here:

Answer 1

I think you're right - entry should not shift the SRAC in a normal model of a market. The textbook pages that you include read very oddly to me. (It is possible that this is because the context being described is somehow very specific, so I may be unfair in the criticism that follows).

A potential entrant compares price to min of ATC and sees that there is a profit to be made. It enters and produces the quantity such that it achieves min ATC. What does this do at the level of the market? At the clearing price before entry, there is now excess supply equal to the output of the new firm. Somehow ... and introductory models of perfect competition and price formation are usually silent on exactly how this happens ... the price must fall for there not to be unsold goods in the market.

There are lots of unsaid things in the above story. What do other firms do in response to the entry? How exactly are prices formed. One natural answer is Bertrand price competition: the new firm undercuts the clearing price; all customers switch to it; it supplies the whole market ... upon which other firms react by undercutting it ... until undercutting no longer makes sense because each firm is at the min of the ATC. Again, there is lots here that is unsaid (are fixed costs sunk or not? what determines capacity constraints ...)

But in a way, the root of the problem with the story told in your textbook is the upward sloping aggregate supply curve. How is this compatible with a micro foundation of perfect competition? If each firm can produce according to the ATC function given, then the aggregate supply curve is actually flat. Under the assumptions of perfect competition, you can enter without constraint. So it is always possible and profitable to produce at the min ATC. So the aggregate supply curve is not upward sloping. I think the text book is trying to bring together two models that don't really fit coherently, and you have picked up the inconsistency involved.

So here's what I think about your question and the answer you have posed: 1. you're right & the book's wrong 2. don't take economic models too literally as representations of reality.