My textbook says the following:
"Perfectly competitive markets only achieve productive efficiency if you assume that there are no economies of scale in the industry."
Why is this the case? And by "in the industry" does it mean external economies of scale?
Put simply, a competitive market is one with a larger number of smaller firms — but economies of scale are best exploited by a smaller number of larger firms.
This doesn’t mean that a competitive market can’t take advantage of economies of scale, just that such a market is only guaranteed to reach productive efficiency if the size of the firm isn’t a factor.
Adding to Bill Clark's answer, imagine that in addition to entry/exit decisions, you add to your model the decision to merge with another firm. When economies of scale are present, there are circumstances in which two companies that are considering exit might instead choose to merge, yielding a firm that has a relative advantage in terms of marginal costs.
As soon as this happens, the equilibrium condition for a perfectly competitive market vanishes (entry and exit until demand is satisfied by a market of firms for which MC = p). A simple way to see this work would be to set up a Bertrand price competition model with three firms: one whose marginal cost is lower than the other two. In addition to the entry/exit decision, allow for a merger decision if the resulting combined marginal cost function results in winning the Bertrand game (which it may do if economies of scale are present). You'll be able to show that there are conditions for which the two inferior companies find it optimal to merge rather than exit, capturing the whole market.
Q1: In the long run, firms operate at the lowest cost point on LRAC curve in perfectly competitive market. That's to say that all scale economies are competed away. Thus, perfectly competitive firms produce at the lowest cost point (on LRAC) which results in productive efficiency.
If there are scale economies to make, firms have yet to achieve productive efficiency by undercutting competition.
Q2: Yes, there are no internal/external economies of scale.
The story of perfect competition is that companies achieve some "peak efficiency" (i.e. lowest LRAC) and the market demand is satisfied by firm entry until all demand is met at this lowest cost point. This story doesn't make sense in an economies of scale situation since economies of scale are defined by declining marginal cost (and hence average cost).
Formally this can be seen with an $n$ player Cournot game with various cost structures (linear gives perfect competition, what happens if you take square root cost?) and letting $n \to \infty$ to study competition in this structure.
