Vertical integration in perfect competition?

Over at the SSRN website there is a new working paper out on the question Do Prices Determine Vertical Integration? Evidence from Trade Policy by Laura Alfaro, Paola Conconi, Harald Fadinger and Andrew F. Newman. The abstract of the paper reads:

What is the relationship between product prices and vertical integration? While the literature has focused on how integration affects prices, this paper shows that prices can affect integration. Many theories in organizational economics and industrial organization posit that integration, while costly, increases productivity. If true, it follows from firms' maximizing behavior that higher prices cause firms to choose more integration. The reason is that at low prices, increases in revenue resulting from enhanced productivity are too small to justify the cost, whereas at higher prices, the revenue benefit exceeds the cost. Trade policy provides a source of exogenous price variation to assess the validity of this prediction: higher tariffs should lead to higher prices and therefore to more integration. We construct firm-level indices of vertical integration for a large set of countries and industries and exploit cross-section and time-series variation in import tariffs to examine their impact on firm boundaries. Our empirical results provide strong support for the view that output prices are a key determinant of vertical integration.

How if I'm getting this right the argument is that under perfect competition, if integration increases productivity, then a price-taking firm will integrate more when the price of its product is higher. At a low price, the increment in revenue resulting from integration is too small to justify any fixed cost integration may incur but at a high price, the incremental revenue is large enough to make integration worthwhile.

My issue with this is that we can't give meaning to integration within perfect competition.

As Nicolai Foss has noted

With perfect and costless contracting, it is hard to see room for anything resembling firms (even one-person firms), since consumers could contract directly with owners of factor services and wouldn't need the services of the intermediaries known as firms.

So if consumers can do it all and there is no need for firms then you have to ask, How can nonexistent firms integrate with one another?

In the neoclassical theory, the firm is a 'black box' there to explain how changes in inputs lead to changes in outputs. It is a black box in the sense that inputs go in and outputs come out, without any explanation of how one gets turned into the other. The firm is taken as given; no attention is paid to how it came into existence, the nature of its internal organisation, where the boundary between one firm and another is or between a firm and the market; or whether anything would change if two firms merged and called themselves a single firm, i.e. integration. The neoclassical production function is a way of representing the black box conversion of inputs into outputs but tells us little about the inner workings of the black box. The production function is independent of the institutional framework of output creation. Thus it represents the 'firm' without explaining the 'firm'.

That the theory cannot explain the boundaries of the firm - that is, explain integration - has been noted by several authors including the Nobel winner Oliver Williamson. He asks,

What determines which activities a firm chooses to do for itself and which it procures from others?

A simple answer to that question is that the natural boundaries of the firm are defined technology-economies of scale, technological nonseparabilities, and the like. The firm-as-production function is in this tradition. [ ... ] In mundane terms, the issue is that of make-or-buy. What is it that determines which transactions are executed how?

That posed a deep puzzle for which the firm-as-production function approach had little to contribute.

So I'm a little confused as to how the authors of the paper can discuss integration within perfect competition.