Rothbard on the size of firms

I have been rereading Murray N. Rothbard's essay "Ludwig von Mises and Economic Calculation Under Socialism". In this Rothbard argues that Mises's argument about the impossibility of socialism can be applied to the problem of the size of firms. Rothbard argues that

There is one vital but neglected area where the Mises analysis of economic calculation needs to be expanded. For in a profound sense, the theory is not about socialism at all! Instead, it applies to any situation where one group has acquired control of the means of production over a large area - or, in a strict sense, throughout the world. On this particular aspect of socialism, it doesn't matter whether this unitary control has come about through the coercive expropriation brought about by socialism or by voluntary processes on the free market.

In other words it can apply to a firm.

Rothbard continues

[ ... ] Mises analysis also supplies us the answer to the age-old criticism leveled at the unhampered, unregulated free-market economy: what if all firms banded together into one big firm that would exercise a monopoly over the economy equivalent to socialism? The answer would be that such a firm could not calculate because of the absence of a market, and therefore that it would suffer grave losses and dislocations. Hence, while a Socialist Planning Board need not worry about losses that would be made up by the taxpayer, One Big Firm would soon find itself suffering severe losses and would therefore disintegrate under this pressure. We might extend this analysis even further. For it seems to follow that, as we approach One Big Firm on the market, as mergers begin to eliminate capital goods markets in industry after industry, these calculation problems will begin to appear, albeit not as catastrophically as under full monopoly. In the same way the Soviet Union suffers calculation problems, albeit not so severe as would be the case were the entire world to be absorbed into the Soviet Union with the disappearance of the world market. If, then, calculation problems begin to arise as markets disappear, this places a free-market limit, not simply on One Big Firm, but even on partial monopolies that eradicate markets. Hence, the free market contains within itself a built-in mechanism limiting the relative size of firms in order to preserve markets throughout the economy.

Rothbard then notes that this argument is related to Coase's argument about the size of firms.

This point also serves to extend the notable analysis of Professor Coase on the market determinants of the size of the firm, or of the relative extent of corporate planning within the firm as against the use of exchange and the price mechanism. Coase pointed out that there are diminishing benefits and increasing costs to each of these two alternatives, resulting, as he put it, in ah " 'optimum' amount of planning" in the free market system. Our thesis adds that the costs of internal corporate planning become prohibitive as soon as markets for capital goods begin to disappear, so that the free market optimum will always stop well short not only of One Big Firm throughout the world market but also of any disappearance of specific markets and hence of economic calculation in that product or resource. Coase stated that the important difference between planning under socialism and within business firms on the free market is that the former "is imposed on industry while firms arise voluntarily because they represent a more efficient method of organizing production." if our view is correct, then, this optimal free-market degree of planning also contains within itself a built-in safeguard against eliminating markets, which are so vital to economic calculation

While Rothbard makes a telling point about the limits of monopoly in free markets, the issue of the size of the firm that Coase was interested in, I would argue, was what determined the size of the firm well before the firm gets to the point where it takes over entire input markets. There must be other factors limiting the size of the firm that become relevant before it gets to the point of controlling its input markets.