Replication alone does not yield a smooth constant-returns-to-scale production function as those usually assumed in the literature. However, such a function arises endogenously with replication, driven by profit-maximization, if the efficiency of the underlying production process varies with the intensity it is operated at, and reaches a maximum at a stationary point. The result applies when the number of production processes must be discrete, thus overcoming the so-called integer problem. When inputs are non-rival, public goods or generated by externalities, replication can lead to increasing or decreasing returns to scale.
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That the managerial input can generate decreasing returns to scale is an old argument (Marshall 1890;Kaldor 1934;Hicks 1939). The idea is that even when management increases proportionally with all other inputs, it becomes overstretched due to the more than proportional complexity of the organization. The same applies for communication.
As an example let with , so that f is strictly concave. Note that while , becomes infinitely large as for any total input . Intuitively, when and f is strictly concave, its efficiency is maximized when .
H can have multiple global maxima with respect to N.
The two derivatives  and  are also equal if , but if this held for all and , for all , since is homogeneous of degree one, and production would always be zero.
While must satisfy , it does not need to be a global maximum of H, any stationary point will make the production function with a discrete number of replications converge toward constant returns to scale.
The Lambert W function, also called the Omega function, or the product logarithm, is the inverse relation of the function . It has no representation in terms of elementary functions, but can be approximated numerically, as discussed in Corless et al. (1996). It is a multivalued relation, and thus not really a function, with an upper (principal) branch denoted and a lower branch denoted . The second-order condition for maximizing total output  with respect to N reveals that we must use the lower branch, which yields a maximum, since the upper branch yields a minimum.
For , and are complex-valued.
Assuming a continuously differentiable profit function maximized at an interior point characterized by first-order conditions, we would have that for the last unit of output produced by increased input use, the marginal revenue equals the marginal cost, which is assumed to be strictly positive. However, if the unit could instead have been produced without increased input use, marginal profits would have equaled marginal revenue, and been positive, so profits cannot be maximized. Note that with discrete N, total output, and therefore profits, may not be continuously differentiable at all input levels.
This is in line with Romer (1990, 1994) and Jones’ (2005) point that perfect competition is incompatible with increasing returns to scale, though their argument relies on a continuously differentiable production function. In our setup, the result applies also to decreasing returns, since these become increasing with a large enough increase in the scale. If the returns to scale of the production function were decreasing for all levels of production, decreasing returns would be compatible with perfect competition.
For example, when , where is scalar, replication leads to constant returns to scale in . Returns to scale in are by assumption constant, so the joint returns to and are increasing.
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