Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter February 6, 2014

Replication and Returns to Scale in Production

Christian Jensen


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.

JEL Codes: D24; E23; L23


Arrow, K. J. 1962. “The Economic Implications of Learning by Doing.” Review of Economic Studies29:15573.10.2307/2295952Search in Google Scholar

Cassels, J. M. 1936. “On the Law of Variable Proportions.” In Explorations in Economics, Notes and Essays Contributed in Honor of Frank Taussig, edited by F. W.Taussig. New York and London: McGraw-Hill.Search in Google Scholar

Corless, R. M., G. H.Gonnet, D. E. G.Hare, D. J.Jeffrey, and D. E.Knuth. 1996. “On the Lambert W Function.” Advances in Computational Mathematics5:32959.10.1007/BF02124750Search in Google Scholar

Edgeworth, F. Y. 1911. “Contributions to the Theory of Railway Rates.” Economic Journal21:34671 and 55171.10.2307/2222325Search in Google Scholar

Frisch, R. 1964. Theory of Production. Dordrecht: Springer.10.1007/978-94-017-6161-1Search in Google Scholar

Hicks, J. 1939. Value and Capital: An Inquiry Into Some Fundamental Principles of Economic Theory. Oxford: Clarendon Press.Search in Google Scholar

Jensen, C. 2013. “An Endogenously Derived AK-Model of Economic Growth.” MPRA Working Paper No. 45639.Search in Google Scholar

Jones, C. I. 1999. “Growth: With or Without Scale Effects?American Economic Review89:13944.10.1257/aer.89.2.139Search in Google Scholar

Jones, C. I. 2005. “Growth and Ideas.” In Handbook of Economic Growth, edited by A.Aghion and S.Durlauf. North Holland, Amsterdam.Search in Google Scholar

Kaldor, N. 1934. “The Equilibrium of the Firm.” Economic Journal44:6076.10.2307/2224727Search in Google Scholar

Koopmans, T. C. 1957. Three Essays on the State of Economic Science. New York: McGraw-Hill.Search in Google Scholar

Lerner, A. P. 1944. The Economics of Control: Principles of Welfare Economics. New York: Macmillan.Search in Google Scholar

Lucas, R. E. 1988. “On the Mechanics of Economic Development.” Journal of Monetary Economics22:342.10.1016/0304-3932(88)90168-7Search in Google Scholar

Marshall, A. 1890. Principles of Economics: An Introductory Volume. London: Macmillan.Search in Google Scholar

Peretto, P. F., and J. J.Seater. 2013. “Factor-Eliminating Technical Change.” Journal of Monetary Economics60:45973.10.1016/j.jmoneco.2013.01.005Search in Google Scholar

Romer, P. M. 1986. “Increasing Returns and Long-Run Growth.” Journal of Political Economy94:100237.10.1086/261420Search in Google Scholar

Romer, P. M. 1990. “Endogenous Technological Change.” Journal of Political Economy90:S71102.10.1086/261725Search in Google Scholar

Romer, P. M. 1994. “The Origins of Endogenous Growth.” Journal of Economic Perspectives8:322.10.1257/jep.8.1.3Search in Google Scholar

Scarf, H. E. 1981. “Production Sets with Indivisibilities-Part I: Generalities.” Econometrica49:132.10.2307/1911124Search in Google Scholar

Scarf, H. E. 1994. “The Allocation of Resources in the Presence of Indivisibilities.” Journal of Economic Perspectives8:11128.10.1257/jep.8.4.111Search in Google Scholar

Shell, K. 1966. “Toward a Theory of Inventive Activity and Capital Accumulation.” American Economic Review56:6268.Search in Google Scholar

Solow, R. M. 1956. “A Contribution to the Theory of Economic Growth.” Quarterly Journal of Economics70:6594.10.2307/1884513Search in Google Scholar

Sydsæter, K., A.Strøm, and P.Berck. 2005. Economists’ Mathematical Manual. New York: Springer Verlag.10.1007/978-3-540-28518-2Search in Google Scholar

Wicksell, K. 1934. Lectures on Political Economy I. London: Routledge & Sons.Search in Google Scholar

Zuleta, H. 2008. “Factor Saving Innovations and Factor Income Shares.” Review of Economic Dynamics11:83651.10.1016/ in Google Scholar

  1. 1

    Solow (1956) and Lucas (1988) both assume constant returns to scale in human and physical capital jointly, however Solow assumes that human capital is not accumulated endogenously, while Lucas does. Consequently, balanced endogenous growth is feasible in Lucas’ model, but not in Solow’s.

  2. 2

    In order for the degree of specialization to vary with the scale, there must be indivisibilities in production (Edgeworth 1911;Kaldor 1934;Wicksell 1934;Lerner 1944).

  3. 3

    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.

  4. 4

    As an example let f(x)=xα with α(0,1), so that f is strictly concave. Note that while f(0)=0, Y=N1αXα becomes infinitely large as N for any total input X>0. Intuitively, when f(0)0 and f is strictly concave, its efficiency is maximized when x0.

  5. 5

    H can have multiple global maxima with respect to N.

  6. 6

    The two derivatives [9] and [20] are also equal if N˜/Xj=0, but if this held for all j=1,2,3,,q and X+q, N˜(X)=j=1qN˜/Xj×Xj=0 for all X+q, since N˜(X) is homogeneous of degree one, and production would always be zero.

  7. 7

    While N˜(X) must satisfy H/N=0, 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.

  8. 8

    Wicksell (1934) suggests that production processes are generally S-shaped. Frisch (1964) labels it a regular ultra-passum law of production, while Cassels (1936) refers to it as the law of variable proportions, both suggesting its universality.

  9. 9

    The Lambert W function, also called the Omega function, or the product logarithm, is the inverse relation of the function gW=WeW. 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 W0 and a lower branch denoted W1. The second-order condition for maximizing total output [22] with respect to N reveals that we must use the lower branch, which yields a maximum, since the upper branch yields a minimum.

  10. 10

    For b<2, W(e1b) and GX are complex-valued.

  11. 11

    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.

  12. 12

    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.

  13. 13

    For example, when f(X/N,Z)=Z×h(X/N), where Z is scalar, replication leads to constant returns to scale in X. Returns to scale in Z are by assumption constant, so the joint returns to X and Z are increasing.

Published Online: 2014-2-6
Published in Print: 2014-1-1

©2014 by De Gruyter

Scroll Up Arrow