Returns to Scale in One-Factor Production Functions
This Demonstration helps in visualizing relationships between such concepts as returns to scale, economies of scale and marginal factor productivity in the extremely simple setting of only one factor. Five easy tractable archetypical functions are considered, and it is shown why only two of them are often used, that is, a function of the type is used in the context of monopolistic markets (including monopolistic competition or natural monopoly) and a function of the type is used in the context of classical monopoly and (or) classical competition. The "intercept" parameter can only be adjusted for the "" and "" cases of the function . There are five cases for . The first and fifth cases correspond to diminishing returns, the second and fourth correspond to increasing returns and the third corresponds to constant returns.[more]
Labels on the plots are as follows: is the production output, is the total cost, is the average cost, is the marginal product of labor and is labor, which is the only factor used.[less]
Five forms of one-factor production functions are considered to study the returns-to-scale concept. Returns to scale are described as follows (for ):
increasing returns: ;
constant returns: ;
diminishing returns: .
Returns to scale are defined in natural terms. When it comes to nominal terms, the term "economies of scale" is used. When factor markets are competitive (meaning that wages, —the payment for a production factor—are given), there is a direct relationship between returns to scale and economies of scale. Increasing returns correspond to the downward part of the curve while diminishing returns correspond to diseconomies of scale described by the upward parts of the curve . , so we need . In this simple world, total cost is just , where can be expressed from .
For example, in , Krugman's famous model of increasing returns in monopolistic competition uses the function [1, p. 471], which is just an inverse production function . Can a function of the type be used instead (formally it also models increasing returns)? The problem is that it has increasing marginal product of labor , which is an unrealistic assumption (the law of diminishing marginal productivity is violated). The plot that alerts the issue is colored red.
On the other hand, we cannot use increasing returns production functions in the context of perfect competition. To see why that is so, consider the curve of economies of scale that always lie above marginal costs ( approaches the horizontal from above). Given that the competitive price is always , all but one of the producers are forced to quit the market, leaving room for an imperfect market. Diminishing returns of the form have to be used, which is a typical price-taking setting; for example, it leads to the individual producer's profit maximization problem , where . At the same time, the linear diminishing returns function results in an infeasible partly negative curve (which is also colored red).
 P. R. Krugman, "Increasing Returns, Monopolistic Competition, and International Trade," Journal of International Economics, 9(4), 1979 pp. 469–479. doi:10.1016/0022-1996(79)90017-5.