Production functions in Agriculture

Abstract

The purpose of this paper is to investigate the quadratic production functions used by D. Gale Johnson [4] in his investigation of sharecropping and by H. Scott Gordon [3] in his classic analysis of a common property resource. Both Johnson and Gordon use linear average and marginal production functions in the geometric exposition of their respective theories, but neither author has explored the unusual properties of the quadratic production function which yields a linear marginal product and average product for one factor.The first part of this paper will investigate the quadratic production function used by Johnson in his sharecropping study, and the second part of this paper will explore the similar quadratic production function employed by Gordon in a subsequent paper on the economic theory of fisheries.In his seminal study of sharecropping, Johnson finds that when one maximizes with respect to labor, the tenant will rent additional land until the marginal physical product of land is zero. He demonstrates this remarkable result in two ways. First he does so with calculus by using a neoclassical production function, and second he does so with geometry in Figure 1 of this classic paper. [4].Although the calculus is correct, his geometric example raises interesting questions. He says linear functions are shown in Figure 1 of this paper for simplicity, but the example can be generalized for any type of average product function. However, he does not explain what type of production function can generate linear marginal and average product functions, and it is the intention of this paper to do so. In particular, one can ask whether such a production function is neoclassical, or one can ask what is the nature of the marginal and average product functions for the second factor of production? Also, one can ask whether such a production function is linear homogeneous?The properties of a neoclassical production function can be found in Burmeister and Dobell [1]. They list six properties of a neoclassical production function, Y = F(K,L):