Proceedings of World Business and Economics Research Conference
24 - 25 February, 2014, Rendezvous Hotel, Auckland, New Zealand, ISBN: 978-1-922069-45-0
Production and Cost Duality: A New Look
Theodore J. Osborne*
The derivation of the long run cost function and the connections between the characteristics of the
production function and the characteristics of this cost function are relatively complex. Due to this
complexity, duality analysis, which includes the mathematical derivation of the cost function from the
production function as well as the role of returns to scale in those functions, is usually reserved for
presentation in advanced undergraduate and graduate microeconomic theory courses. It is unfortunate
that this analysis is not normally presented in second year courses since it significantly enhances
understanding of all further applications of microeconomic theory. A set of simpler tools specifically for
teaching this analysis, which are easy to present and understand, are clearly needed. This paper
presents just such a set of tools.
The two main elements of this approach are the “scale product curve” and a method, developed in
this paper, for deriving a cost function from a production function without calculus. The scale product
curve, which plots output against scale, shows the long run characteristics of the production function
much as the total product curve does for short run characteristics (Cole (1973), Eaton and Eaton
(1991)). It is used in this paper in order to more fully and clearly examine the connections between
returns to scale and the characteristics of costs. These two elements are then applied to three
production functions—two Cobb-Douglas forms (with constant and increasing returns to scale) and a
relatively simple CES. These three are then combined into one function using their scale product curves.
We refer to this new production function as the Merged CES production function (or MCES). The long
run cost function associated with the MCES is then found by factoring the production function (no
calculus or algebra required!). The scale product, average and marginal products of scale and long run
average and marginal costs curves of this production function are derived and then the relationship
between these curves explored—i.e. an example of duality analysis. In their explanation of the
connection between returns to scale, the production function and costs, virtually all textbook authors
assume that the production function is homothetic which gives the impression that special cases are
general principles. Since the MCES is not homothetic, the conclusions drawn from the duality analysis
presented in this paper are much more general.
*
Department of Economics, King’s University College at Western University, 266 Epworth Ave., London, Ontario, N6A
2M3. (519-433-3491 ext. 4346), email: tosborne@uwo.ca,