Shephard Duality

Bruce C. Dieffenbach ()
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Bruce C. Dieffenbach: Independent author

Chapter 40 in Conjugate Duality in Economic Analysis, 2026, pp 291-298 from Springer

Abstract: Abstract Shephard's book Cost and Production Functions explains and proves the equivalence between the production function and the cost function: either determines the other. Furthermore, the relationship is symmetric: the relationship of cost to product is the same as the relationship of product to cost. This chapter uses conjugate and perturbation duality to develop Shephard's ideas. We study the “classical technology with constant returns to scale.” The unit-isocost set is the set of input prices such that the cost is greater than or equal to one. Its lower boundary—where the cost is one—is the factor-price frontier, the input prices compatible with supply equilibrium with positive production. The unit-isoquant set is the set of all input quantities for which production is one or more. This chapter develops the relationship between the production function, the cost function, the unit-isoquant set, and the unit-isocost set. The relationship of the cost function to the unit-isoquant set is the same as the relationship of the production function to the unit-isocost set. Shephard introduces the distance-function representation of production and cost. The Shephard equality holds if and only if prices and quantities constitute a production equilibrium.

Date: 2026
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DOI: 10.1007/978-3-032-21396-9_40

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