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Lecture20-2011 - Concepts of Subadditivity Composite Cost...

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Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Subadditity of Cost Functions: Lecture XX Charles B. Moss 1 1 University of Florida November 3, 2011 Charles B. Moss Subadditity of Cost Functions: Lecture XX
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Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale 1 Concepts of Subadditivity Mathematics of the Cost Function Subadditivity and the Translog Admissible Region 2 Composite Cost Functions and Subadditivity 3 Economies of Scope 4 Economies of Scale Charles B. Moss Subadditity of Cost Functions: Lecture XX
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Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region Evans, D. S. and J. J. Heckman. 1984. A Test for Subadditivity of the Cost Function with an Application to the Bell System. American Economic Review 74, 615-23. The issue addressed in this article involves the emergence of natural monopolies. Specifically, is it possible that a single firm is the most cost-e cient way to generate the product. In the specific application, the researchers are interested in the Bell System (the phone company before it was split up). Charles B. Moss Subadditity of Cost Functions: Lecture XX
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Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region Mathematics of te Cost Function Until this lecture, we have considered cost functions of single outputs. The the cost frontier is easily extended to consider multiple outputs C ( w 1 , w 2 , y 1 , y 2 ) = α 0 + α 1 w 1 + α 2 w 2 + 1 2 α 11 w 1 w 1 + α 12 w 1 w 2 + α 22 w 2 w 2 + β 1 y 1 + β 2 y 2 + 1 2 β 11 y 1 y 1 + β 12 y 1 y 2 + 1 2 β 22 y 2 y 2 γ 11 w 1 y 1 + γ 12 w 1 y 2 + γ 21 w 2 y 1 + γ 22 w 2 y 2 (1) Charles B. Moss Subadditity of Cost Functions: Lecture XX
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Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region The cost function C ( y ) is subadditive at some output level if and only if C ( y ) < n i =1 C ( y i ) n i =1 q i = q (2) which states that the cost function is subadditive if a single firm could produce the same output for less cost. As a mathematical nicety, the point must have at least two nonzero firms. Otherwise the cost function is by definition the same. Charles B. Moss Subadditity of Cost Functions: Lecture XX
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Concepts of Subadditivity Composite Cost Functions and Subadditivity Economies of Scope Economies of Scale Mathematics of the Cost Function Subadditivity and the Translog Admissible Region Developing a formal test, Evans and Heckman assume a cost function based on two input i C ( a i y 1 , b i y 2 ) > C ( y 1 , y 2 ) i = 1 , · · · n i a i , i b i = 1 , a i 0 , b i 0 (3) Thus, each of i firms produce ai percent of output y 1 and b i percent of the output y 2 .
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