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Lecture 20-2005 - Subadditivity of Cost Functions Lecture...

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Subadditivity of Cost Functions Lecture XX I. Concepts of Subadditivity Evans, D. S. and J. J. Heckman. “A Test for Subadditivity of the Cost Function with an Application to the Bell System.” American Economic Review 74(1984): 615-23. A. The issue addressed in this article involves the emergence of natural monopolies. Specifically, is it possible that a single firm is the most cost- efficient way to generate the product. B. In the specific application, the researchers are interested in the Bell System (the phone company before it was split up). C. Basic concept: 1. The cost function ( ) C q is subadditive at some output level if and only if: ( ) ( ) 1 1 n i i n i i C q C q q q = = < = % % % % which states that the cost function is subadditive if a single firm could produce the same output for less cost. 2. As a mathematical nicety, the point must have at least two nonzero firms. Otherwise the cost function is by definition the same. D. Developing a formal test, Evans and Heckman assume a cost function based on two input: ( ) ( ) 1 2 1 2 , , 1 1 0 0 i i i i i i i i i C a q b q C q q i n a b a b > = = = % % % % L 1, Thus, each of firms produce percent of output and percent of the output . i i a 1 q i b 2 q 1. A primary focus of the article is the region over which subadditivity is tested. a. If the sum of the disaggregated firm’s cost functions are greater than the cost of the aggregated firm, ( ) ( ) 1 2 1 2 , , i i i C a q b q C q q > % % % % The cost function is subadditive, and the technology implies a natural monopoly. b. If the sum of the disaggregated firm’s cost functions are less than the cost of the aggregated firm, ( ) ( ) 1 2 1 2 , , i i i C a q b q C q q < % % % % the cost function is superadditive, and the firm could save money by breaking itself up into two or more divisions. 1
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AEB 6814 – Production Economics Lecture XX Professor Charles Moss Fall 2005 c. Finally, if the sum of the disaggregated cost functions is equal to the cost of the aggregated firm, ( ) ( ) 1 2 1 2 , , i i i C a q b q C q q = % % % % the cost function is additive.
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