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CHAPTER 10 COST FUNCTIONS The problems in this chapter focus mainly on the relationship between production and cost functions. Most of the examples developed are based on the Cobb-Douglas function (or its CES generalization) although a few of the easier ones employ a fixed proportions assumption. Two of the problems (10.7 and 10.8) make use of Shephard's Lemma since it is in describing the relationship between cost functions and (contingent) input demand that this envelope-type result is most often encountered. The analytical problems in this chapter focus on various elasticity concepts, including the introduction of the Allen elasticity measures. Comments on Problems 10.1 Famous example of Viner's draftsman. This may be used for historical interest or as a way of stressing the tangencies inherent in envelope relationships. 10.2 An introduction to the concept of “economies of scope”. This problem illustrates the connection between that concept and the notion of increasing returns to scale. 10.3 A simplified numerical Cobb-Douglas example in which one of the inputs is held fixed. 10.4 A fixed proportion example. The very easy algebra in this problem may help to solidify basic concepts. 10.5 This problem derives cost concepts for the Cobb-Douglas production function with one fixed input. Most of the calculations are very simple. Later parts of the problem illustrate the envelope notion with cost curves. 10.6 Another example based on the Cobb-Douglas with fixed capital. Shows that in order to minimize costs, marginal costs must be equal at each production facility. Might discuss how this principle is applied in practice by, say, electric companies with multiple generating facilities. 10.7 This problem focuses on the Cobb-Douglas cost function and shows, in a simple way, how underlying production functions can be recovered from cost functions. 10.8 This problem shows how contingent input demand functions can be calculated in the CES case. It also shows how the production function can be recovered in such cases. This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 78
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79 Chapter 10: Cost Functions Analytical Problems 10.9 Generalizing the CES cost function. Shows that the simple CES functions used in the chapter can easily be generalized using distributional weights. 10.10 Input demand elasticities. Develops some simple input demand elasticity concepts in connection with the firm’s contingent input demand functions (this is demand with no output effects). 10.11 The elasticity of substitution and input demand elasticities. Ties together the concepts of input demand elasticities and the (Morishima) partial elasticity of substitution concept developed in the chapter. A principle result is that the definition is not symmetric.
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