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Unformatted text preview: Nice to know about CES functions (preliminary) Ferdinand Mittermaier Department of Economics, University of Munich, D-80799 Munich (Germany), Phone: +49 89 2180 6752, E-mail: firstname.lastname@example.org This set of derivations was prepared for the tutorial to the Ph.D./Advanced Master level Economics course Advanced Theory of Taxation at LMU Munich in the summer semester of 2008. It compiles important and very basic features and derivarions that one may find useful when working with constant elasticity of substitution (CES) functions. These are widely used in models of monopolistic competition, macro, new trade and agglomeration/new economic geography, inter alia. 1 Introduction Just a few notes and caveats: This is work in progress, far from being exhaustive and may still contain errors (which I encourage you to point out to me). I hope this fact box proves useful in two ways: For beginners with CES-functions/Dixit-Stiglitz- preferences models and for those who are experienced readers and are interested in basic derivations which never show up in published work. I draw heavily on Chiang/Wainwright (2005), Baldwin et al. (2003) and Fujita et al. (1999). All errors are mine. 1 2 Functional form In models using a CES production or utility function, the basic functional form is always F = A 1 1 x - 1 1 + 1 2 x - 1 2 - 1 (1) or similar, where A is an efficiency parameter which is dropped in the following for simplicity as I focus on utility function (monotone transformation); the s serve as distribution parameters [cf. exponents in Cobb-Douglas (C-D) functions!]; and the x s are the actual variables of interest [e.g., goods produced or consumed]. has no logical counterpart in the C-D function. It is the substitution parameter giving the CES func- tion its name. 1 At least for the utility representation, one may wonder what the outer exponent is needed for. The reason is little surprising: It renders the function linearly homogeneous (this is easily demonstrated: Multiplying x 1 and x 2 by , F also grows -fold.) 3 Elasticity of substitution First of all, it is worthwhile deriving that the elasticity of substitution is indeed con- stant. Let us work with the following utility function for the time being: U = h 1 1 x - 1 1 + 1 2 x - 1 2 i - 1 (2) This leads to the following partial derivatives: U x 1 = - 1 h 1 1 x - 1 1 + 1 2 x - 1 2 i - 1- 1 - 1 1 1 x - 1 - 1 1 = 1 1 x- 1 1 h 1 1 x - 1 1 + 1 2 x - 1 2 i 1 - 1 . (3) Note that the last factor equals U 1 , so that the expression reduces to U x 1 = 1 1 U x 1 1 , (4) 1 The widely used C-D function is a special case of the CES function, where the elasticity of substitution is not only constant, but also equal to one. The proof is not entirely trivial as setting = 1 in equ. (1) would imply dividing by zero. For a demonstration that the general CES function= 1 in equ....
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