= Factor productivity = Share parameter of input i, = Quantities of factors of production (i = 1,2...n) = Elasticity of substitution. Extending the CES (Solow) functional form to accommodate multiple factors of production creates some problems. However, there is no completely general way to do this. Uzawa showed the only possible n-factor production functions (n>2) with constant partial elasticities of substitution require either that all elasticities between pairs of factors be identical, or if any differ, these all must equal each other and all remaining elasticities must be unity. This is true for any production function. This means the use of the CES functional form for more than 2 factors will generally mean that there is not constant elasticity of substitution among all factors. Nested CES functions are commonly found in partial equilibrium and general equilibrium models. Different nests (levels) allow for the introduction of the appropriate elasticity of substitution. CES utility function The same CES functional form arises as a utility function in consumer theory. For example, if there exist types of consumption goods , then aggregate consumption could be defined using the CES aggregator: Here again, the coefficients are share parameters, and is the elasticity of substitution. Therefore, the consumption goods are perfect substitutes when approaches infinity and perfect complements when approaches zero. In the case where approaches one is again a limiting case where L'Hôpital's Rule applies. The CES aggregator is also sometimes called the Armington aggregator, which was discussed by Armington (1969). CES utility functions are a special case of homothetic preferences. The following is an example of a CES utility function for two goods, and , with equal shares: The expenditure function in this case is: The indirect utility function is its inverse: The demand functions are: A CES utility function is one of the cases considered by Dixit and Stiglitz (1977) in their study of optimal product diversity in a context of monopolistic competition. Note the difference between CES utility and isoelastic utility: the CES utility function is an ordinal utility function that represents preferences on sure consumption commodity bundles, while the isoelastic utility function is a cardinal utility function that represents preferences on lotteries. A CES indirect (dual) utility function has been used to derive utility-consistent brand demand systems where category demands are determined endogenously by a multi-category, CES indirect (dual) utility function. It has also been shown that CES preferences are