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Unformatted text preview: Chapter 4. The CES Production Function as a General Mean The discoverers of the CES production function, K. Arrow, H. Chenery, B. Minhas and R. Solow had observed that it was a linear transformation of a general mean of order & & . It turns out that income per person, the central variable of economic growth theory, is also a general mean. Its properties will prove to be of great importance. In this chapter, we will &rst recall the de&nition of a general mean and its main properties. We will then examine how they apply directly to the CES function and the resulting income per person. 1 Notation From Chapter 2, we know that & & = 1 & 1 =¡ . To simplify notation, we will denote & & = p . We thus have p = 1 & 1 ¡ (1) where p , the order of the general mean, is an increasing function of ¡ . We thus have between ¡ and p a positive correspondence to which we will often refer. It is also very useful to visualize this relationship (see Figure 1). Notice that in the vicinity of ¡ = 1 , the linear approximation of p is ¡ & 1 i.e. a straight line with slope equal to +1 . We will use this observation later. Table 1 gives the correspondence between ¡ and p for some important values of ¡ . Total income, expressed as the index number Y=Y , is a general mean of order p = 1 & 1 =¡ of the indices K=K and L=L : Y Y = & ¢ ¡ K K ¢ p + (1 & ¢ ) ¡ L L ¢ p £ 1 p : (2) Consider now income per person expressed as the index y=y ; it is also a general mean. Indeed, it is the mean of order p of the index r=r and 1 : 1 2 Y=L Y =L = y y = & & ¡ r r ¢ p + (1 & & ) £ 1 p (3) where r = K=L and r = K =L . Table 1. Relationship between the elasticity of substitution ¡ and the order of the mean p = 1 & 1 =¡ . Elasticity of substitution Order of the mean ¡ p &1 1 1 1 1.1 The concept of the general mean of order p , and its fundamental properties The familiar arithmetic, geometric and harmonic means are just particular cases of the &general mean of order p ¡, de¢ned as follows. 1.1.1 De&nition Let x 1 ;:::;x n be positive numbers; let f 1 ;:::;f n positive numbers adding up to 1. We call f i &weights¡. The mean of order p of x 1 ;:::;x n is de¢ned by M ( p ) = " n X i =1 f i x p i # 1 =p : (4) 3 1.1.2 Important particular cases It can be easily seen that the means of order 1 and 1 are the arithmetic and the harmonic means, respectively. We have M (1) = n X i =1 f i x i ; (5) the arithmetic mean of the x i &s, and M ( & 1) = " n X i =1 f i x & 1 i # & 1 ; (6) the harmonic mean of the x i &s. In order to show that the mean of order 0 is the geometric mean, ¡rst take the logarithm of (4): log M ( p ) = 1 p log n X i =1 f i x p i ! : (7) When p ! , the ratio (0/0) can be determined by applying L&Hospital&s rule. We get: lim p !...
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This note was uploaded on 06/16/2010 for the course MS&E 249 taught by Professor Olivierdelagrandville during the Fall '08 term at Stanford.
 Fall '08
 OLIVIERDELAGRANDVILLE

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