This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 , dened 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 dened 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 !...
View Full
Document
 Fall '08
 OLIVIERDELAGRANDVILLE

Click to edit the document details