Elasticity of Substitution and Growth: Normalized Ces in the Diamond Model

The Normalized CES Production Function in the Solow Model

Oliver de La Grandville (1989) suggested that a meaningful examination of the properties of di®erent

members of the same family of CES production functions requires the following normalization.

Given the standard intensive-form CES production function f(kt) = A[±k½

t + (1 ¡ ±)]1

½ , where kt

is the capital per worker at time t, choose arbitrary baseline values for capital per worker (¹k),

output per worker (¹y) and the marginal rate of substitution between capital and labor de¯ned by

¹m = [f(¹k) ¡¹kf0(¹k)]=f0(¹k) (primes denote derivatives with respect to k). Then, use those baseline

values to solve for the normalized e±ciency parameter A(¾) = ¹y

³¹k1¡½+¹m

¹k

+¹m

'1=½

, and the normalized

distribution parameter ±(¾) = ¹k1¡½

¹k

1¡½+¹m as a function of ¾ = 1

1¡½ , the elasticity of substitution.

Substituting these normalized parameters into the initial equation yields the normalized CES production

function:1

f¾(kt) = A(¾) f±(¾)k½

t +[1 ¡ ±(¾)]g1

½ : (1)

Figure 1 illustrates the de La Grandville normalization. Despite disparate values for ¾, all the

isoquants for a given initial level of output (¹y) are shown to go through the common point (point

A) de¯ned by ¹k(given by ray OA) and ¹m (given by line BAC). As shown by Pitchford (1960),

an increase in ¾ without the normalization causes not only an increase in the curvature of the

1For extensive discussions on the normalized CES function see de La Grandville (1989, p.476), and Klump and

Preissler (2000, pp.44-45).

1

isoquant for a given level of output; it also causes the isoquant to shift inward by making factors

more e±cient. The de La Grandville normalization prevents such dispersions.

The