15.4.1 CES substitution elasticities

The CES- production function or CES- utility function is a production function for which the substitution elasticity always assumes the same value. Here, CES stands for c onstant e BeginExpansion > EndExpansion XXX lasticity of substitution. This property is advantageous in many economic applications. The symmetrical XXX form is: n,f(v) = a 0 j=1nc jvjρ h ρ ,n 1, where the elasticity of substitution is σ = 1 1+ρ. By variation of ρ the type of utility function can be changed from Leontief to Cobb-Douglas to perfect substitution (linear utility function). h indicates the degree of homogeneity. If h = 1, the function is linearly homogeneous, i.e., if all input factors are doubled, the output is also doubled. For h > 1 positive economies of scale apply, for h > 1 XXX negative economies of scale apply.
u represents the production- or utility- level, a the technology factor, c1 and c2 the relative weights of the two input factors x and y.
In the above graph, for n=2 a graph of the CES function is

u = a c1xρ + c 2yρ h ρ .

Since this representation is overparameterized, the parameter was set c2 = 1 , so that the relative weight of the two goods is represented only by c1.
A selection of graphical illustrations can be found hier.

(c) by Christian Bauer
Prof. Dr. Christian Bauer
Chair of monetary economics
Trier University
D-54296 Trier
Tel.: +49 (0)651/201-2743
E-mail: Bauer@uni-trier.de
URL: https://www.cbauer.de