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: ${ℝ}^n\mapsto ℝ,f\left(v\right)=a_0{\left({\sum <!--\; FUNCTION\; APPLICATION\; -->}_{j=1}^n{c}_j{v}_j^{-\rho }\right)}^{-\frac{h}{\rho }},n\ge 1$, where the elasticity of substitution is $\sigma =\frac{1}{1+\rho }$. By variation of $\rho$ 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, $c_1$ and $c_2$ 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

Since this representation is overparameterized, the parameter was set $c_2=1$ , so that the relative weight of the two goods is represented only by $c_1$.
A selection of graphical illustrations can be found hier.