15.4.2 Elasticities of substitution: Cobb-Douglas

The Cobb-Douglas production function or Cobb-Douglas utility function is a special element of the CES- family for $\sigma =1$. This property is advantageous in many economic applications. The general form is: ${ℝ}^n\mapsto ℝ,f\left(v\right)=a\left({\prod <!--\; FUNCTION\; APPLICATION\; -->}_{j=1}^n{x}_j^{{\alpha }_j}\right),n\ge 1$, where ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{j=1}^n{\alpha }_j$ indicates the degree of homogeneity. u represents the production- or utility- level and a the technology factor. In the above graph, a graph of the Cobb-Douglas function $u=a{x}^{{\alpha }_1}{y}^{{\alpha }_2}$ is drawn for n=2.