Returns to scale and substitution Although much of our discussion of measuring the elasticity of substitution for various production functions has assumed constant returns to scale, often that assumption is not necessary. This problem illustrates some of these cases.

a. In footnote 6 we pointed out that, in the constant returns-to-scale case, the elasticity of substitution for a two-input production function is given by

σ 5 fk fl f # fkl

.

Suppose now that we define the homothetic production function F as F 1k, l2 5 3 f 1k, l2 4 γ
, where f 1k, l2 is a constant returns-to-scale production function and γ is a positive exponent. Show that the elasticity of substitution for this production function is the same as the elasticity of substitution for the function f.

b. Show how this result can be applied to both the Cobb–
Douglas and CES production functions.