Neutral Inventions and CES Production Functions

Abstract

In the literature of economic theory the most frequently used classes of (macroeconomic) production functions with completely substitutable factors are the CD functions ²) and ACMS functions ³) defined respectively (for two arguments, usually interpreted as capital and labour) by

$$F\left( {K,L} \right) = c{K^a}{L^b}\quad \quad \quad \left\{ \begin{gathered} a,b,c\;positive,\;const.; \hfill \\ K>0,\;L>0 \hfill \\ \end{gathered} \right)$$

(0.1)

and

$$ F\left( {K,L} \right) = {\left( {{c_1}{K^{{ - \rho }}}} \right)^{{ - \frac{1}{\rho }}}}\quad \left\{ \begin{gathered} {c_1},{c_2}\;pos.,\;\rho \ne 0,\;const.; \hfill \\ K>0,\;L>0, \hfill \\ \end{gathered} \right)$$

(0.2)

or, if a time variable t is involved⁴, by

$$F\left( {K,L,t} \right) = c(t){K^a}{L^b}\quad \quad \quad \left\{ \begin{gathered} a,b\;pos.,const.; \hfill \\ c(t)>0;\;t>0,\;K>0,\;L>0 \hfill \\ \end{gathered} \right)$$

(0.1′)

and

$$F\left( {K,L,t} \right) = \bar{c}(t){\left( {{c_1}{K^{{ - \rho }}} + {c_2}{L^{{ - \rho }}}} \right)^{{ - \frac{1}{\rho }}}}\quad \left\{ \begin{gathered} {c_1},{c_2}\;pos.,\;\rho \ne 0, \hfill \\ const.;\;\bar{c}(t)>0; \hfill \\ t>0,\;K>0,\;L>0. \hfill \\ \end{gathered} \right)$$

(0.2′)

.