# Economic interpretation of a CES production function

I am following a paper where a production function of this type is used.

$$Y=\left [\beta K^{- \rho}+\alpha \eta \left (\frac{K}{L} \right )^{-c(1+\rho)}L^{- \rho} \right ]^{-1/\rho}$$

It is a modified version of a CES where in addition there is the term $$\alpha \eta \left (\frac{K}{L} \right )^{-c(1+\rho)}$$ which multiply the $$L$$ factor.

I was therefore wondering. What may be the economic justification interpretation for adding this factor?

• May be to nest the Cobb-Douglas (if $\beta=0$) or the allow for nonconstant returns to scale (if $c=0$ we have CRTS). The reason should actually be given in the paper... – Bertrand Jan 14 at 12:37
• Yep.. if c=0 it nest a standard CES. But from a merely economic point of view, what's the sense that the production depends not just on K and L but also on their ratio? – Alessandro Jan 14 at 12:40
• The meaning is unclear. But is the dependence wrt $K/L$ not spurious as we can rewrite this term $\alpha \eta K^{-c(1+\rho)}L^{- \rho + c + c\rho }$ ? – Bertrand Jan 14 at 12:49
• Yes, of course, you can rewrite the term that way. Mathematically adding $K/L$to multiply $L$. This means that if K increase (holding L constant) Y increase more than if we increase L (keepening K constant), because in the second case the $K/L$ ratio decrease and hence the $MP$ of Capital is sistematically higher than that of labor. Therefore it's like imagine that the production process is biased toward capital, right? – Alessandro Jan 14 at 13:01
• The production function is still homothetic and has CRTS (I made a mistake earlier), and as there is no technological change occurring here, I would not say that it is biased. – Bertrand Jan 14 at 13:10