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?

  • $\begingroup$ 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... $\endgroup$ – Bertrand Jan 14 at 12:37
  • $\begingroup$ 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? $\endgroup$ – Alessandro Jan 14 at 12:40
  • $\begingroup$ 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 } $ ? $\endgroup$ – Bertrand Jan 14 at 12:49
  • $\begingroup$ 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? $\endgroup$ – Alessandro Jan 14 at 13:01
  • 1
    $\begingroup$ 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. $\endgroup$ – Bertrand Jan 14 at 13:10

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