Endogeneity-corrected stochastic frontier with market imperfections

Abstract

While the product and labour market imperfections reveal efficiency losses, they may influence technology adoption and its change, raising the endogeneity issue of productivity and efficiency estimates. Using a two-step approach, this work offers the endogeneity-corrected stochastic frontier for such a contemporaneous relation and accounts for efficiency and productivity losses due to market imperfections. A modified frontier function, defined as the residue per capital unit, has been drawn from the Cobb–Douglas function to estimate the terms containing the product and labour market imperfections along with other factors capturing the levels of technology, scale and technical efficiency. First, a standard frontier panel model estimates technology and technical efficiency terms with a proxy function in polynomials of market imperfection terms used for the contemporaneous relation, and then a GMM approach applies to the residue to estimate the parameters containing market imperfections. The estimated results using the three-digit industries across 17 major Indian states for 2008–2016 reveal a strong presence of product and labour market imperfections and associated efficiency losses. The efficiency in the product market has been lower and has further deteriorated in most industries, but not in the labour market.

Appendix

1.1 Nash bargaining solution

The Nash bargaining expression is as follows:

$$\begin{aligned} \max _{w,L}\Omega =(Lw+(\bar{L}-L)w_0-\bar{L}w_0)^\theta (PY-wL)^{1-\theta } \end{aligned}$$

(A.24)

Differentiating with respect to wage and employment and then arranging terms, it can be expressed as follows:

$$\begin{aligned} w= & {} (1-\theta )w_0+\theta \frac{pY}{L} \end{aligned}$$

(A.25)

$$\begin{aligned} w= & {} \frac{\theta }{1-\theta }\frac{pY-wL}{L}+\frac{\partial (pY)}{\partial L} \end{aligned}$$

(A.26)

One can find that $\frac{\partial (pY)}{\partial L}=\frac{\partial (pY)}{\partial Y} \frac{\partial (Y)}{\partial L}=\frac{p}{m}\frac{\partial Y}{\partial L}$. where, $\frac{\partial (pY)}{\partial Y}= p(1-\frac{1}{|e_p|}); |e_p|= -\frac{\partial p}{\partial Y}\frac{Y}{p}>1; m=\frac{|e_p|}{|e_p|-1}$ and $|e_p|=\frac{\partial Y}{\partial p}\frac{p}{Y}$

Table 5 Results of simple stochastic frontier model

Table 6 Results of endogeneity-corrected stochastic frontier model

Multiply by L and then dividing by pY of (A.26), we get

$$\begin{aligned} \frac{wL}{pY}=\frac{\theta }{1-\theta }\big (1-\frac{wL}{pY}\big )+\frac{1}{m}\frac{\partial Y}{\partial L} \frac{L}{Y} \end{aligned}$$

It is assumed that $s^U_L=\frac{w L}{pY}$ and $\beta _L=\frac{\partial Y}{\partial L} \frac{L}{Y}$. Applying these expressions and rearranging the terms, it can expressed as follows:

$$\begin{aligned} \alpha _L=s_L=m\left[ s^U_L +\frac{\theta }{1-\theta } (s^U_L-1)\right] \end{aligned}$$

(A.27)

Use that $m=\frac{1}{1-\beta }$

$$\begin{aligned} \alpha _L=s_L=\frac{1}{1-\beta }\left[ S^U_L +\frac{\theta }{1-\theta } (s^U_L-1)\right] \end{aligned}$$

(A.28)

Using the above expression and ignoring subscripts from Eq. (4), we get

$$\begin{aligned} (y-k)-\frac{1}{1-\beta }\left[ s^U_L +\frac{\theta }{1-\theta } (s^U_L-1)\right] (l-k)=\lambda k +a+v-u \end{aligned}$$

Rearranging the terms, we find the following expression.

$$\begin{aligned}{} & {} (y-k)- s^U_L (l-k)= \beta (y-k)+\frac{\theta }{1-\theta } (s^U_L-1)](l-k)\nonumber \\{} & {} \quad +\frac{\lambda }{\mu } k +(1-\beta ) (a+v-u) \end{aligned}$$

(A.29)

1.2 Frontier regression results