Modeling the effect of competition on US manufacturing sectors’ efficiency: an order-m frontier analysis

Abstract

The study applies the probabilistic framework of nonparametric frontier estimation to model the effect of competitive conditions on sectors’ production efficiency levels. We utilize conditional order-m robust frontiers to model the dynamic effects of competition on a sample of U.S. manufacturing sectors over the period 1958–2009. Contrary to the existing studies, we apply for the first time in the Industrial Organization literature the latest advances of robust nonparametric frontier analysis to disentagle the dynamic effects alongside the effects of competition on sectors’ productive efficiency levels. The results derived from the time-dependent robust conditional estimators unveil a non-linear relationship between product market competition and productive efficiency. Our findings suggest that for higher competition levels the effect is positive up to a certain threshold after which the effect becomes negative.

Appendix

A1: Computational aspects among full and partial frontiers

Under the probabilistic framework (Daraio and Simar 2005, 2007a, 2007b), the effect of the competitive conditions W and time T, can be represented as:

$${\it{\varrho }}_t\left( {\left. {x,\,{\mathrm{y}}} \right|{\mathrm{w}}} \right) = {\mathrm{sup}}\left\{ {\left. {\it{\varrho }} \right|{\mathrm{\Phi }}_{\left. Y \right|X,W}^t\left( {\left. y \right|x,w} \right)} \right\}$$

(16)

Then we can estimate sectors’ time-dependent conditional efficiency level applying the following LP of the DEA estimator under the variable returns to scale (VRS) assumption as:

$${\hat \varrho _t}(x,{\rm{y}}|{\rm{w}}) = \left\{ (x,y) \in {\Bbb R}_ + ^p \times {\Bbb R}_ + ^q|y\, \le\, \sum\limits_{j \in \Im (w,t)} {{\omega _j}{y_j};\,x\, \ge \,\sum\limits_{j \in \Im (w,t)} {{\omega _j}{x_j};\,\omega \, \ge \,0\,s.\,t.\,\sum\limits_{j \in \Im (w,t)} {\omega \, = \,1} } } \right\}$$

(17)

Where ℑ(w,t) = {j = (i, v)|w − h_w < w_i,v < w + h_w; t − h_t < v < t+h_t}. Based on Mastromarco and Simar (2015, p.829) we can estimate the conditional distribution $\widehat \Phi _{\left. Y \right|X,W}^t\left( {\left. y \right|x,w} \right)$ as:

$$\widehat \Phi _{\left. Y \right|X,W}^t\left( {\left. y \right|x,w} \right) = \frac{{\mathop {\sum }\nolimits_{j = \left( {i,\upsilon } \right)} {\it{{\rm{I}}}}\left( {x_j \le x,y_j \ge y} \right)K_{hw}\left( {w_j - w} \right)K_{ht}\left( {\upsilon - t} \right)}}{{\mathop {\sum }\nolimits_{j = \left( {i,\upsilon } \right)} {\it{{\rm{I}}}}\left( {x_j \le x} \right)K_{hw}\left( {w_j - w} \right)K_{ht}\left( {\upsilon - t} \right)}}.$$

(18)

It must be noted that h represents the smoothing and K(·) is a kernel with compact support (for Matlab codes see Bădin et al. 2010, 2019). The DEA estimator presented in Eq. (17) has been applied by Polemis and Tzeremes (2019) assuming that the technology is convex (concave). When assuming non-convex (non-concave) technology the time-dependent conditional order-m estimator can be computed as:

$$\begin{array}{ll}\widehat \vartheta _{t,m}\left( {\left. {x,\,{\mathrm{y}}} \right|{\mathrm{w}}} \right) &= \widehat {\it{\varrho }}_t\left( {\left. {x,\,{\mathrm{y}}} \right|{\mathrm{w}}} \right) - \\ &\quad{\int}_0^{\widehat {\it{\varrho }}_t\left( {\left. {x,\,{\mathrm{y}}} \right|{\mathrm{w}}} \right)} {\left( {1 - \widehat \Phi _{\left. Y \right|X,W}^t\left( {\left. y \right|x,w} \right)} \right)^mdu} \end{array}$$

(19)

Daraio and Simar (2007b) provide a Monte Carlo algorithm to solve the conditional order-m estimators, however, for large m values the estimator can be calculated via the numerical integration (for computational issues also see the Matlab codes provided by Daraio et al. 2020).

A2: Comparison of the results among the full and partial frontiers

Figure 7 presents the density plots of the conditional estimators under which imply convexity (concavity) assumption and those which imply the non-convexity (non-concavity) assumption. Specifically, as in the study by Polemis and Tzeremes (2019) the left panel represent the $\widehat {\it{\varrho }}_t\left( {\left. {x,\,{\mathrm{y}}} \right|{\mathrm{w}}} \right)$ estimator (red line) whereas, the black line represents the estimator used in this study $\widehat \vartheta _{t,m}\left( {\left. {x,\,{\mathrm{y}}} \right|{\mathrm{w}}} \right)$. Finally, the right panel represents the Q-ratios under the full frontiers $Q_{vrs} = \frac{{\widehat {\it{\varrho }}_t\left( {\left. {x,\,{\mathrm{y}}} \right|{\mathrm{w}}} \right)}}{{\widehat {\it{\varrho }}\left( {x,y} \right)}}$ and under the partial frontiers $Q_m = \frac{{\widehat \vartheta _{t,m}\left( {\left. {x,\,{\mathrm{y}}} \right|{\mathrm{w}}} \right)}}{{\widehat \vartheta \left( {{\mathrm{x}},{\mathrm{y}}} \right)}}$.

Fig. 7

Comparison between full and partial estimators