Estimating the shadow price of undesirable outputs is an important topic in production and environmental economics, as it provides information on the economic cost of environmental regulation. Directional distance function (DDF) approaches have been widely employed for this purpose; however, conventional DDF estimation methods are largely limited to point estimation, which constrains statistical inference. This study proposes a bootstrap procedure integrated into a parametric linear programming-based DDF framework that explicitly incorporates undesirable outputs. The proposed methodology is applied to Korean firm-level data. By allowing interval estimation of DDF values, the approach makes it possible to conduct statistical inference on the estimated parameters. The empirical results indicate that DDF estimates obtained using the bootstrap approach differ from those derived from the non-bootstrap method. The resulting shadow prices of CO2 are economically interpretable and comparable in magnitude to observed emission permit prices, suggesting their relevance for policy analysis. The results indicate that incorporating bootstrapping into a parametric DDF framework enhances statistical inference in the estimation of shadow prices for undesirable outputs.
Lim et al. (Tue,) studied this question.