As the book has pointed out in its earliest chapters, economic theory provides the most reasoned, and often the most powerful and leveraged, guidance for econometric modeling of productivity. The primal and dual relationships that are specified and estimated by functional representations in the form of the production, cost, revenue, and profit functions derive their interpretability from the regularity conditions that were utilized in specifying the production sets, distance functions, and in deriving cost, revenue, and profit functions. These regularity conditions are often difficult to impose with many of the flexible parametric functional forms we discussed in Chapter 6 and may be even more difficult to impose when the functional relationships are specified nonparametrically using kernel smoothers or other classical nonparametric methods. In the production setting monotonicity is often required, analogous to its requirement in models with rational preferences. Concavity of production functions have analogs in convex preferences and risk aversion in utility theory. Demand theory results in downward sloping demand curves for normal goods (Matzkin, 1991; Lewbel, 2010; Blundell et al., 2012), while production theory and duality provide us with implications of profit-maximizing behavior that require profit functions to be concave in output prices. Cost minimization yields cost functions that are monotonically increasing and concave in input prices. Auction theory and optimal bidding strategies that vary across auction formats and bidders’ preferences are based on monotonicity in bidders valuations. Derivative pricing models are highly leveraged on convex function estimation (Broadie et al., 2000; Aıt-Sahalia and Duarte, 2003; Yatchew and Härdle, 2006). Such considerations are ubiquitous in economics and it is essential that we address them in the context of the topic that our book in part endeavors to address, empirical productivity analysis.

In this chapter, we discuss several methods to deal with estimation of the primal production function utilizing semi- and nonparametric econometric specifications under monotonicity and curvature constraints. General reviews of this material can be found in Matzkin (1994) and Yatchew (2003, Chapter 6). Work that speaks to relatively recent extensions can be found in Hall and Huang (2001), Groeneboom et al. (2001), Horowitz et al. (2004), Carroll et al. (2011), Shively et al. (2011), Blundell et al. (2012), and Pya and Wood (2015), among others.