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.

In previous chapters we were focusing on measuring production efficiency in various ways. We now know that one should use the technical efficiency measure if one is concerned with how well the technology potential is used (yet, recall that one still needs to choose an appropriate orientation of measurement – input or output or a mix of these). Furthermore, we learned that one should use cost or revenue (or profit) efficiency if, in addition, one is interested in how well different inputs or outputs (or both) are chosen or allocated with respect to the corresponding prices. The goal of this chapter is to discuss a closely related and, in fact, more general concept – the concept of productivity.

A roadmap for this chapter is useful. We will start by clarifying the differences and relationships between the two main themes of our book: efficiency, which we explored in detail in previous chapters, and productivity, which we will focus on in this chapter. We then consider different approaches to productivity measurement. We will start with the classical growth accounting approach and then move on to the economic approach using index numbers, where we will first consider price indexes, then quantity indexes and then productivity indexes. We also examine some of their decompositions and the relationships among them. After considering a wide range of approaches within the economic approach to index numbers, we will then show that the growth accounting approach can be considered as a restrictive special case. We will finish the chapter with a discussion of transitivity (or circularity) of indexes, what it means in general and for indexes in particular, and how desirable or critical and restrictive this particular property is for an economic index number. We then discuss the sacrifices one must make in order to preserve transitivity and how to mitigate problems with the index number approach when transitivity is not imposed. We conclude with brief remarks on the literature, which will be further discussed in Chapter 7.

PRODUCTIVITY VS. EFFICIENCY

While a lot has been done on efficiency measurement in production, it is a relatively modern area in economics and long before its academic origins, people already used, and still use, the notion of productivity.