The Components of Total Factor Productivity Change

Abstract

This chapter delves into the components of (total factor) productivity change. By making minimal assumptions about underlying technologies it appears that productivity change, here defined as output quantity change divided by input quantity change, can be seen as the combined result of (technical) efficiency change, technological change, a scale effect, and input and output mix effects. Given a certain functional form for the productivity index, the problem is then how to decompose such an index into factors corresponding to these five components. A basic insight offered in this chapter is that meaningful decompositions of productivity indices can only be obtained for indices that are transitive in the main variables. Using a unified approach, decompositions for the classes of Malmquist, Moorsteen-Bjurek, Lowe, and Cobb-Douglas productivity indices are obtained. A unique feature of this chapter is that all the decompositions are applied to the same dataset, a real-life panel of decision-making units, so that the extent of the differences between the various decompositions can be judged.

Appendices

Appendix A: Components of $\check {M}_{o}^{0}(x^1,y^1,x^0,y^0)$ Along Path A

Recall that productivity change, going from (x, y) to (x′, y′), is measured by $\check {M}_{o}^{0}(x',y',x,y) = \check {D}_{o}^{0}(x',y')/\check {D}_{o}^{0}(x,y)$. Along the first segment of Path A productivity change is

$$\displaystyle \begin{aligned} \frac{\check{D}_{o}^{0}(x^0,y^0/D_{o}^{0}(x^0,y^0))}{\check{D}_{o}^{0}(x^0,y^0)} = \frac{1}{D_{o}^{0}(x^0,y^0)}. \end{aligned}$$

Along the second segment

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\check{D}_{o}^{0}(\lambda x^0,y^0/D_{o}^{0}(\lambda x^0,y^0))}{\check{D}_{o}^{0}(x^0,y^0/D_{o}^{0}(x^0,y^0))} & = &\displaystyle \frac{\check{D}_{o}^{0}(\lambda x^0,y^0)}{D_{o}^{0}(\lambda x^0,y^0)} \frac{D_{o}^{0}(x^0,y^0)}{\check{D}_{o}^{0}(x^0,y^0)} \\ & = &\displaystyle \frac{\mathit{OSE}^{0}(\lambda x^0,y^0)}{\mathit{OSE}^{0}(x^0,y^0)} = \mathit{SEC}_{o,M}^{0}(\lambda x^0,x^0,y^0).{} \end{array} \end{aligned} $$

Along the third segment

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\check{D}_{o}^{0}(x^1,y^0/D_{o}^{0}(x^1,y^0))}{\check{D}_{o}^{0}(\lambda x^0,y^0/D_{o}^{0}(\lambda x^0,y^0))} & = &\displaystyle \frac{\check{D}_{o}^{0}(x^1,y^0)}{D_{o}^{0}(x^1,y^0)} \frac{D_{o}^{0}(\lambda x^0,y^0)}{\check{D}_{o}^{0}(\lambda x^0,y^0)} \\ & = &\displaystyle \frac{\mathit{OSE}^{0}(x^1,y^0)}{\mathit{OSE}^{0}(\lambda x^0,y^0)} = \mathit{SEC}_{o,M}^{0}(x^1,\lambda x^0,y^0).{} \end{array} \end{aligned} $$

Along the fourth segment

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\check{D}_{o}^{0}(x^1,y^1/D_{o}^{0}(x^1,y^1))}{\check{D}_{o}^{0}(x^1,y^0/D_{o}^{0}(x^1,y^0))} & = &\displaystyle \frac{\check{D}_{o}^{0}(x^1,y^1)}{D_{o}^{0}(x^1,y^1)} \frac{D_{o}^{0}(x^1,y^0)}{\check{D}_{o}^{0}(x^1,y^0)} \\ & = &\displaystyle \frac{\mathit{OSE}^{0}(x^1,y^1)}{\mathit{OSE}^{0}(x^1,y^0)} = \mathit{OME}_{M}^{0}(x^1,y^1,y^0).{} \end{array} \end{aligned} $$

Along the fifth segment

$$\displaystyle \begin{aligned} \frac{\check{D}_{o}^{0}(x^1,y^1/D_{o}^{1}(x^1,y^1))}{\check{D}_{o}^{0}(x^1,y^1/D_{o}^{0}(x^1,y^1))} = \frac{D_{o}^{0}(x^1,y^1)}{D_{o}^{1}(x^1,y^1)} = \mathit{TC}_{o}^{1,0}(x^1,y^1). \end{aligned}$$

Along the sixth segment

$$\displaystyle \begin{aligned} \frac{\check{D}_{o}^{0}(x^1,y^1)}{\check{D}_{o}^{0}(x^1,y^1/D_{o}^{1}(x^1,y^1))} = D_{o}^{1}(x^1,y^1). \end{aligned}$$

Multiplication of the left-hand sides of these equations delivers productivity change between (x ⁰, y ⁰) and (x ¹, y ¹), $\check {M}_{o}^{0}(x^1,y^1,x^0,y^0)$, whereas multiplication of the right-hand sides establishes the decomposition. Recall thereby that

$$\displaystyle \begin{aligned} \frac{1}{D_{o}^{0}(x^0,y^0)} \times D_{o}^{1}(x^1,y^1) = \mathit{EC}_{o}(x^1,y^1,x^0,y^0). \end{aligned}$$

Appendix B: Data Envelopment Analysis (DEA)

Suppose we are given panel data (w ^kt, x ^kt, p ^kt, y ^kt) for production units k = 1, …, K and time periods t = 0, 1, …, T. These data allow us to calculate any productivity index we wish, cross-sectionally,³¹ intertemporally, or a combination of the two. If only quantity data are given, the choice is restricted to Malmquist indices. As we have shown in the foregoing sections, several options are available for a decomposition of productivity change.

Although expressions such as (10.48) look quite intimidating, their computation should not be much of a problem, if one has knowledge of the functions involved. For example, for the computation of the various parts of expression (10.48) only knowledge of the output distance functions $D_o^t(x,y)$ and $\check {D}_o^t(x,y)$ is required, and for the various parts of expression (10.61) only knowledge of the input distance functions $D_i^t(x,y)$ and $\check {D}_i^t(x,y)$, in both cases for t = 0, 1.

Using the technique of Data Envelopment Analysis the period t technology S ^t is approximated by

$$\displaystyle \begin{aligned} \begin{array}{rcl} S^{t} & \approx &\displaystyle \{(x,y) \mid \sum^{K}_{k=1}z_{k}x^{kt} \leq x, y \leq \sum^{K}_{k=1}z_{k}y^{kt}, {}\\ & &\displaystyle z_{k} \geq 0 \: (k=1,\ldots ,K), \sum^{K}_{k=1}z_{k} = 1 \}, \end{array} \end{aligned} $$

(10.100)

whereas the associated cone technology $\check {S}^t$ is approximated by

$$\displaystyle \begin{aligned} \begin{array}{rcl} \check{S}^{t} & \approx &\displaystyle \{(x,y) \mid \sum^{K}_{k=1}z_{k}x^{kt} \leq x, y \leq \sum^{K}_{k=1}z_{k}y^{kt}, {}\\ & &\displaystyle z_{k} \geq 0 \: (k=1,\ldots ,K) \}. \end{array} \end{aligned} $$

(10.101)

These approximations satisfy the axioms P.1–P.9. The right-hand side of expression (10.100) exhibits variable returns to scale (VRS). The right-hand side of expression (10.101) exhibits CRS. For proofs the reader is referred to Färe et al. (2015, Chapter 1).

Based on these approximations, the computation of the required output or input distance functions is reduced to the solution of linear programming problems.