Productivity indexes under Hicks neutral technical change

Abstract

The Malmquist and Hicks–Moorsteen productivity indexes are the two most widely used theoretical indexes for measuring productivity growth of firms, industries and countries. We indicate that these two indexes coincide under constant returns to scale technology and Hicks neutral technical change. While the conditions on production technology under which two indexes coincide have been examined before, this is the first study on the types of technical change under which they coincide. We shed new insight on the relationship between these popular indexes.

1 Introduction

Productivity indexes are applied to measure changes in productivity of firms, industries and countries between two periods.¹ Malmquist and Hicks–Moorsteen are the two most popular theoretical productivity indexes.

The Malmquist productivity index was originally introduced by Caves et al. (1982) to measure productivity growth by the radial distance of the observed output and input vectors of two periods relative to a reference technology. Technologies of the two periods that are being compared can be used as a reference. Thus, the geometric mean of the productivity index relative to technology for one period and the productivity index relative to technology of another period define the Malmquist productivity index.

Caves et al. (1982) also propose Malmquist output (input) quantity by the radial distances of the observed output (input) vectors of two periods relative to a reference technology. Like the Malmquist productivity index, the geometric mean of two quantity indexes relative to different reference technologies defines the Malmquist output (input) quantity index. Hicks–Moorsteen productivity index, which was first formulated by Bjurek (1996), is the ratio of the Malmquist output quantity index over Malmquist input quantity index.² By construction, it is also regarded as the geometric mean of two alternative productivity indexes relative to different technologies.

Färe et al. (1996) indicate that Malmquist and Hicks–Moorsteen productivity indexes coincide under two conditions of technology: (i) inverse homotheticity and (ii) constant returns to scale. While they are often considered to advocate that this pair of conditions is necessary and sufficient for these two indexes to coincide, they only deal with the productivity indexes relative to a specific reference technology, rather than the geometric mean of two productivity indexes relative to different reference technologies. Thus, their result does not exclude the possibility that Malmquist and Hicks–Moorsteen productivity indexes, which are defined by the geometric mean of two productivity indexes to different technologies, coincide under different conditions.

This study indicates that the Malmquist and Hicks–Moorsteen productivity indexes coincide under two conditions: (i) constant returns to scale technology and (ii) Hicks neutral technical change. While the former imposes conditions on the underlying technology, the latter restricts the type of technical change. To the best of our knowledge, this study is the first attempt to compare two productivity indexes under Hicks neutral technical change.³

There are studies showing equality between the two productivity indexes under more special cases. O’Donnell (2012) states that they coincide under constant returns to scale technology and no technical changes and Karagiannis and Lovell (2016) indicate the equality result under constant inputs or outputs. Although we do not exclude such cases, we deal with more general circumstances that productivity growth and changes in inputs and outputs are possible, which is considered by Färe et al. (1996).⁴

The study is organised as follows. Section 2 explains the analytical framework and introduces the definitions of two productivity indexes. Section 3 discusses the necessary and sufficient conditions by Färe et al. (1996). Section 4 includes the main results, which characterise the types of technical change that allow these two indexes to coincide. Section 5 concludes the study.

2 Malmquist and Hicks–Moorsteen productivity indexes

Consider a firm produces M types of outputs ${\boldsymbol{y}} = \left( {{y_1}, \ldots ,{y_M}} \right) \in {\Bbb R}_ + ^M$ from N types of inputs ${\boldsymbol{x}} = \left( {{x_1}, \ldots ,{x_N}} \right) \in {\Bbb R}_ + ^N$. The firm’s technology available at period t is characterised by the technology set T ^t, which is the set of all feasible combinations of inputs and outputs and defined as follows:

$${T^{t}} \equiv \left\{ {\left( {{\boldsymbol{x}},{\boldsymbol{y}}} \right) \in {\Bbb R}_ + ^{N + M}:{\boldsymbol{x}}\,{\rm{can}}\,{\rm{produce}}\,{\boldsymbol{y}}\,{\rm{in}}\,{\rm{period}}\,t} \right\} \cdot$$

(1)

The production technology represented by T ^t can be alternatively expressed by the following output set P ^t(x), which is the set of all output y that is attainable from x:

$${P^t}\left( {\boldsymbol{x}} \right) \equiv \left\{ {{\boldsymbol{y}} \in {\Bbb R}_ + ^M:\left( {{\boldsymbol{x}},{\boldsymbol{y}}} \right) \in {T^t}} \right\} \cdot$$

(2)

We assume that the technology satisfies the following regularity conditions: (T.1) no free lunch: y ∉ P ^t(0 _N ); (T.2) no production is possible with given input: 0 _M  ∈ P ^t(x); (T.3) strong disposability of outputs: if y ∈ P ^t(x) and y ^* ≤ y, then y ^* ∈ P ^t(x); (T.4) strong disposability of inputs: if y ∈ P ^t(x) and x ^* ≥ x, then y ∈ P ^t(x ^*); (T.5) P ^t(x) is closed and (T.6) P ^t(x) is bounded. These conventional axioms on the technology guarantee the existence of the distance functions (Färe and Primont 1995). The distance function is a convenient tool for characterising the technology. We deal with two types of distance function.

The period t output distance function $D_o^t:{\Bbb R}_ + ^{N + M} \to {{\Bbb R}_ + } \cup \left\{ { + \infty } \right\}$ characterises the technology of period t. It is defined as follows:

$$D_o^t\left( {{\boldsymbol{x}},{\boldsymbol{y}}} \right) \equiv \inf \left\{ {\theta > 0{\rm{:}}\left( {{\boldsymbol{x}},{\boldsymbol{y}}{\rm{/}}\theta } \right) \in {T^t}} \right\} \cdot$$

(3)

The period t input distance function $D_i^t:{\Bbb R}_ + ^{N + M} \to {{\Bbb R}_ + } \cup \left\{ { + \infty } \right\}$ characterises the technology of period t. It is defined as follows:

$$D_i^t\left( {{\boldsymbol{y}}{\rm{,}}\,{\boldsymbol{x}}} \right) \equiv {\rm{sup}}\left\{ {\theta >0:\left( {{\boldsymbol{x}}/\theta {\rm{,}}\,{\boldsymbol{y}}} \right) \in {T^t}} \right\} \cdot$$

(4)

We consider productivity growth takes place between periods 0 and 1 by applying the Malmquist and Hicks–Moorsteen productivity indexes. Both indexes are defined by distance functions.

The Malmquist productivity index compares the radial distances of input and output vectors in two periods relative to the reference technology. Output-oriented Malmquist productivity index with respect to the period t technology is defined as follows:

$${M^t}\left( {{{\boldsymbol{x}}^0},{{\boldsymbol{x}}^1},{{\boldsymbol{y}}^0},{{\boldsymbol{y}}^1}} \right) \equiv \frac{{D_o^t\left( {{{\boldsymbol{x}}^1},{{\boldsymbol{y}}^1}} \right)}}{{D_o^t\left( {{{\boldsymbol{x}}^0},{{\boldsymbol{y}}^0}} \right)}} \cdot$$

(5)

The technologies that are available at periods 0 and 1 are equally reasonable candidates for the reference technology. Thus, the geometric mean of the two productivity indexes relative to the technologies of two periods are adopted in the following definition of the output-oriented Malmquist productivity index M by Caves et al. (1982):

$$M \equiv \sqrt {{M^0} \times {M^1}} \cdot$$

(6)

In the single input–single output case, productivity is the ratio of output over input y/x. Thus, productivity growth can be interpreted as the growth rate of output over that of input such that (y ¹/x ¹)/(y ⁰/x ⁰) = (y ¹/y ⁰)/(x ¹/x ⁰). The Hicks–Moorsteen productivity index generalises this ratio for multiple inputs and outputs utilising the Malmquist quantity index. The Hicks–Moorsteen productivity index with respect to the period t technology is defined as follows:

$$H{M^t}\left( {{{\boldsymbol{x}}^0},{{\boldsymbol{x}}^1},{{\boldsymbol{y}}^0},{{\boldsymbol{y}}^1}} \right) \equiv \frac{{D_o^t\left( {{{\boldsymbol{x}}^t},{{\boldsymbol{y}}^1}} \right)/D_o^t\left( {{{\boldsymbol{x}}^t},{{\boldsymbol{y}}^0}} \right)}}{{D_i^t\left( {{{\boldsymbol{y}}^t},{{\boldsymbol{x}}^1}} \right)/D_i^t\left( {{{\boldsymbol{y}}^t},{{\boldsymbol{x}}^0}} \right)}} \cdot$$

(7)

While the numerator is the Malmquist output quantity index relative to the input and the technology that are available at period t, the denominator is the Malmquist input quantity index relative to the output and the technology available at period t.⁵ The technologies that are available at periods 0 and 1 are equally reasonable candidates for reference. Thus, the geometric mean of the two productivity indexes relative to technologies and input and output vectors of two periods are adopted in the following definition of the Hicks–Moorsteen productivity index HM by Bjurek (1996):

$$HM \equiv \sqrt {H{M^0}{\rm{ \times }}H{M^1} \cdot }$$

(8)

This study explores the relationship between the output-oriented Malmquist and Hicks–Moorsteen productivity indexes defined by Eqs. (6) and (8).⁶ In this study, the output-oriented Malmquist and Hicks–Moorsteen productivity indexes refer to these definitions.

3 Necessary and sufficient condition for equivalence result by Färe et al. (1996)

Färe et al. (1996) indicate that Malmquist and Hicks–Moorsteen productivity indexes coincide under two conditions of technology: (i) constant returns to scale and (ii) inverse homotheticity. We introduce the formal definitions as follows:

Definition 1

The technology exhibits constant returns to scale if

$${P^t}\left( {\lambda {\boldsymbol{x}}} \right) = \lambda {P^t}\left( {\boldsymbol{x}} \right)\,{\rm{for}}\,{\rm{all}}\,\lambda > 0 \cdot$$

(9)

It allows us to compute the output distance function from the input distance function and vice versa as follows:

$$D_o^t\left( {{\boldsymbol{x}},{\boldsymbol{y}}} \right) = 1{\rm{/}}D_i^t\left( {{\boldsymbol{y}},{\boldsymbol{x}}} \right) \cdot$$

(10)

Definition 2

The technology is inverse homothetic if

$$D_o^t\left( {{\boldsymbol{x}}{\rm{,}}\,{\boldsymbol{y}}} \right) = D_o^t\left( {\overline {\boldsymbol{x}} {\rm{,}}\,{\boldsymbol{y}}} \right){\rm{/}}F\left( {D_i^t\left( {{\boldsymbol{\overline y}} ,{\boldsymbol{x}}} \right)} \right),$$

(11)

where F is a strictly positive, increasing and invertible function and $\left( {{\boldsymbol{\overline x}} ,{\boldsymbol{\overline y}} } \right)$ are arbitrary input-output vectors.⁷

Färe et al. (1996) have been often misunderstood to claim that these two conditions are necessary and sufficient for equivalence between the output-oriented Malmquist and Hicks–Moorsteen productivity indexes.⁸ However, the productivity indexes they deal with are those relying on a specific reference technology, such as those defined by Eqs. (5) and (7). They are not the Malmquist and Hicks–Moorsteen productivity indexes defined by Eqs. (6) and (8), which are the geometric means of a productivity index relative to period 0 technology and a productivity index relative to period 1 technology. This implies that the output-oriented Malmquist and Hicks–Moorsteen productivity indexes defined by Eqs. (6) and (8) might coincide under different circumstances.

The following homotheticity condition is less demanding than inverse homotheticity.

Definition 3

The technology is output homothetic⁹ if

$$D_o^t\left( {{\boldsymbol{x}},{\boldsymbol{y}}} \right) = D_o^t\left( {{\bf 1},{\boldsymbol{y}}} \right)/{G^t}\left( {\boldsymbol{x}} \right),$$

(12)

where G ^t is a non-increasing function consistent with the regularity conditions T.1–T.6¹⁰ and 1 = (1,…,1) is a constant input vector.¹¹

Inverse homotheticity implies output homotheticity. Thus, the condition where the technology exhibits constant returns to scale and output homotheticity is weaker than the condition where the technology exhibits constant returns to scale and inverse homotheticity, which is adopted by Färe et al. (1996). Since a weaker condition allows a higher number of technologies to exist, the possibility of two productivity indexes coinciding is lower than when under a more stringent condition. However, the following proposition shows that the two productivity indexes will still coincide under such weaker conditions.¹²

Proposition 1

Assume the technology exhibits constant returns to scale and output homotheticity. The output-oriented Malmquist and Hicks–Moorsteen productivity indexes will then coincide: M = HM.¹³

Proof

$$\begin{array}{ccccc}\\ M = & {\left( {\frac{{D_o^0\left( {{\bf 1},\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^0\left( {{\bf 1},\,{{\boldsymbol{y}}^0}} \right)}}\frac{{D_o^1\left( {{\bf 1},\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^1\left( {{\bf 1},\,{{\boldsymbol{y}}^0}} \right)}}} \right)^{1/2}}/{\left( {\frac{{{G^0}\left( {{{\boldsymbol{x}}^1}} \right)}}{{{G^0}\left( {{{\boldsymbol{x}}^0}} \right)}}\frac{{{G^1}\left( {{{\boldsymbol{x}}^1}} \right)}}{{{G^1}\left( {{{\boldsymbol{x}}^0}} \right)}}} \right)^{1/2}} \;{\rm{from}}\;{\rm{(12)}},\hfill \\ \\ = & {\left( {\frac{{D_o^0\left( {{\bf 1},\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^0\left( {{\bf 1},\,{{\boldsymbol{y}}^0}} \right)}}\frac{{D_o^0\left( {{\bf 1},\,{{\boldsymbol{y}}^0}} \right)}}{{D_o^0\left( {{\bf 1},\,{{\boldsymbol{y}}^0}} \right)}}\frac{{D_o^1\left( {{\bf 1},\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^1\left( {{\bf 1},\,{{\boldsymbol{y}}^0}} \right)}}\frac{{D_o^1\left( {{\bf 1},\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^1\left( {{\bf 1},\,{{\boldsymbol{y}}^1}} \right)}}} \right)^{1/2}}/{\left( {\frac{{{G^0}\left( {{{\boldsymbol{x}}^1}} \right)}}{{{G^0}\left( {{{\boldsymbol{x}}^0}} \right)}}\frac{{{G^1}\left( {{{\boldsymbol{x}}^1}} \right)}}{{{G^1}\left( {{{\boldsymbol{x}}^0}} \right)}}} \right)^{1/2}} \hfill \\ \\ = & {\left( {\frac{{D_o^0\left( {{\bf 1},\,{{\boldsymbol{y}}^1}} \right)/{G^0}\left( {{{\boldsymbol{x}}^0}} \right)}}{{D_o^0\left( {{\bf 1},\,{{\boldsymbol{y}}^0}} \right)/{G^0}\left( {{{\boldsymbol{x}}^0}} \right)}}\frac{{D_o^0\left( {{\bf 1},\,{{\boldsymbol{y}}^0}} \right)/{G^0}\left( {{{\boldsymbol{x}}^1}} \right)}}{{D_o^0\left( {{\bf 1},\,{{\boldsymbol{y}}^0}} \right)/{G^0}\left( {{{\boldsymbol{x}}^0}} \right)}}\frac{{D_o^1\left( {{\bf 1},\,{{\boldsymbol{y}}^1}} \right)/{G^1}\left( {{{\boldsymbol{x}}^1}} \right)}}{{D_o^1\left( {{\bf 1},\,{{\boldsymbol{y}}^0}} \right)/{G^1}\left( {{{\boldsymbol{x}}^1}} \right)}}\frac{{D_o^1\left( {{\bf 1},\,{{\boldsymbol{y}}^1}} \right)/{G^1}\left( {{{\boldsymbol{x}}^1}} \right)}}{{D_o^1\left( {{\bf 1},\,{{\boldsymbol{y}}^1}} \right)/{G^1}\left( {{{\boldsymbol{x}}^0}} \right)}}} \right)^{1/2}} \hfill \\ \\ = & {\left( {\frac{{D_o^0\left( {{{\boldsymbol{x}}^0},\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^0\left( {{{\boldsymbol{x}}^0},\,{{\boldsymbol{y}}^0}} \right)}}\frac{{D_o^0\left( {{{\boldsymbol{x}}^1},\,{{\boldsymbol{y}}^0}} \right)}}{{D_o^0\left( {{{\boldsymbol{x}}^0},\,{{\boldsymbol{y}}^0}} \right)}}\frac{{D_o^1\left( {{{\boldsymbol{x}}^1},\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^1\left( {{{\boldsymbol{x}}^1},\,{{\boldsymbol{y}}^0}} \right)}}\frac{{D_o^1\left( {{{\boldsymbol{x}}^1},\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^1\left( {{{\boldsymbol{x}}^0},\,{{\boldsymbol{y}}^1}} \right)}}} \right)^{1/2}}\;{\rm{from}}\; {\rm{(12)}}, \hfill \\ \\ = & {\left( {\frac{{D_o^0\left( {{{\boldsymbol{x}}^0},\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^0\left( {{{\boldsymbol{x}}^0},\,{{\boldsymbol{y}}^0}} \right)}}\frac{{D_i^0\left( {{{\boldsymbol{y}}^0},\,{{\boldsymbol{x}}^0}} \right)}}{{D_i^0\left( {{{\boldsymbol{y}}^0},\,{{\boldsymbol{x}}^1}} \right)}}\frac{{D_o^1\left( {{{\boldsymbol{x}}^1},\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^1\left( {{{\boldsymbol{x}}^1},\,{{\boldsymbol{y}}^0}} \right)}}\frac{{D_i^1\left( {{{\boldsymbol{y}}^1},\,{{\boldsymbol{x}}^0}} \right)}}{{D_i^1\left( {{{\boldsymbol{y}}^1},\,{{\boldsymbol{x}}^1}} \right)}}} \right)^{1/2}}\;{\rm{from}} \;{\rm{(10)}}, \hfill \\ \\ = & {\left( {\frac{{D_o^0\left( {{{\boldsymbol{x}}^0},\,{{\boldsymbol{y}}^1}} \right)/D_o^0\left( {{{\boldsymbol{x}}^0},\,{{\boldsymbol{y}}^0}} \right)}}{{D_i^0\left( {{{\boldsymbol{y}}^0},\,{{\boldsymbol{x}}^1}} \right)/D_i^0\left( {{{\boldsymbol{y}}^0},\,{{\boldsymbol{x}}^0}} \right)}}\frac{{D_o^1\left( {{{\boldsymbol{x}}^1},\,{{\boldsymbol{y}}^1}} \right)/D_o^1\left( {{{\boldsymbol{x}}^1},\,{{\boldsymbol{y}}^0}} \right)}}{{D_i^1\left( {{{\boldsymbol{y}}^1},\,{{\boldsymbol{x}}^1}} \right)/D_i^1\left( {{{\boldsymbol{y}}^1},\,{{\boldsymbol{x}}^0}} \right)}}} \right)^{1/2}} = HM.\;{\rm{QED}} \hfill \\ \end{array}$$

As we demonstrate that the output-oriented Malmquist productivity index coincides with the Hicks–Moorsteen productivity index under output homotheticity, similar reasoning can be applied for showing that the reciprocal of the input-oriented Malmquist productivity index coincides with the Hicks–Moorsteen productivity index under input homotheticity, which is the input counterpart of output homotheticity.¹⁴

4 Malmquist and Hicks–Moorsteen productivity indexes under Hicks neutral technical change

So far, we have dealt with several conditions on the underlying technology. Two productivity indexes are shown to coincide under some restrictions on technology of every period. This section considers the following restriction on the types of technical change.

Definition 4

Technical change is Hicks output-neutral¹⁵ if

$$D_o^t\left( {{\boldsymbol{x}},\,{\boldsymbol{y}}} \right) = A\left( t \right){\overline D _o}\left( {{\boldsymbol{x}},\,{\boldsymbol{y}}} \right)$$

(13)

We demonstrate that two productivity indexes coincide under Hicks neutral technical change even when we do not impose strong conditions on technology, such as inverse or output homotheticity.

Proposition 2

Assume the technology exhibits constant returns to scale and technical change is Hicks output-neutral. Then, the output-oriented Malmquist and the Hicks–Moorsteen productivity indexes coincide: M = HM.¹⁶

Proof

$$\begin{array}{ccccc}\\ M = & {\left( {\frac{{A\left( 0 \right){{\bar D}_o}\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^1}} \right)}}{{A\left( 0 \right){{\bar D}_o}\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^0}} \right)}}\frac{{A\left( 1 \right){{\bar D}_o}\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^1}} \right)}}{{A\left( 1 \right){{\bar D}_o}\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^0}} \right)}}} \right)^{1/2}} = \frac{{{{\bar D}_o}\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^1}} \right)}}{{{{\bar D}_o}\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^0}} \right)}} \;{\rm{from}}\;(13),\hfill \\ \\ = & {\left( {\frac{{{{\bar D}_o}\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^1}} \right)}}{{{{\bar D}_o}\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^0}} \right)}}\frac{{{{\bar D}_o}\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^0}} \right)}}{{{{\bar D}_o}\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^0}} \right)}}\frac{{{{\bar D}_o}\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^1}} \right)}}{{{{\bar D}_o}\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^0}} \right)}}\frac{{{{\bar D}_o}\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^1}} \right)}}{{{{\bar D}_o}\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^1}} \right)}}} \right)^{1/2}} \hfill \\ \\ = & {\left( {\frac{{A\left( 0 \right){{\bar D}_o}\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^1}} \right)}}{{A\left( 0 \right){{\bar D}_o}\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^0}} \right)}}\frac{{A\left( 0 \right){{\bar D}_o}\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^0}} \right)}}{{A\left( 0 \right){{\bar D}_o}\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^0}} \right)}}\frac{{A\left( 1 \right){{\bar D}_o}\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^1}} \right)}}{{A\left( 1 \right){{\bar D}_o}\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^0}} \right)}}\frac{{A\left( 1 \right){{\bar D}_o}\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^1}} \right)}}{{A\left( 1 \right){{\bar D}_o}\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^1}} \right)}}} \right)^{1/2}} \hfill \\ \\ = & {\left( {\frac{{D_o^0\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^0\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^0}} \right)}}\frac{{D_o^0\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^0}} \right)}}{{D_o^0\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^0}} \right)}}\frac{{D_o^1\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^1\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^0}} \right)}}\frac{{D_o^1\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^1\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^1}} \right)}}} \right)^{1/2}}\;{\rm{from}}\;(13), \hfill \\ \\ = & {\left( {\frac{{D_o^0\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^0\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^0}} \right)}}\frac{{D_i^0\left( {{{\boldsymbol{y}}^0},\,{{\boldsymbol{x}}^0}} \right)}}{{D_i^0\left( {{{\boldsymbol{y}}^0},\,{{\boldsymbol{x}}^1}} \right)}}\frac{{D_o^1\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^1}} \right)}}{{D_o^1\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^0}} \right)}}\frac{{D_i^1\left( {{{\boldsymbol{y}}^1},\,{{\boldsymbol{x}}^0}} \right)}}{{D_i^1\left( {{{\boldsymbol{y}}^1},\,{{\boldsymbol{x}}^1}} \right)}}} \right)^{1/2}}\;{\rm{from}}\;(10), \hfill \\ \\ = & {\left( {\frac{{D_o^0\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^1}} \right)/D_o^0\left( {{{\boldsymbol{x}}^0},\,\,{{\boldsymbol{y}}^0}} \right)}}{{D_i^0\left( {{{\boldsymbol{y}}^0},\,{{\boldsymbol{x}}^1}} \right)/D_i^0\left( {{{\boldsymbol{y}}^0},\,{{\boldsymbol{x}}^0}} \right)}}\frac{{D_o^1\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^1}} \right)/D_o^1\left( {{{\boldsymbol{x}}^1},\,\,{{\boldsymbol{y}}^0}} \right)}}{{D_i^1\left( {{{\boldsymbol{y}}^1},\,{{\boldsymbol{x}}^1}} \right)/D_i^1\left( {{{\boldsymbol{y}}^1},\,{{\boldsymbol{x}}^0}} \right)}}} \right)^{1/2}} = HM.\;{\rm{QED}}\\ \end{array}$$

One implication from the proof is that both Malmquist and Hicks–Moorsteen productivity indexes are independent of the reference period under the condition of Proposition 2 so that $M = HM = {\bar D_o}\left( {{{\boldsymbol{x}}^1},\,{{\boldsymbol{y}}^1}} \right)/{\bar D_o}\left( {{{\boldsymbol{x}}^0},\,{{\boldsymbol{y}}^0}} \right)$. It guarantees that both indexes satisfy transitivity under these conditions such as ${\bar D_o}\left( {{{\boldsymbol{x}}^2},\,{{\boldsymbol{y}}^2}} \right)/{\bar D_o}\left( {{{\boldsymbol{x}}^0},\,{{\boldsymbol{y}}^0}} \right) = <$> <$> \left( {{{\bar D}_o}\left( {{{\boldsymbol{x}}^2},\,{{\boldsymbol{y}}^2}} \right)/{{\bar D}_o}\left( {{{\boldsymbol{x}}^1},\,{{\boldsymbol{y}}^1}} \right)} \right) \times \left( {{{\bar D}_o}\left( {{{\boldsymbol{x}}^1},\,{{\boldsymbol{y}}^1}} \right)/{{\bar D}_o}\left( {{{\boldsymbol{x}}^0},\,{{\boldsymbol{y}}^0}} \right)} \right)$. As emphasised by O’Donnell (2012, 2016), violating transitivity is an well-known shortcoming of the Malmquist and Hicks-Moorsteen productivity indexes. However, while these indexes possibly violate transitivity, they do not necessarily do so. They are shown to always satisfy transitivity as long as technical change is Hicks output-neutral.¹⁷

As we show that the output-oriented Malmquist productivity index coincides with the Hicks–Moorsteen productivity index under Hicks output-neutral technical change, similar reasoning can be applied for showing that the reciprocal of the input-oriented Malmquist productivity index coincides with the Hicks–Moorsteen productivity index under Hicks input-neutral technical change, which is the input counterpart of Hicks output-neutral technical change.¹⁸

5 Conclusions

This study examines the relationship between the Malmquist and Hicks–Moorsteen productivity indexes. We find that they coincide either when technology exhibits constant returns to scale and, at the same time, technical change is Hicks-neutral or when the technology exhibits constant returns to scale as well as output (input) homotheticity.

Distances between multiple isoquants measured along different reference vectors constitute both productivity indexes. Hicks neutrality and homotheticity induce the radial expansion of isoquants. Thus, under these assumptions, the distances between some isoquants become constant, regardless of the selected reference vector. Thus, some components of the two productivity indexes, which adopt different reference vectors for comparing isoquants, coincide, which yields equivalence between the Malmquist and Hicks–Moorsteen productivity indexes.

Our result also reaffirms that the constant returns to scale technology is an indispensable condition for deriving equality between the two productivity indexes.¹⁹ While the Malmquist productivity index is thoroughly defined by either the output or input distance function, the Hicks–Moorsteen productivity index is formulated by output distance function as well as input distance function. Thus, we need to reformulate the Hicks–Moorsteen productivity index based on either the output distance function only or the input distance function only. The assumption of constant returns to scale technology plays a crucial role for transferring from input distance function to output distance function or from output distance function to input distance function.²⁰

The present study shows that the Malmquist and Hicks–Moorsteen productivity indexes are more likely to coincide with each other than previously thought. One of the advantages of the equality result is to allow us to apply theoretical results derived for one index to the other. For example, Zelenyuk (2006) and Mayer and Zelenyuk (2014) propose the procedure of aggregating the Malmquist productivity index. Their results can be directly adopted for aggregating the Hicks-Moorsteen productivity index under the conditions examined in this study.