On Luenberger input, output and productivity indicators

Accepted Manuscript On Luenberger input, output and productivity indicators Rolf Färe, Valentin Zelenyuk

PII: DOI: Reference:

S0165-1765(19)30103-X https://doi.org/10.1016/j.econlet.2019.03.024 ECOLET 8406

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Economics Letters

Received date : 26 February 2019 Revised date : 19 March 2019 Accepted date : 20 March 2019 Please cite this article as: R. Färe and V. Zelenyuk, On Luenberger input, output and productivity indicators. Economics Letters (2019), https://doi.org/10.1016/j.econlet.2019.03.024 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

On Luenberger Input, Output and

Productivity Indicators

Rolf Färe

∗

and Valentin Zelenyuk

†

March 22, 2019

Abstract In this paper we consider justications for the equally-weighted arithmetic averaging for the Luenberger indicators with respect to two dierent references, introduced in Chambers (2002).

Keywords: Luenberger Productivity Indicators. JEL Code: D24.

Department of Economics, Oregon State University and Department of Agricultural and Resource Economics, University of Maryland, USA † . School of Economics and Centre for Eciency and Productivity Analysis (CEPA) at The University of Queensland, Australia; 530, Colin Clark Building (39), St Lucia, Brisbane, Qld 4072, AUSTRALIA; tel: + 61 7 3346 7054; e-mail: [email protected]. ∗

Corresponding author

Address:

1

1

Introduction

Based on directional distance functions,

1

Chambers (2002) introduced input, output and

productivity indicators as an additive alternative to the corresponding Malmquist indexes.

2

Each indicator is the arithmetic mean of two indicators. These may be indicators of dierent time periods or technologies in general. We derive the equally-weighted arithmetic mean as the only aggregator that satises certain desired properties.

While we limit ourselves to Chambers' input indicator to be

concise, our derivations also apply to the output analogue and to the productivity indicators, which are made as the combination of the input and output indicators. This paper is in the spirit of Diewert and Fox (2017) for the context of theoretically justifying the square-root of two Hicks-Moorsteen/Bjurek total factor productivity indexes.

2

The Luenberger Input Indicator

Let

L t (y)

be an input requirement set representing the technology at time

t,

dened in

general terms as

L t (y) = {x : x can where

x = (x1 , . . . , xN )0 ∈ RN +

produce

y

denotes an input vector and

at time

t}

y = (y1 , . . . , yM )0 ∈ RM +

(1)

denotes an

output vector. We assume the production technology satises standard regularity conditions,

3

including strong disposability of inputs and outputs. The

t-period

directional input distance function is given by

→ −t D i (x, y; gx ) = max{β ∈ R : (x − βgx ) ∈ L t (y)}, β

See Chambers et al. (1996). See Caves et al. (1982). 3 See Färe and Primont (1995). 1 2

2

(2)

where

gx ∈ RN +

and

gx 6= 0N

direction in which input

x

is the directional vector in the input space that denes the

is contracted towards the boundary of

Given two situation or states,

r-indicator

and the

3

is dened as

s-indicator

r

and

s

L t (y).4

(which may be time period or technologies), the

5

→ − → − X r (xs , xr , y r ) = D ri (xs , y r ; gx ) − D ri (xr , y r ; gx )

(3)

→ − → − X s (xs , xr , y s ) = D si (xs , y s ; gx ) − D si (xr , y s ; gx ).

(4)

is

Aggregation of Luenberger Indicators

In general, the indicators discussed in the previous section, (3) and (4), may provide dierent numbers for the same data set, possibly radically dierent in the sense that they may imply dierent conclusions or related policy implications. It is therefore very important to either have some solid justications for using either one of them or to have a well-justied way to aggregate the two. The latter approach is indeed a natural solution researchers frequently take in such circumstances. An important theoretical question is: What aggregation function (or what type of averaging) is to be taken? Below we consider two alternative justications that lead to the same conclusion.

3.1

Functional equations approach

Due to the additive nature of these indicators, arithmetic averaging appears as the most natural to be used here.

Indeed, Aczél (1966, p.234) proved that the weighted arithmetic

mean is the most general solution to a functional equation that meets the translation and

The idea of this function goes back to at least Allais (1943), Diewert (1983) and Luenberger (1992) and thoroughly developed by Chambers et al. (1996). 5 For its properties, see Chambers et al. (1996) or Chambers (2002). 4

3

homogeneity conditions, i.e., a function

f : R2 → R

satisfying

f (p + γ, q + γ) = f (p, q) + γ, γ ∈ R

(5)

f (λp, λq) = λf (p, q), ∀λ > 0.

(6)

and

This leads us to take the weighted arithmetic mean of the two indicators,

X(xs , xr , y s , y r ) = ω1 X r (xs , xr , y r ) + ω2 X s (xs , xr , y s )

where the weights or shares If the indicator

ω1 , ω2 = 0

X(xs , xr , y s , y r )

and

(7)

ω1 + ω2 = 1.

is also symmetric in the sense that

X(xs , xr , y s , y r ) = ω1 X r (xs , xr , y r ) + ω2 X s (xs , xr , y s ) = ω1 X s (xs , xr , y s ) + ω2 X r (xs , xr , y r )

then, to encompass all the cases including when true thatω1

= ω2 .

Furthermore, since

X s (xs , xr , y s ) 6= X r (xs , xr , y r ),

ω1 + ω2 = 1,

(8)

it must be

we have

ω1 = ω2 = 1/2

and thus we found the Luenberger input indicator dened by Chambers (2002), namely

X(xs , xr , y s , y r ) =

1 (X r (xs , xr , y r ) + X s (xs , xr , y s )) . 2

4

(9)

3.2

Anchoring on the time-reversal property

Here we present another way of justifying the  1/2 in the aggregation of the indicators measured with respect to the dierent references. To do so, we develop an additive version of the approach proposed by Diewert and Fox (2017), where they derived the 

√

 for the

Hicks-Moorsteen/Bjurek total factor productivity index. We now dene an additive version of the time-reversal test for an indicator a vector

x0

to a vector

x1

I that compares

as a property that requires that

I(x0 , x1 ) = −I(x1 , x0 ).

(10)

To mimic the Diewert-Fox proof, we introduce a new notation in their spirit. In particular, the Laspeyres-Luenberger input indicator is dened as

→ − → − LL (x0 , x1 ) = D 0i (x1 , y 0 ; gx ) − D 0i (x0 , y 0 ; gx )

(11)

and the Paasche-Luenberger input indicator is dened as

→ − → − LP (x0 , x1 ) = D 1i (x1 , y 1 ; gx ) − D 1i (x0 , y 1 ; gx ) These indicators are obtained from above by setting

r=0

and

s = 1.6

(12)

The Luenberger

input quantity indicator of Chambers (2002) is then written as

L(x0 , x1 ) =


1 LL (x0 , x1 ) + LP (x0 , x1 ) 2

(13)

and again we want to theoretically justify this type of aggregation. To do so, denote

a = LL (x0 , x1 )

and

b = LP (x0 , x1 )

and, similar to Diewert and Fox

(2017), the problem reduces to nding/justifying a suitable aggregation function

m(a, b)

Also note that we simplify the notation by denoting LL (x0 , x1 ) for LL(x0, x1|y0 , gx) and LP (x0, x1) for LP (x0 , x1 |y 1 , gx ). 6

5

that will aggregate the two indicators measured with respect to dierent technologies, i.e., nd a function

m : R2 → R,

such that


m(a, b) = m LL (x0 , x1 ), LP (x0 , x1 ) = L(x0 , x1 ), for all relevant compinations of

x0

and

x1

and satisfying some desirable properties.

Unlike in Diewert and Fox (2017), here we do not (and should not) assume that

m(a, b) > 0

because by its very nature the aggregate indicators should be allowed to be positive or negative, reecting the progress or regress, respectively.

What we do need in our case of

aggregating the indicators, again due to their additive nature, is the translation property, i.e.,

m(a + λ, b + λ) = m (a, b) + λ,

λ ∈ R.

Now, let us reverse the order of time in our denitions of the indicators, to obtain

→ − → − LL (x1 , x0 ) = D 1i (x0 , y 1 ; gx ) − D 1i (x1 , y 1 ; gx ) →  − → − = − D 1i (x1 , y 1 ; gx ) − D 1i (x0 , y 1 ; gx ) = −LP (x0 , x1 ) = −b.

and

→ − → − LP (x1 , x0 ) = D 0i (x0 , y 0 ; gx ) − D 0i (x1 , y 0 ; gx ) →  − → − = − D 0i (x1 , y 0 ; gx ) − D 0i (x0 , y 0 ; gx ) = −LL (x0 , x1 ) = −a.

6

(14)

And therefore, the Luenberger indicators are given by


L(x0 , x1 ) = m LL (x0 , x1 ), LP (x0 , x1 ) = m(a, b)

and


L(x1 , x0 ) = m LL (x1 , x0 ), LP (x1 , x0 ) = m(−b, −a).

Now, note that the time reversal test (10) applied to the Luenberger indicators imply that

L(x0 , x1 ) = −L(x1 , x0 ),

(15)

and so combining it together with the previous two expressions, we also must have

m(a, b) = −m(−b, −a), and therefore, using the translation property (14) we get

0 = m (a, b) + m (−b, −a) = m (a − a, b − a) + a + m (−b + b, −a + b) − b = m (0, b − a) + a + m (0, b − a) − b

7

(16)

or

1 (b − a). 2

m (0, b − a) = Now, adding

a

to both sides of the last equation, we get

m (0, b − a) + a =

1 (b − a) + a 2

and again applying the translation property (14) to the left hand side of the last equation and simplifying the right hand side of the equation, we get the required form of the aggregation function

m

given by

m (a, b) =

1 (b + a), 2

which is the unique solution to the stated problem.

4

Concluding Remarks

We obtained two theoretical justications for the equally-weighted arithmetic averaging for the Luenberger input indicators that are measured with respect to dierent references. One justication is based on functional equations approach (e.g., Aczél (1966)), involving translation, homogeneity and symmetry conditions.

The other approach is based on adapting

ideas from Diewert and Fox (2017), involving translation property and an additive version of the time-reversal test, which we introduced in this paper as an adaptation of the usual (multiplicative) time reversal test that is popular in the literature on index numbers. While here we focused on the Luenberger input indicators, analogous developments can be outlined for the Luenberger output indicators and, combining both of them, for the Luenberger productivity indicators.

8

Acknowledgements The authors thank the Editor and anonymous referee as well as other colleagues (especially Erwin Diewert, Duc Manh Pham, Bao Hoang Nguyen) for their fruitful feedback on this paper.

The authors also acknowledge the nancial support from ARC Future Fellowship

grant (FT170100401).

9

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Allais, M., 1943. Traitè D'Èconomie Pure. Vol. 3. Paris, FR: Imprimerie Nationale.

Caves, D. W., Christensen, L. R., Diewert, W. E., 1982. The economic theory of index numbers and the measurement of input, output, and productivity. Econometrica 50 (6), 13931414.

Chambers, R., 2002. Exact nonradial input, output, and productivity measurement. Economic Theory 20, 751765.

Chambers, R. G., Chung, Y., Färe, R., 1996. Benet and distance functions. Journal of Economic Theory 70 (2), 407419.

Diewert, W. E., 1983. The measurement of waste within the production sector of an open economy. The Scandinavian Journal of Economics 85 (2), 159179.

Diewert, W. E., Fox, K. J., 2017. Decomposing productivity indexes into explanatory factors. European Journal of Operational Research 256 (1), 275291.

Färe, R., Primont, D., 1995. Multi-Output Production and Duality: Theory and Applications. New York, NY: Kluwer Academic Publishers.

Luenberger, D. G., 1992. Benet functions and duality. Journal of Mathematical Economics 21 (5), 461481.

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