MTT Economic Research 
    Discussion Papers 
    Number 2004:17 
 
 
On the Anatomy of      
Productivity Growth:       
A Decomposition of the Fisher 
Ideal TFP Index 
 
Kuosmanen, T. 
Sipiläinen, T. 
  October 2004 
On the Anatomy of Productivity Growth: 
A Decomposition of the Fisher Ideal TFP Index 
 
Timo Kuosmanen1, Timo Sipiläinen2 
 
1) Wageningen University, Environmental Economics and Natural Resources Group, P.O. Box 8130, 6700 EW 
Wageningen, The Netherlands, Tel. +31 317 484 738. Fax. +31 317 484 933, E-mail. Timo.Kuosmanen@wur.nl 
 
2) Agrifood Research Finland, MTT Economic Research, Luutnantintie 13, 00410 Helsinki, Finland, Tel. +358 9 
5608 6221, Fax: + 358 9 563 1164, Email Timo.Sipilainen@mtt.fi 
 
 
Abstract 
Decompositions of productivity indices contribute to our understanding of what drives the observed productivity 
changes by providing a detailed picture of their constituents. This paper presents the most comprehensive 
decomposition of total factor productivity (TFP) to date. Starting from the Fisher ideal TFP index, we systematically 
isolate the productivity effects of changes in production technology, technical efficiency, scale efficiency, allocative 
efficiency, and the market strength. The three efficiency components further decompose into input- and output-side 
effects. The proposed decomposition is illustrated with an empirical application to a sample of 459 Finnish farms 
over period 1992-2000.  
 
Key Words: index numbers and aggregation, Total Factor Productivity (TFP) measurement, Fisher ideal index, 
Malmquist index, decompositions, agriculture 
 
JEL classification: C43, D24, O30 
 
 
 
 
__________________________________________________________________________________________ 
Suggested citation: Kuosmanen, T. & Sipiläinen, T.  (2004). On the Anatomy of Productivity Growth: 
A Decomposition of the Fisher Ideal TFP Index. MTT Economic Research, Agrifood Research Finland. 
Discussion Papers 2004/17. 
1. Introduction 
Traditionally, productivity has been viewed as a synonym to technical progress, following the influential work of 
Solow (1957). Today, development of more sophisticated methods for measuring total factor productivity (TFP) 
enables us to distinguish technological development from other sources of productivity changes, most notably, 
changes in technical and scale efficiency (see e.g. Färe et al., 1994a). As a consequence, many recent studies 
interpret productivity in broader terms as a welfare measure, which includes the effect of technical change among 
other components (e.g. Basu and Fernald, 1997; Kumar and Russell, 2002). From the welfare perspective, it is 
interesting to investigate the relative importance of the different effects that influence productivity. Decomposing 
the overall productivity index into sub-components can provide more detailed information about the underlying 
sources of productivity changes.  
Decompositions of productivity date back at least to the seminal work of Farrell (1957), who expressed the 
overall economic efficiency as a product of the technical efficiency and allocative (price) efficiency components. 
Farrell restricted to a static framework, which left no room for technical progress. Nishimizu and Page (1982) 
proposed a first dynamic extension of Farrell’s decomposition, which included technical change and technical 
efficiency components. Färe et al. (1994a,c) generalized and developed the decomposition further, introducing the 
scale efficiency component. These decompositions are based on the Malmquist productivity index by Caves et al. 
(1982), which has gained momentum after Färe et al. presented their decomposition, especially in the firm-level 
applications. Indeed, the decomposability of the Malmquist index is generally seen as its main advantage to other 
productivity indices, together with its weaker data requirements. Similar decompositions have been extended to 
variants of the Malmquist index such as the Hicks-Moorsten type productivity index of Bjurek (1994) (see Lovell, 
2003) and to the Malmquist-Luenberger type index (see Färe and Primont, 2003). However, the classic productivity 
indices such as the Fisher ideal TFP index and the Törnqvist TFP index currently lack such decompositions.  
This paper intends to fill this gap at least partly by proposing an intuitive decomposition for the Fisher ideal 
TFP index (henceforth ‘the Fisher index’). Following the approach of Färe et al. (1994a,c), we express the Fisher 
index as a product of five components that represent changes in 1) production technology, 2) technical efficiency, 
and 3) scale efficiency. In addition to these standard components, our decomposition includes 4) allocative 
efficiency, and 5) market strength components. Components 2) – 4) can be further decomposed into separate sub-
components for inputs and outputs, offering a detailed picture of the constituents of productivity change as 
measured by the Fisher index. 
 2 
Our decomposition contributes to the productivity literature in at least four important ways. Firstly, the 
decomposition further enhances our general understanding about how the Fisher index works (i.e., which effects it 
captures, and which ones not). While a number of decompositions of the Fisher ideal price and quantity indices 
have been proposed (see Balk, 2003, for a recent review), we are unaware of other decompositions in the present 
context of productivity indices. Secondly, the decomposition provides a more detailed picture about the driving 
forces behind productivity growth (or decline) in empirical applications (i.e., which effects have improved 
productivity, and which ones have deteriorated it). For example, in the application of Section 9 we will find that the 
decline in the market strength explains the slow productivity growth in the Finnish agriculture in the aftermath of 
Finland’s EU accession in 1995. Thirdly, our decomposition recognizes price effects as important elements of 
productivity change. From the welfare perspective, not only the quantity of output is important – also quality 
matters. By attributing changes in economic value of inputs and outputs to the productivity index, we hope to 
capture at least some quality aspects of productivity. Fourthly, better understanding of the anatomy of the Fisher 
index also enables one to tailor the productivity index for the purposes of the study. If some components capture 
effects that do not fit in a given definition of productivity, the decomposition enables one to correct for (or eliminate) 
these undesirable components from the overall productivity index.  
Since our decomposition is closely related to (and inspired by) the decomposition of the Malmquist index (in 
particular the version by Färe et al., 1994c; see Appendix 1 for details), it is useful to compare the two in more 
detail. We consider the classic Fisher index to be a useful alternative for the nowadays widely used Malmquist 
approach: the Fisher index is simple to calculate from the data; it does not require empirical estimation of 
production technology; it is not influenced by arbitrary assumptions about the technology or its functional form; it 
has a well-established axiomatic foundation (see Diewert, 1992); and it is consistent with the index number 
formulae widely used for price and quantity indices by statistical agencies around the world. By contrast, the 
Malmquist index relies on the technology distance functions, which must be estimated empirically. This estimation 
requires a number of assumptions to be made. To begin with, the direction or orientation of measurement must be 
chosen; the choice of input distance function can lead to very different results than the choice of output distance 
function. Next, the choice of the estimation method and the assumptions involved may influence the results 
considerably.  
Of course, these disadvantages should be balanced against the two major advantages of the Malmquist 
index. The first concerns the data requirements: the Malmquist index only requires data on input and output 
 3 
quantities, whereas the Fisher index requires the complete price data in addition to the quantities. While the data 
requirements do favour the Malmquist index, it is worth to ask if it is easier to estimate the prices or the production 
technology when both are unknown. Moreover, if the assumption of allocative efficiency can be maintained, then 
the quantity data suffices for estimating bounds for the Fisher index, as shown by Kuosmanen et al. (2004) (who 
build on the work of Färe and Grosskopf, 1992; and Balk, 1993). The second benefit has been the decomposability 
of the Malmquist index. A number of alternative decompositions have been proposed (e.g., Färe et al. 1994a,c; 
Ray and Desli 1997). It should be noted, however, that the decompositions of the Malmquist index have also 
attracted critical comments. We refer to Lovell (2003) and Grosskopf (2003) for recent surveys of this debate. By 
starting from the Fisher ideal index, we hope to overcome some of the difficulties experienced with the Malmquist 
decompositions.   
The remainder of this paper is organized as follows. Section 2 introduces the standard notation and 
terminology used in production theory, and presents a formal duality theorem regarding profitability function. 
Sections 3 to 6 introduce the components of productivity change in a step-by-step manner, comparing with the 
analogous components available for the Malmquist index. Section 3 starts with the technical efficiency component 
followed by technical change in section 4. Section 5 discusses the definition of scale efficiency component. Section 
6 introduces the allocative efficiency component of the Fisher index and section 7 presents the market strength 
component.  Having introduced all components, in Section 8 we are finally equipped to fit all the components 
together in our main theorem, which proves that the Fisher ideal TFP index is obtained as a product of these 
intuitive components. Section 9 provides an empirical application on Finnish farm data. The paper finishes with 
some concluding remarks. 
  
2. Preliminaries 
Productivity growth is the change in output not explained by change in input use. To measure productivity changes 
in the general multiple input multiple output setting, we must aggregate the inputs and outputs in one way or 
another. Inputs (outputs) measured in different units do not add up as such, but must first be converted to some 
common scale of measurement, a natural choice being money. Indeed, the classic index theory uses unit prices (or 
cost shares) as weights of quantity index. Using prices as weights has the added advantage that the more precious 
or important commodities are given a greater weight than the inexpensive ones. In this vein, the conventional index 
theory typically uses the prices (or cost/revenue) shares as the weights of quantity indexes. Of course, the prices 
 4 
often change from one period to another, so we inevitably face the question of which set of prices provide the most 
appropriate weights (consider e.g. difference between the classic Paashe and Laspeyres indexes). Fisher (1922) 
solved this question in an ingenious way evaluating productivity at the weights of the base period and at the 
weights of the target period, and taking the geometric average of the two.  
 We limit attention to two-period comparisons, and denote the base period as period 0 and the target period 
as period 1. Adopting the standard notation,  represents the output quantity vector of period +∈\ty s ∈ {0,1}t  
and  the associated price vector. Similarly,  denotes the input quantity vector of period t and 
 the associated price vector. The Fisher ideal output and input quantity indices are defined as  
++∈\t sp
++∈\t r
+∈\t rx
w
(1) ≡0,1 0,1( , )yF p y  ⋅ ⋅⋅ ⋅ ⋅ 
1
0 1 1 1 2
0 0 1 0
p y p y
p y p y
 and ≡0,1 0,1( , )xF w x  ⋅ ⋅⋅ ⋅ ⋅ 
1
0 1 1 1 2
0 0 1 0
w x w x
w x w x
, 
respectively.  
Total factor productivity is usually defined as the ratio of the output quantity index to the input quantity 
index (e.g. Diewert, 1992; Bjurek, 1996). Using the Fisher ideal indexes to aggregate both inputs and outputs, the 
Fisher ideal TFP index is obtained simply as the ratio of the aggregate output to the aggregate input  
(2) ≡
0,1 0,1
0,1 0,1 0,1 0,1
0,1 0,1
( , )
( , , , )
( , )
y
TFP
x
F
F
F
p y
p w y x
w x
. 
This productivity measure is easily calculated given the necessary price and quantity data. It does not require 
estimation of any sort. Unlike the Malmquist index, the Fisher TFP index for a firm is independent on performance 
of other firms in the sample. 
To get a more detailed picture of the underlying sources of productivity changes, we next proceed to 
decompose the Fisher index (2) into subcomponents along the lines of Färe et al. (1994a,c). Our decomposition 
uses a number of alternative equivalent representations of technology, which will be introduced next. The 
production possibility set of period t is defined as  
(3) { }++= ∈\( , )  can produce  in period t r sT tx y x y .  
These sets are assumed to be non-empty, closed and satisfy the scarcity and no-free-lunch assumptions (see Färe 
and Primont, 1995).  Alternative set representations of technology are the input set 
{ }+= ∈ ∈\( ) ( , )t rL Ty x x y t  
 5 
and the output set { }+= ∈ ∈\( ) ( , )t sx y x y tP T . 
Input-output vector (x,y) is considered technically efficient if it lies on the boundary of set T. The degree of 
inefficiency is traditionally measured using Shephard’s (1953, 1970) distance functions. Slightly deviating from the 
original definition,1 we define the input distance function as 
(4) { }θ θ≡ ∈( , ) min ( , )t txD Tx y x y , 
and the output distance function as  
(5) { }θ θ≡ ∈( , ) min ( , / )t tyD Tx y x y . 
Distance functions can also be seen as representations of technology: we can recover Tt from input (output) 
distance function  when inputs (outputs) are weakly disposable (Färe and Primont, 1995).  
The minimum cost of producing a given target output y at given input prices w is given by the cost function 
(6) { }≡ ⋅ ∈( , ) min ( , )t t
x
C Tw y w x x y . 
Analogously, the maximum revenue that can be obtained given inputs x and output prices p is given by the 
revenue function  
(7) { }≡ ⋅ ∈( , ) max ( , )t t
y
R Tx p p y x y . 
By duality theory, these monetary expressions can also be viewed as technology representations. Specifically, cost 
function Ct enables us to recover the convex monotonic hull of the input set Lt, and analogously, we can recover 
the convex monotonic hull of the output set Pt from the revenue function Rt.  
As the only non-standard concept we introduce the profitability function 
(8) ρ  ⋅≡ ∈ ⋅ ,( , ) max ( , )
t t
x y
Tp yw p x y
w x
, 
which indicates the maximum return to the dollar achievable with the given non-negative input-output prices. The 
following duality theorem shows that (like profit function) profitability function can be taken as a representation of 
technology: 
 
                                                 
1 In our definition, the values of both the input distance function (4) and the output distance function (5) are conveniently restricted to the half-open interval 
(0,1]. Since in this paper we restrict to measuring efficiency of observed firm behaviour (which must be technically feasible to be observable to begin with), 
and since the production possibility sets Tt are assumed to be closed, the minimum of (4) and (5) will always exists. 
 6 
Theorem 1: If production possibility set Tt satisfies the assumptions of free disposability, convexity, and constant 
returns to scale, then profitability function  is a complete characterisation of the technology. In particular,  tρ
( , ) ( , ) ( , )t r s tT ρ+ ++ + ⋅= ∈ ≤ ∀ ∈ ⋅ 
p yx y w p w p
w x
\ \ r s . 
Proof. See Appendix 2.  
 
Under the assumptions of Theorem 1, the input distance function can be expressed in terms of profitability as  
(9)   
( , )
( , ) max ( , ) 1
r s
t t
xD ρ++∈
⋅ = ≤ ⋅ w p
p yx y w p
w x\
.  
This result will be used in the developments of the subsequent sections. 
 As a final remark, it should be noted that production possibility sets, distance functions, cost and revenue 
functions, and profitability functions are usually not known and must be estimated from empirical data. A number of 
techniques for such estimation are available, including Data Envelopment Analysis (DEA) and Stochastic Frontier 
Analysis (SFA) (e.g. Färe et al., 1994b, Kumbhakar and Lovell, 2000). This paper abstracts from the estimation 
issues and focuses solely on the decomposition. Recall that no estimations are needed for calculation of the Fisher 
input, output or TFP indices. Estimation is only needed for the decomposition. 
 
3. Technical efficiency  
Following Farrell (1957), we measure technical efficiency as a distance from the production frontier, using distance 
functions (4) and (5). In theory, operational inefficiency signals under-utilization of inputs, which in turn may depend 
on a number of factors such as managerial competence (Leibenstein, 1966) or incentive structure (Bogetoft, 1994, 
1995). Changes in input-utilization over time may also be attributed to the gradual diffusion of innovations (Färe et 
al., 1994c) or to the business cycle. Though we usually cannot isolate the different effects that influence the input 
utilization, it is still interesting to measure how much its changes contribute to productivity changes.    
Technical efficiency component is an integral part of the Malmquist decomposition. Decompositions 
following Färe et al. have traditionally oriented either towards inputs or towards outputs. The input oriented 
measure of technical efficiency change 1 1 1 0 0 0( , ) / ( , )x xDxD y x y  gauges the movements towards or away from the 
frontier in the direction of inputs, holding the output levels as constant. By contrast, the output oriented measure of 
 7 
technical efficiency change 1 1 1 0 0 0( , ) / ( , )y yxD Dy x y  gauges the movements towards or away from the frontier in 
the direction of outputs, holding the input levels as constant. The well-known result of Färe and Lovell (1978) 
implies that these two approaches are equivalent if and only if the production sets T0 and T1 exhibit constant 
returns to scale. 
 ⋅   
1
1 1 1 21
0 0 0 0
1 1
0 0
( , )( , )
( , ) ( , )
y
x y
D
D D
x yx y
x y x y
)
If the technology exhibits variable returns to scale, the choice of orientation does matter. Since we have no 
particular reason to prefer input or output orientation over another, we propose to resolve the problem in the 
traditional fashion (of Irving Fisher) by measuring efficiency change by the geometric mean of the ratios of input 
distance functions and output distance functions. This gives the following overall measure of technical efficiency 
change 
(10) ∆ =TEff  xD
Under constant returns to scale, this measure is equivalent to the usual technical efficiency component of the 
Malmquist index. Attractively, under variable returns to scale this measure is invariant to the choice of orientation. If 
the orientation makes a difference, then we can distinguish between the sub-components of input efficiency 
( 11 1 1 0 0 0 2( , ) / ( , )x xD Dx y x y  and the output efficiency ( ) 11 1 1 0 0 0 2( , ) / ( , )y yD Dx y x y .  In our decomposition, both 
will be accounted for with the equal weight. 
Measures of technical efficiency change are graphically illustrated in Figure 1. Figure 1a presents input 
oriented technical efficiency component in input-output space and Figure 1b in input-input space. In the figures, the 
input oriented technical efficiency change is =
1
1 1, 1 0 0, 0
0( , ) / ( , )x x
OB Ox
D x y D x y
OA Ox
. Similar relationship can be 
found in the output orientation, which is presented in Figures 1c and 1d as 
=
1
1 1, 1 0 0, 0
0( , ) / ( , )y y
Oy OD
D x y D x y
Oy OC
. Taking a geometric mean of these two provides our weighted average for 
technical efficiency. 
 8 
  
(x1, y1) 
(x0, y0) 
T0 
T1 
x1 B  x0 A 
Y 
O X 
 
Figure 1a. Input oriented technical efficiency change in input-output space. 
 
X2 
x0 
A 
L0 
x1 
L1 B 
O X1 
 
Figure 1b. Input oriented technical efficiency change in input-input space. 
 
 9 
 10 
 
 
Figure 1c. Output oriented technical efficiency change in input-output space.  
 
 
Figure 1d. Output oriented technical efficiency change in output-output space. 
 
 
 
Y2 
y1 
P1 P0 
D 
C 
y0 
D 
 y1 
y0 
(x1, y1) 
(x0, y0) 
T0 
T1 
C 
X O 
Y 
O Y1 
 4. Technical change 
The traditional perspective on technical progress is that of Hicks neutral frontier shift considered by Solow (1957). 
Recent economic literature has recognised the importance of biases in technical progress, which means that 
productivity of different inputs increases at different rates (e.g., Acemoglu, 2002). Both neutral and biased technical 
changes are accounted for by the technical change component of the Malmquist index, which measures the shifts 
in the production frontier by distance functions. By duality theory, input and output distance functions can be seen 
as equivalent representations of technology as production sets Tt.  
 In this paper we resort to another representation of technology, the profitability function . By Theorem 
1,  is an equally valid representation of technology as the distance function when T
ρ t
ρ t t satisfies free disposability, 
convexity, and constant returns to scale. It is common to measure technical change in terms of the constant-
returns-to-scale benchmark technology (e.g., Färe et al., 1994c). Thus, measuring technical change in terms of 
profitability functions  is equally legitimate as it is with distance functions.   ρ t
Consider ratio ρρ
1 0 0
0 0 0
( , )
( , )
w p
w p
, which represents the change of maximum profitability from the base period 
to the target period, at the prices of the base period. As the same price vectors appear both in the nominator and 
the denominator, any change in profitability is inevitably due to the change of technology. Technical progress 
would tend to increase profitability, and hence this ratio. Technical regress would decrease this ratio. We could 
similarly measure technical changes using the prices of the target period. Again, as we do not have any reason to 
prefer the prices of the base or target period, we express the technical change component as a geometric mean of 
these two: 
(11) ∆Tech ρ ρρ ρ
 = ⋅  
1
1 0 0 1 1 1 2
0 0 0 0 1 1
( , ) ( , )
( , ) ( , )
w p w p
w p w p
. 
We should note that the technical change component of Färe et al. (1994a,c) also measures technical 
change in terms of profitability, as the dual formulation of the distance function relative to a constant returns to 
scale technology can be expressed as profitability measure (see equality (9) above). Our measure for technical 
change can differ from that of Färe et al. to some extent because we measure profitability in terms of observed 
input-output prices, whereas Färe et al. use shadow prices. Like the component of Färe et al., our technical change 
 11 
measure does not capture technical progress that occurs in the part of the frontier exhibiting increasing or 
decreasing returns to scale if the frontier does not shift at the most productive scale size. In particular, our 
component only captures frontier shifts that have an impact on profitability; thus, it could be more precisely 
qualified as “profitability enhancing technical progress”. Other types of frontier shifts are attributed to the allocative 
or scale efficiency components (as in Färe et al. decomposition). For example, suppose a technical innovation 
helps to improve productivity of small firms, but has no effect on the profitability of the leading firms operating at the 
most productive scale size. Such a local frontier shift makes the small-scale production more attractive, and thus 
improves scale efficiency of small firms even if these firms do not expand their scale. Even though this productivity 
improvement was ultimately due to a technical innovation, it does not show up in the technical change component 
because the same productivity could have been achieved with the earlier technology if economies of scale were 
appropriately utilised. Thus, excluding these local frontier shifts from the technical change components seems 
justified.  
This measure is illustrated in Figure 2 where and  represent the maximum return per dollar (most 
profitable) lines in period 0 and 1. The change in maximum profitability is simply given by the following geometric 
expression: 
0ρ 1ρ
β
α∆ =
tan
tan
Tech . 
 
 
T1 
β α 
ρ1 Y 
ρ0 
T0 
O X 
Figure 2. Technical change component, measured with profitability functions 
 12 
5. Scale efficiency 
Economies (and diseconomies) of scale can obviously contribute to productivity. Indeed, varying utilisation of scale 
economies in different stages of the business cycle is considered as one plausible explanation for pro-cyclical 
productivity (e.g. Basu and Fernald, 1997). Therefore, it is interesting to isolate the effect of scale economies 
component in the productivity change. While technical and allocative inefficiencies relate to managerial and 
organisational aspects within the firm, scale inefficiency typically signals structural inefficiency in the operating 
environment of the firm. Possible reasons for an ineconomically small scale include credit constraints, trade 
barriers, and shortage of essential resources, while imperfect competition and monopoly power may lead firms to 
produce on ineconomically large scale.   
  The decomposition of Färe et al. (1994c) utilises the intuitive scale efficiency measure by Färe et al. 
(1983), which compares the distance functions gauged relative to a variable returns to scale and constant returns 
to scale reference technologies. However, applying this conventional scale measure as such in the present 
framework is problematic, because the Färe et al. (1983) decomposition depends on the order in which its 
components are calculated, as pointed out by McDonald (1996) in the case of technical efficiency, congestion 
efficiency, and scale efficiency components (see also Färe and Grosskopf, 2000). The same problem extends to 
the present case with allocative and scale efficiency, where allocative efficiency may be different if we measure it 
relative to the variable returns to scale technology or the constant returns to scale benchmark that already 
accounts for scale inefficiency. Only when the input sets L and output sets P are homothetic the order of 
decomposition does not influence the result.   
To avoid this arbitrariness, we propose to define scale efficiency in economic rather than technical terms. In 
line with our treatment of technical change, we adopt the dual perspective and characterise the optimal scale size 
in terms of the profitability function . Thus, the most natural measure of scale efficiency is the ratio of the 
maximum profitability at the current scale size (scale constrained profitability function) to the overall (global) 
maximum profitability (unconstrained profitability function). In essence, this definition of scale efficiency assumes 
constraints as the reason for scale inefficiency. 
tρ
The current scale size can be measured either in terms of inputs or outputs. If we fix the output level to yt, 
the maximum profitability (or return on the dollar) in period t is given by the ratio ⋅
( , )
t t
t t tC
p y
w y
. Thus, the input 
 13 
oriented profitability based scale efficiency measure is the ratio of output constrained maximum profitability to the 
unconstrained maximum profitability, that is, 
(12) ρ ⋅  
( , )
( , )
t t
t t t
t t tC
p y w p
w y
. 
Similarly, if we fix the inputs to xt, the maximum profitability is ⋅
( , )t t t
t t
R x p
w x
. Thus, the output oriented profitability 
based scale efficiency measure is  
(13) ρ  ⋅ 
( , ) ( , )
t t t
t t t
t t
R x p w p
w x
 
In the case of the Malmquist index, the scale efficiency measure is especially dependent on the choice of 
orientation. To avoid this problem, we again follow the Fisher approach and take the geometric mean of the input 
oriented and output oriented scale efficiency measures. This results as the following expression for the scale 
efficiency component: 
(14) ∆SEff ρ ρ ρ ρ
        ⋅ ⋅        ⋅ ⋅        = ⋅ ⋅        
1 1
2 21 1 1 1 1 0 0 0 0 0
1 1 1 1 1 0 0 0 0 0
1 1 1 1 1 1 0 0 0 0 0 0
( , ) ( , )
( , ) ( , )
( , ) ( , ) ( , ) ( , )
R R
C C
p y x p p y x p
w y w x w y w x
w p w p w p w p
  
. 
Input oriented scale efficiency measure and its change is illustrated in the Figure 3a. Input scale efficiency 
∆ = OK OBISEff
OJ OA
 is calculated as the change of input scale efficiency between periods 0 ( /OJ OA ) and 1 
( /OK OB ). Similarly, Figure 3b depicts the output oriented scale efficiency measure and its change. Output 
scale efficiency ∆ = OD OMOSEff
OC OC
 is the change in output scale efficiency between periods 0 and 1, 
respectively. Note that in period 0 the output scale efficiency equals 1. The value is bigger (smaller) than one if the 
unit gets closer (further away) to (from) the most profitable scale of production.  
 14 
  
 
 
ρ1 
ρ0 (x
1, y1) 
(x0, y0) 
T0 
T1 
O X 
Y 
  K B x0 x1 J A 
Figure 3a. Input oriented scale efficiency change.  
 
 
 
Y 
ρ1 
M 
T1 D 
y1 (x
1, y1) ρ0 
T0 
C 
(x0, y0) y0 
O X 
 
Figure 3b. Output oriented scale efficiency change. 
 15 
6. Allocative efficiency 
Thus far, allocative efficiency components have not appeared in the decompositions of productivity change, 
although it is a component of Farrell’s (1957) static decomposition of productive efficiency.  Introducing allocative 
efficiency into the productivity index may appear unorthodox if we consider productivity and technical progress to 
be the same. Since the standard neoclassical assumption of rational profit maximizing firms leaves no room for 
allocative inefficiency, changes in allocative efficiency do not rank among the most standard explanations for 
productivity changes. Yet, in the real world, markets are far from perfectly competitive, incomplete information and 
other frictions are present, and adjustments to price changes can occur with considerable delay. Since changes of 
allocative efficiency do occur in the real life, especially in more turbulent times, it is well in line with the welfare 
interpretation of productivity to account for its changes. Ignoring such a potentially important component can only 
contribute to mis-measurement of welfare.  
2
 While technical efficiency captures the radial input reduction and output expansion potential, Farrell’s 
(1957) allocative efficiency measure represents the non-radial productivity improvement potential, which is 
obtainable by restructuring production (i.e., changing the input-output mix). As Kuosmanen and Post (2001, Eq. 
2.8) note, allocative efficiency can be expressed solely in terms of technology distance functions. Indeed, allocative 
efficiency measure can be viewed as a distance measure in the input-output quantity space, reflecting the 
technically feasible distance to the economic iso-cost (iso-revenue) surfaces representing the maximum aggregate, 
quality adjusted input (output) quantity. In this perspective, introducing allocative efficiency component to the TFP 
index, defined as a ratio of two quantity indices, seems fully legitimate.  
Allocative efficiency can be measured in terms of inputs or outputs (costs or revenues). Input allocative 
efficiency is the ratio of the cost function and the cost of the technically efficient input vector that can be obtained 
by proportionately scaling the observed input vector downward to the efficient frontier, that is,  
(10) tIAEff ≡ ⋅
( , )
( ( , ) )
t t t
t t t t t
x
C
D
w y
w x y x
. 
Analogously, output allocative efficiency is the ratio of the hypothetical revenue of the technically efficient output 
vector, obtained by equiproportionate augmentation of the observed output vector, and the revenue function, that 
is, 
                                                 
2 A notable exception is the decomposition of profitability by Grifell-Tatjé and Lovell (1999).  
 16 
(11) tOAEff
⋅= ( / ( , ))
( , )
t t t t t
y
t t t
D
R
p y x y
x p
. 
The overall allocative efficiency of the firm should take into account the changes of mix both in the input and the 
output side. As before, we measure the overall change of allocative efficiency as the geometric mean of the 
changes in allocative efficiencies in inputs and outputs, that is, 
(12)   
 ∆ = ⋅  
1
1 1 2
0 0
IAEff OAEffAEff
IAEff OAEff
. 
This is in line with fact that changes in input and output allocative efficiency contribute to profitability with equal 
weight and in multiplicative form. 
Input and output oriented allocative efficiency changes are illustrated graphically in Figure 4. Input 
oriented allocative efficiency change in Figure 4a is depicted as ∆ = . The measure compares 
the deviation from the minimum cost due to wrong allocation of inputs between subsequent periods when given 
output is produced. Similarly, output oriented allocative efficiency change is illustrated in Figure 4b as 
OF OB
IAEff
OE OA
∆ = OD OHOAeff
OC OG
. It compares the deviations from maximum revenue due to wrong allocation of outputs 
given inputs between two periods, respectively. Again, we can take a geometric mean of these two, which provides 
our weighted average for allocative efficiency. 
 17 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
X2 
F 
E 
B 
A 
x0 
x1 
L0 
L1 
C0 (y0) 
C1 (y1) 
X1 O 
Figure 4a. Input allocative efficiency change. 
 
 
 
Y2 
G 
C 
H 
y0 D 
y1 
R0 (x0) R1 (x1) 
P1 P0 
Y1 O 
 
Figure 4b. Output allocative efficiency change. 
 
 
 18 
7. Market strength  
The market strength measure, to be introduced next, is the most atypical component of all. Therefore, we need to 
provide some motivating examples. Suppose a new technological innovation or learning by doing improves the 
quality of some outputs more than other outputs, such that the same inputs produce the same output quantities as 
before, but the high-quality outputs yield a higher price. Alternatively, quality of some inputs may improve such that 
a smaller amount of the same input is needed for producing the same output (consider e.g. the effect of education 
and training on labour input). Or an external change in the market environment (e.g. tariffs or trade barriers) or the 
government policies (e.g. taxes and subsidies) influences the competitiveness of the firms. As a result of these 
kinds of changes, firms may become more or less profitable, but the traditional technical change measures as well 
as technical, allocative, and scale efficiency components fail to account for this effect.  
The increased capacity of firms to produce aggregate output with the same aggregate input could be 
measured by ratio 
0 1 1
0 0 0
( , )
( , )
ρ
ρ
w p
w p
, representing the increase in profitability of the leading firms in the industry, given 
the base year technology. Alternatively, we could use the target year technology as the benchmark, and consider 
the ratio 
1 1 1
1 0 0
( , )
( , )
ρ
ρ
w p
w p
. Since we have no reason to prefer either base or target year technology, we can take the 
geometric mean of these ratios: 
(17) ρ ρρ ρ
 ⋅  
1
0 1 1 1 1 1 2
0 0 0 1 0 0
( , ) ( , )
( , ) ( , )
w p w p
w p w p
.  
This geometric mean effectively captures the change in maximum profitability at given technology attributable to 
the price changes. However, this measure fails to account for possible profitability effects of changes in nominal 
prices due to inflation. To convert the profitability changes from nominal to real terms, we need an appropriate 
price deflator. The most natural deflator for profitability change is the ratio of Fisher output price index to the input 
price index, 0,1 0,1 0,1 0,1( , ) / ( , )p wFxF y x y , where the input and output vectors of the evaluated unit are used as the 
index weights. This deflator essentially measures the change of output prices relative to the change of input prices. 
If average output prices increase more (less) than the input prices, then the value of the deflator is greater 
(smaller) than one. If output and input prices change by the same factor, then this deflator is equal to one. To 
measure the change in market strength in terms of real prices, we propose the following measure:  
 19 
(18) MS∆ ρ ρρ ρ
 = ⋅  
1 0,1 0,10 1 1 1 1 1 2
0 0 0 1 0 0 0,1 0,1
( , )( , ) ( , )
( , ) ( , ) ( , )
p
w
F
F
x yw p w p
w p w p x y
.  
  The market strength measure does not change if all prices change by the same percentage. In fact, as 
the nominal profitability changes are adjusted by the price indices, the market strength remains constant even if 
output prices change by a different factor than input prices, if price changes are uniform across input factors and 
output goods. In essence, this component measures changes in the production possibilities of the quality-adjusted 
aggregate output due to changes in relative prices, measured in real terms. As these relative price changes are 
typically beyond the influence of an individual firm but rather depend on the external operating environment, we 
find market strength an appropriate name for this component. While non-uniform changes in quality of inputs and 
outputs may be one important reason for the change of market strength, as such, this component cannot be 
interpreted as a measure of quality change. For example, this component would likely fail to capture output quality 
improvements in the computer industry, where the quality improvement of outputs has coincided with decreasing 
output prices. If possible, the input and output quantity measures and indicators should be directly adjusted for 
their quality.  
In the previous section we interpreted allocative efficiency change as a qualitative, non-radial, price-based 
component of efficiency change. In analogy with that interpretation, we may consider the market strength 
component as a qualitative, price-driven frontier shift. In other words, the market strength component can be 
viewed as the change in the aggregate production possibilities attributable to price changes. In this respect, it is 
instructive to multiply the market strength component by the associated technical change component to obtain      
(19) ∆ ×Tech MS∆
0,1 0,11 1 1
0 0 0 0,1 0,1
( , )( , )
( , ) ( , )
p
w
F
F
ρ
ρ=
x yw p
w p x y
.  
That is, the product of the technical change and market strength components is the change of maximum 
profitability measured in real terms (i.e., the nominal change of profitability function divided by the Fisher price 
index deflator).   
The market strength component is difficult to illustrate graphically, but an attempt towards this is made in 
Figure 5. Figure 5a represents the input and Figure 5b the output orientation. In the form of cost and return function 
the market strength component can be presented as ∆ = ⋅
0 *0 1 *0 1 1 *1 1 1 *1 1
0 *0 0 *0 0 1 *1 0 1 *1 0
( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
o
o
R C R CMS
R C R C
x p y w x p y w
x p y w x p y w
 
where x* and y * refer to optimal inputs and outputs given prices. 
 20 
  
X2 L0 L1 
C0 (y*0,w0) 
C1 (y*1,w1) 
C1 (y*0,w0) 
C1 (y*1,w1) 
X1 O 
Figure 5a. Market strength (input orientation). 
 
 
Y2 
P1 
P0 
R0 (x*1,p 1) R1 (x*1,p 1) 
R0 (x*0,p 0) R1 (x*0,p 0) 
Y1 O 
 
Figure 5b. Market strength (output orientation). 
 
 21 
 8. The main result 
We are now ready to present the main result of the paper. It turns out that by multiplying the five components 
introduced above, we obtain the classic Fisher index. 
 
Theorem 2: The Fisher ideal TFP index can be expressed as the product of the technical efficiency, technical 
change, scale efficiency, allocative efficiency, and the market strength components: 
(20)  TFPF = ∆ ×TEff ∆ ×Tech ∆SEff ×∆AEff MS×∆
 
Proof. See Appendix 3  
 
This result is interesting at least for the following reasons. Firstly, the decomposition further enhances our 
general understanding about how the Fisher index works. For example, we note that changes in allocative 
efficiency contribute to the Fisher index, but not to the Malmquist index (compare also with Färe and Grosskopf, 
1992; Balk, 1993, 1998; and Kuosmanen et al., 2004). Secondly, the decomposition provides a more detailed, 
anatomical picture about the driving forces behind productivity changes. By distinguishing between input and 
output oriented sub-components, and by introducing allocative efficiency and a new market strength component, 
the present decomposition is by far the most detailed one ever presented. Thirdly, our decomposition recognises 
price-based allocative efficiency and market strength components as important elements of productivity change. By 
attributing changes in economic value of inputs and outputs to the productivity index, we hope to capture at least 
some quality aspects of productivity. Fourthly, better understanding of the anatomy of the Fisher index also 
enables one to tailor the productivity index for the purposes of the study. Even though the Fisher TFP index may 
not be the most appropriate index number formula for some applications, we believe our decomposition can 
provide insight for developing a tailored index that captures those effects that are important for the purposes of the 
study. If some components represent effects that do not fit in a given definition of productivity, the decomposition 
enables one to correct for (or eliminate) these undesirable components from the overall productivity index.  
As a direct corollary of Theorem 2, we obtain an alternative decomposition of profitability (compare with 
Grifell-Tatjé and Lovell, 1999). In particular, the change of revenue can be expressed as the product of the Fisher 
output quantity and output price indices, and the change of costs can be expressed as the product of the Fisher 
 22 
input quantity and input price indices (Diewert, 1992). Note that the Fisher price indices were included as the 
deflator in the market strength component. Multiplying both sides of (18) with 0,1 0,1 0,1 0,1( , ) / ( , )p wFxF y x y  we obtain 
that the change in profitability is the product of our technical efficiency, technical change, scale efficiency, and 
allocative efficiency components and the nominal price version of the market strength component. 
 
9. Application to Finnish farms 
9.1 Motivation 
We next apply the proposed decomposition to study productivity development in 459 Finnish bookkeeping farms 
through 1992-2000. The period under investigation is interesting because Finland joined the European Union (EU) 
in the middle of the study period in 1995. The EU accession meant increased international competition and a fall in 
price support, which together resulted as a drastic fall in output prices. Figure 6 illustrates the development of price 
indices for milk, meat and other animal products, and crops. The drop in prices of meat and other animal products 
and crop products was more than 50 percent, but the price of milk decreased only by 15 percent. Input prices have 
not fallen as dramatically as some output prices, although prices in input categories like land, animals, fertilizers, 
purchased feed, and materials have decreased. The fall has been more than 30 percent in purchased feed and 
fertilizers and even more than 40 percent in prices of purchased animals. These dramatic price changes have led 
farms to adjust their input-output mix over time, the effects of which should show up in the technical, allocative, and 
scale efficiency measures. 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1992 1993 1994 1995 1996 1997 1998 1999 2000
Milk
Other animal
Crop
 
Figure 6. Development of average output prices (1992 = 1).  (Source: Statistics Finland)  
 
 23 
9.2 Data 
Our balanced panel data includes all types of farms varying from specialized animal production and crop farms to 
more conventional farms engaged both in animal husbandry and crop farming. A typical farm produces several 
outputs including milk, meat and/or several types of plant products using a number of different inputs, but not all 
farms produce all products. For example, less than half of the sample farms produce milk. Thus, it was necessary 
to aggregate specific inputs and outputs into larger categories to avoid dimensionality problems in the empirical 
efficiency analysis.  
Table 1 lists the input and output categories and presents the descriptive statistics of costs and revenues in 
1992, 1996 and 2000. Four observations are worth noting. First, Finnish farms are typically small. Table 1 shows 
that the mean arable land area of sample farms in 1992 was only 37 hectares, the number of animal units being 
31. Since 1992 output per farm has increased; the maxima and the standard deviations have also increased 
considerably. Second, the average crop output of farms did not change much, despite the drastic changes in price 
relations (see Table 1). Third, the capital stock per farm has increased more rapidly than output. Considerable 
public investment aids boosted investment activities. In year 2000, the capital stock of machinery was in real terms 
85 percent and that of buildings 70 percent higher than in 1992. Capital intensity increased since the annual growth 
of labour input was only about 5 percent during the time period under study. Although the total labour input in 
agriculture has decreased because of the decreasing number of farms, simultaneously, the farms increasing in 
size have had to increase their total labour input, but relatively less than other inputs. Fourth, use of other inputs 
per farm has increased during the research period. For example, fertiliser use on sample farms has increased 25 
percent together with the increase in arable land area. However, the use of fertilisers per hectare has decreased. 
Although the intensity of crop production seems to have decreased, relative price changes have contributed to the 
intensification of production with respect to purchased feed, for example, in milk production. 
 
 24 
Table 1. Inputs and outputs. Descriptive statistics of revenues and costs (in €): 1992, 1996 and 2000. 
 1992 1996 2000 
 Mean Std Dev Mean Std Dev Mean Std Dev 
Outputs:       
Milk  23354 25628 26732 30291 32403 40987 
Other animal products  18331 30707 19984 40541 24710 55487 
Crops  11272 18076 11109 18623 14311 21146 
Inputs:       
Labour  27057 11283 29420 13147 28634 14415 
Animal units 1242 1334 1385 1653 1500 2062 
Land 4820 2842 5496 3223 6401 3711 
Machinery  28917 20017 33749 21620 54025 39749 
Buildings  25843 29903 29451 31168 42117 52952 
Fertiliser  3785 2409 4552 3114 4742 3924 
Energy  5411 3146 5933 3615 4802 3443 
Purchased feed  9879 13487 10118 14195 14455 22899 
Materials  10370 6280 12964 8361 16854 12225 
 
9.3 The Fisher indices 
Calculation of the Fisher quantity indices requires full price and quantity data on all inputs and outputs. 
Unfortunately, both prices and quantities were documented only for labour and land inputs, and for milk output. For 
other input and output categories only cost and revenue data are available; a common feature in farm-level 
production data. Since all farms are relatively small and their market power is low, the farms are assumed to take 
the prevailing market prices for inputs and outputs as given. All farms are assumed to face the same market prices 
for specific inputs and outputs (i.e., the law of one price is assumed to hold).3 Changes in market prices over time 
were measured by price indices documented by Statistics Finland. For aggregated input and output categories 
(such as crops), farm-specific Fisher price indices were constructed from more specific price indices (e.g., indices 
for wheat, barley, oats, and so forth) and the farm-level quantity data. For input and output variables with missing 
quantity data, implicit quantity indices were calculated by dividing the costs or revenues by the respective price 
indices. These implicit quantities capture possible quality differences associated with higher returns or costs due to 
higher prices in quantities.  
Given thus constructed farm-specific quantity and price data, the Fisher ideal output and input quantity 
indices could now be calculated. The total factor productivity index was subsequently obtained as the ratio of the 
two. Figure 7 presents cumulative cost-share weighted averages of Fisher input (Fx) and output (Fy) quantity 
indices and the Fisher index (F).   
                                                 
3 Note that the same assumption has to be made when cost and/or revenue data are used as proxies for respective input and output 
quantities, as is often done in the Malmquist index approach. For more detailed discussion on the one price assumption, see Cross and Färe 
(2003) and Kuosmanen et al. (2005). 
 25 
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1992 1993 1994 1995 1996 1997 1998 1999 2000
F
Fy
Fx
 
Figure 7. Cumulative Fisher index (F), Fisher input (Fx) and output (Fy) quantity indices.  
 
The aggregate input has grown steadily over the whole research period. The output growth has followed the input 
growth but there have been two major deviations: in 1992/93 the output growth deviated downwards and in 
1999/2000 upwards from the input growth. These deviations can also be observed in the Fisher index, which at 
first falls but then recovers, although very slowly until 1998. After 1998 the growth rate has increased being 14.6 
percentage units from 1999 to 2000.  
 
9.5 Decomposition of the Fisher TFP index 
To implement the decomposition of Section 8, we utilise the nonparametric Data Envelopment Analysis (DEA) 
approach for estimating the production technology. We resort to the most standard convex, variable returns to 
scale benchmark technology. The values of distance functions as well as cost and revenue functions relative to 
DEA technology are obtained as optimal solution to specific Linear Programming (LP) problems. Detailed 
descriptions of these LP problems are presented e.g. in Färe et al. (1985, 1994b). The only non-standard measure, 
the profitability function, was calculated by simply enumerating through all observed combinations of price and 
quantity vectors to calculate the full profitability distribution at all observed prices. To make this critical profitability 
measure less sensitive to data errors and outliers, we used the 95 percentile of the profitability distribution as our 
empirical estimator for the profitability function.  
 
 26 
9.5.1 Technical efficiency and technical change components 
Figure 8 describes the development of technical efficiency and technical change components in cumulative 
fashion. The weighted annual averages of the input and output oriented sub-components of technical efficiency 
change are presented separately. Their geometric mean (the solid thick line) is the overall technical efficiency 
measure of our decomposition. Technical efficiency has grown slowly until 1998. Since then the growth of technical 
efficiency has accelerated considerably. Figure 8 shows that output oriented technical efficiency indices indicate 
approximately twice as high growth as input oriented ones. On the other hand, their geometric mean is close to the 
input oriented index. This is probably due to the fact that output orientation yields a larger variation in individual 
efficiencies. 
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1992 1993 1994 1995 1996 1997 1998 1999 2000
TEff
OTEff
ITEff
Tech
 
Figure 8. Cumulative input oriented (ITEff) and output oriented (OTEff) technical efficiency change, their 
geometric mean (TEff), and the technical change (Tech) components. 
 
The technical change component shows a surprisingly steady growth path. The black broken line of Figure 8 
represents the cumulative weighted average of farms' technical change components. This figure shows a 
considerable technical regress in the beginning of the sample period, which can be attributed to extensive set-
aside programs and adverse weather conditions in 1993 and 1994. After these first adverse years, technological 
progress is rapid until 1998, showing impressive average cumulative growth of 25 percent in a five-year period. 
However, the progress slows down in 1999 and turns to regress in 2000, in spite of the fact that overall productivity 
growth is at its highest in that year. As we noted earlier, there was almost no change in technical efficiency but 
since 1998 technical efficiency improves 15 percentage units. Thus, average farms are getting closer to the frontier 
but partially due to the regress on the most profitable farms. 
 27 
9.5.2 Scale efficiency component 
Figure 9 presents the scale efficiency components analogous to the previous figure. The scale efficiency 
component shows large annual fluctuations, showing an increasing trend before the EU accession in 1995 and 
until 1996, then decline until 1999, and a sudden 20 percentage points jump in 2000. The fluctuations are 
especially large in the output-oriented measure of scale efficiency. This is surprising because farms' production 
scale exhibited relatively steady growth throughout the study period (compare with Figure 7), while the composition 
of input-output mix changed much more dramatically. The great fluctuations in scale efficiency must therefore arise 
from rapid changes in the most profitable scale size itself. It seems that the expansion of scale size was profitable 
in the market conditions prevailing before the EU accession, but the continuing expansion after 1995 was at odds 
with the deteriorated market conditions. Note that the input oriented component showed greater variation prior to 
1995, while the output oriented sub-component fluctuates drastically after 1995.  
 
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1992 1993 1994 1995 1996 1997 1998 1999 2000
SEff
OSEff
ISEff
 
Figure 9. Cumulative input oriented (ISEff) and output oriented (OSEff) scale efficiency change and their 
geometric mean (SEff).  
 
 
9.5.3 Allocative efficiency and market strength components 
The decomposition of the Fisher index also includes allocative efficiency change. Analogous to Figure 8, Figure 9 
presents the weighted annual averages of the input and output oriented sub-components and their geometric 
mean. This figure shows that allocative efficiency has on average improved less than 10 percent during the 
research period. At the end of the research period output oriented allocative efficiency has increased relatively 
more than input oriented allocative efficiency. A change in development can be observed at the time of the EU 
accession, and it is probably related to different incentives and constraints set by the changed agricultural policy. 
 28 
Before that point the input oriented allocative efficiency improved faster than the efficiency in output orientation. 
Instead, after the EU accession output oriented allocative efficiency has shown faster growth than input oriented 
efficiency.  
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1992 1993 1994 1995 1996 1997 1998 1999 2000
AEff
OAEff
IAEff
MS
 
Figure 10. Cumulative input oriented (AECi) and output oriented (AECo) technical efficiency change and 
their geometric mean (AEC), and market strength change (MS). 
 
The weighted average of the market strength components is represented by the broken line in Figure 10. There 
was little change in the market strength in the beginning of the study period. In 1995-1996, the first two years after 
the EU accession, the market strength dropped by almost 15 percentage points, reflecting the adverse effect of the 
price changes to farms' potential for producing aggregate output. After this sharp downfall, market strength 
recovered slowly, but never reached the original price conditions.  
It is interesting to note that overall Fisher TFP index showed slow but steady growth throughout the study 
period, supported by both technical efficiency and technical change components. The effect of dramatic prices 
changes was revealed most clearly by the market strength component. The negative effect of the market strength 
component was countered at least to some extent by improved allocative efficiency.  
 
9.4 Comparison of the Fisher and Malmquist TFP indices 
For comparison, we also calculated the Malmquist productivity index of Färe et al. (1994c) using the same DEA 
reference technology and the same input and output quantity data as in the Fisher indices. While the Malmquist 
index does not require any price data, the assumptions of price taking behaviour and the law of one price are still 
 29 
required for recovering quantity data from the observed cost/revenue aggregates. Thus, the data requirements do 
not favour either approach in this application.  
Figure 11 illustrates the average productivity growth measured by the Malmquist index (denoted by M100; 
the thin broken line) and the Fisher index (F100, solid black line). We observe that the weighted average 
Malmquist index shows considerably higher cumulative productivity growth than the Fisher ideal TFP index: the 
Malmquist index indicates an average productivity growth of 68.7 percent during the research period of eight years, 
which is three times higher than the average growth according to the Fisher index. Such a productivity growth 
seems incredibly high in the context of agricultural production.  
 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1992 1993 1994 1995 1996 1997 1998 1999 2000
M100
M95
M90
F100
 
Figure 11. Cumulative Fisher index (F) versus Malmquist indices (M). The number after capital letter 
indicates the percentage of observations taken into account. 
 
To investigate the distribution of productivity growth figures across farms, we clipped the extreme tails of the 
distribution, and plotted in Figure 12 the average values of the Malmquist productivity index for farms falling within 
the (2.5, 97.5) and (5, 95) percentile ranges (denoted by M95 and M90 respectively). After clipping the tails, the 
average Malmquist index follows closely the path of the average Fisher TFP index. This suggests that for the large 
majority of farms the two index formulae offer relatively similar results. For a small percentage of farms in our 
sample, the Malmquist index produces considerably higher index values than the Fisher TFP index. In our 
interpretation, the Fisher TFP index proves a more stable and robust index number formula of the two, providing 
more credible results in this application. Recall that the Fisher TFP index for an individual farm only depends on its 
own performance, while the Malmquist index compares the performance of each farm relative to the farms defining 
the best practice frontier. 
 30 
We next compare the results component-wise to see where the differences between the Malmquist and 
Fisher TFP indices may occur. We focus on the most standard decomposition of the Malmquist index by Färe et al. 
(1994c), specified in detail in Appendix 1.  
Firstly, technical efficiency change components of the Malmquist index are equal to their input- and output-
oriented counterparts in the Fisher index. Applications of the Malmquist index typically assume either input or 
output orientation, while our Fisher decomposition incorporates both orientations applying the geometric mean of 
the two. Secondly, allocative efficiency change and the market strength components appear in our decomposition, 
but are not included in the Malmquist index decompositions. Thus, we can only compare the scale efficiency and 
technical change components empirically.  
Figure 12 presents a comparison between the input oriented scale efficiency components of the Malmquist 
(MISEff) and the Fisher index (ISEff). As Figure 12 shows, the scale components of the Fisher index are typically 
larger than the corresponding Malmquist component. Moreover, the Fisher components exhibit more variation over 
time.  
 
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1992 1993 1994 1995 1996 1997 1998 1999 2000
ISEff
MISEff
 
Figure 12. Input oriented scale efficiency change component of the Malmquist index (MISEff) and inputs 
oriented scale efficiency component (ISEff) and weighted scale efficiency component (SEff) of the Fisher 
index. 
 
Figure 13 compares the technical change components of the Malmquist (MTech100: the thin broken line) 
and Fisher indices (Tech100: the solid black line). The Malmquist component suggests considerably faster 
technical development than the Fisher component, with the cumulative difference exceeding 20 percentage points 
at the end of the research period. Thus, the difference in the technical change components explains about half of 
the difference in the overall index. As in the case of the overall index (Figure 12), we also considered the technical 
 31 
change component of the mid-95 percentile of the farm distribution clipping away 5 percent of the extreme 
observations. The resulting average is illustrated by the broken line with diamonds (MTech95). Like in the case of 
the overall index, the majority of farms exhibited similar technical progress according to both Fisher and Malmquist 
components. 
 
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1992 1993 1994 1995 1996 1997 1998 1999 2000
Tech100
MTech100
MTech95
 
Figure 13. Technical change components of the Malmquist and the Fisher index. 
 
10. Concluding remarks 
We have shown that the Fisher ideal TFP index can be decomposed in the similar fashion to the numerous 
decompositions of the Malmquist index. We believe this decomposition enhances both our understanding about 
the different sources of productivity growth captured by the Fisher index and the position of the Fisher index as a 
useful index number formula for productivity analysis. The decomposition is readily implementable in empirical 
applications using the standard parametric or nonparametric frontier estimation techniques.  
The proposed decomposition includes two new components – allocative efficiency and market strength – 
which have not been accounted for explicitly in the productivity decompositions before. The allocative efficiency 
component is based on the classic Farrell allocative efficiency measure, which indicates the nonradial efficiency 
improvement potential in production of quality-adjusted aggregate output. The market strength component 
captures changes in the market conditions, transmitted through changes in relative prices, which influence the 
firms’ capability to produce the quality-adjusted aggregate output. While both these components represent 
economic rather than technical efficiency, we find it meaningful to account for these qualitative components in the 
productivity index (which is essentially a quantity index), because usually it is difficult to draw a sharp distinction 
 32 
between the technical input-output quantities and the associated economic costs and revenues. For example, the 
capital input is almost always measured by some kind of a cost aggregate.  
The proposed decomposition may also provide useful insights for other decompositions. While the existing 
Malmquist decompositions usually assume either input or output orientation, our decomposition builds on 
geometric means of both input and output oriented sub-components. Of course, the same approach could be 
adapted to the Malmquist decompositions in a straightforward manner. On the other hand, we employed the dual 
representation of the technology, the profitability function, for our measure of technical change. By duality theory, it 
is equally legitimate to measure technical change by means of monetary profitability data as with more traditional 
input/output technology distance functions.  
The usefulness of the new decomposition was illustrated by an empirical application where productivity 
developments in a large sample of Finnish farms were studied over the period 1992-2000. This period is interesting 
because of the drastic price changes due to Finland's EU accession in 1995, which increased international 
competition and led to major revisions in the agricultural policy. Despite major restructuring of production, the 
average productivity growth was found to be surprisingly stable, on the average about 2.5 percent per annum. 
Technical change and technical efficiency followed a similar stable growth path. The impacts of the EU accession 
presented themselves most clearly in the market strength component, which showed a sudden downfall in 1994-
1996. This negative market strength effect was offset most importantly by improved allocative efficiency. The scale 
efficiency component showed relatively large annual fluctuations and proved more difficult to interpret.      
We hope this decomposition might inspire debate about the relative merits of different index number 
formulae used in productivity measurement. We believe there is no single superior index number for all empirical 
studies, but different index formulae may be appropriate depending on the purposes of the analysis and the 
interpretation of productivity. Our decomposition suggests that it is possible to extend the approach of Färe et al. 
(1994a,c) from the domain of Malmquist type indices towards the more classic, price-weighted indices. Further 
research could consider other important indices such as the widely used Törnqvist productivity index.   
 
Acknowledgements  
We thank participants of the 4th International DEA Symposium (Aston University, the United Kingdom, September 
2004) for comments. This study forms a part of the research program on Nonparametric Methods in Economics of 
Production, Natural Resources, and the Environment. For further information, see the program homepage: 
 33 
http://www.sls.wageningen-ur.nl/enr/staff/kuosmanen/program1/  
The financial support for this program from the Emil Aaltonen Foundation, Finland, is gratefully acknowledged. 
 
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 36 
Appendix 1: The Malmquist index and its decomposition 
For brevity, we focus on the input oriented Malmquist index of Färe et al. (1994b). Let ( , )t t txD x y  denote the input 
distance function defined relative to the constant returns to scale reference technology (i.e., the conical hull 
 rather than T ). Given this CRS input distance function, the Malmquist index is defined as  ,tTλ λ ≥ 0 t
≡0,1 0,1( , )xM x y  ⋅  
1
0 1 1 1 1 1 2
0 0 0 1 0 0
( , ) ( , )
( , ) ( , )
x x
x x
D D
D D
x y x y
x y x y
. 
The Malmquist index decomposes into components of efficiency change (Eff) and technical change (Tech) as  
=0,1 0,1( , )xM x y ⋅0,1 0,1 0,1 0,1( , ) ( , )Eff Techx y x y ,  
where  
=
1 1 1
0,1 0,1
0 0 0
( , )( , )
( , )
x
x
DEff
D
x yx y
x y
  
and  
≡0,1 0,1( , )Tech x y  ⋅  
1
0 1 1 0 0 0 2
1 1 1 1 0 0
( , ) ( , )
( , ) ( , )
x x
x x
D D
D D
x y x y
x y x y
.  
Efficiency change can be further decomposed into ’pure’ technical efficiency change (OEff) and scale efficiency 
change (SEff) as  
= ⋅0,1 0,1 0,1 0,1 0,1 0,1( , ) ( , ) ( , )Eff OEff SEffx y x y x y   
where  
=
1 1 1
0,1 0,1
0 0 0
( , )( , )
( , )
x
x
DOEff
D
x yx y
x y
  
and   
 = ⋅  
1
1 1 1 0 0 0 2
1 1 1 0 0 0
( , ) ( , )
( , ) ( , )
x x
x x
D DSEff
D D
x y x y
x y x y
. 
 37 
Appendix 2: Proof of Theorem 1 
This is a variant of the duality result on profit functions (e.g., Färe and Primont, 1995: Proposition 6.1.4). The result 
follows directly from Lemma 1 in Kuosmanen et al. (2004), which shows that if Tt satisfies free disposability, 
convexity, and constant returns to scale, then the input distance function can be expressed as  
(i) 
( , )
( , ) max 1 ( , )
r s
t r
xD ++
+
+∈
′ ⋅ ⋅ ′ ′= ≤ ∀ ′⋅ ⋅ w p
p y p yx y x y
w x w x\
\ s∈ .  
By the definition of function , this further implies that  tρ
(ii) 
( , )
( , ) max ( , ) 1
r s
t t
xD ρ++∈
⋅ = ≤ ⋅ w p
p yx y w p
w x\
.  
Furthermore, we know that under weak disposability of inputs  
(iii) { }( , ) ( , ) 1t r s tT D++= ∈ ≤xx y x y\ . 
Substituting (ii) in (iii), we have that 
(iv) 
( , )
( , ) max ( , ) 1 1
r s
t r s tT ρ++
+
+ ∈
 ⋅ = ∈ ≤ ≤  ⋅  w p
p yx y w p
w x\
\    
(v) 
( , )
( , ) max ( , ) 1
r s
r s tρ++
+
+ ∈
  ⋅  = ∈ ≤   ⋅    w p
p yx y w p
w x\
\  
(vi) ( , ) ( , ) ( , )r s t r sρ+ ++ + ⋅= ∈ ≤ ∀ ∈ ⋅ 
p yx y w p w p
w x
\ \ .,  
 38 
Appendix 3: Proof of Theorem 2 
We start from the definition of the Fisher TFP index  
(i)  ≡0,1 0,1 0,1 0,1( , , , )TFPF p w y x
 ⋅  
 ⋅  
1
0 1 1 1 2
0 0 1 0
1
0 1 1 1 2
0 0 1 0
p y p y
p y p y
w x w x
w x w x
,  
and modify it in order to isolate the specific components. (To simplify the expressions, the arguments of FTFP will be 
suppressed.) As the first step, we divide the revenue terms by the corresponding revenue functions and the cost 
terms by the corresponding cost functions as  
(ii)  =TFPF
    ⋅ ⋅        ⋅    ⋅ ⋅         
   ⋅ ⋅      ⋅   ⋅ ⋅       
1
20 1 1 1
0 0 0 1 1 1
0 0 1 0
0 0 0 1 1 1
1
20 1 1 1
0 0 0 1 1 1
0 0 1 0
0 0 0 1 1 1
( , ) ( , )
( , ) ( , )
( , ) ( , )
( , ) ( , )
R R
R R
C C
C C
p y p y
x p x p
p y p y
x p x p
w x w x
w y w y
w x w x
w y w y


. 
We then substitute the numerators of the second-level ratios to obtain 
(iii)  =TFPF
    ⋅ ⋅        ⋅    ⋅ ⋅         
    ⋅ ⋅        ⋅    ⋅ ⋅         
1
21 1 0 1
1 1 1 0 0 0
0 0 1 0
0 0 0 1 1 1
1
21 1 0 1
1 1 1 0 0 0
0 0 1 0
0 0 0 1 1 1
( , ) ( , )
( , ) ( , )
( , ) ( , )
( , ) ( , )
R R
R R
C C
C C
p y p y
x p x p
p y p y
x p x p
w x w x
w y w y
w x w x
w y w y
. 
 
Next, we further decompose the left-hand side ratios by introducing revenues and costs of the technically efficient 
reference points as  
 39 
(iv) =TFPF
  ⋅⋅ ⋅  ⋅  ⋅  ⋅ ⋅  ⋅ ⋅⋅  ⋅   ⋅  
⋅ ⋅⋅⋅
1
21 1 1 1 11 1
1 1 1 1 1 1 1 1 0 1 1 1 1
1 0 0 0 00 0 0 0 00 0
0 0 0 0 0 0 0 0
1 1 1 1 1
1 1 1 1 0
( / ( , ))
( / ( , )) ( , ) ( , )
( , )( / ( , ))
( / ( , )) ( , )
( ( ,
( ( , ) )
o
o
o
o
i
i
D
D R R
RD
D R
D
D
p y x yp y
p y x y x p p y x p
p y x pp y x yp y
p y x y x p
w x w x
w x y x
      ⋅  ⋅ ⋅  ⋅ ⋅ ⋅  ⋅   ⋅  
1
21 0
1 1 1 0 1 1 1 1
1 0 0 0 00 0 0 0 0 1 0
0 0 0 1 0 0 0 0
) )
( , ) ( , )
( , )( ( , ) )
( ( , ) ) ( , )
i
i
C C
CD
D C
y x
w y w x w y
w x w yw x w x y x
w x y x w y
. 
 
Reorganizing the left-hand components of (iv), we  
(v)  =TFPF
            ⋅ ⋅ ⋅ ⋅                 
1
21 1 11 1 1 1 1 0 1
1 1 1 1 1 11 1 1 11 1 1 1 1 1 2
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
( , )( / ( , ))
( ( , ) )( , )( , ) ( , )
( , ) ( , ) ( / ( , )) ( , )
( , ) ( ( , ) )
o
io i
o i o
i
CD
DRD D
D D D C
R D
w yp y x y p y
w x y xx p px y x y
x y x y p y x y w y
x p w x y x
 ⋅  
 ⋅  
1
1 1 1 2
0 0 0 0
1
0 1 1 1 1 2
1 0 0 0 0
( , )
( , )
( , )
( , )
R
R
C
C
x p
y x p
w x w y
w x w y
 
The first component of (v) is the change in technical efficiency ( ), the second component is the change in 
allocative efficiency ( ), so we rewrite (v) as 
∆TEff
∆AEff
(vi)  =TFPF ∆ ×TEff ∆ ×AEff
 ⋅  
 ⋅  
1
0 1 1 1 1 2
1 0 0 0 0
1
0 1 1 1 1 2
1 0 0 0 0
( , )
( , )
( , )
( , )
R
R
C
C
p y x p
p y x p
w x w y
w x w y
. 
Next, we turn attention to the remaining undecomposed element. Firstly, we expand the revenue and cost ratios as 
(vii)  =TFPF ∆ ×TEff ∆ ×AEff
    ⋅        ⋅ ⋅
    ⋅        
1 1
0 0 0 1 1 1 1 1 12 2
1 0 1 1 0 0 0 0 0
1 1
0 0 0 1 1 1 1 1 12 2
1 0 1 1 0 0 0 0 0
( , )
( , )
( , )


1
2
1
2
( , )
R
R
C
C
p y p y p y x p
p y p y p y x p
w x w x w x w y
w x w x w x w y
. 
The third component is the ratio of the inverse of Fisher output price index and the input price index:  
(viii)  =TFPF ∆ ×TEff ∆ ×AEff
         ⋅ ⋅
         
1 1
1 1 1 1 12 2
0 0 0 0 0
1 1
1 1 1 1 12 2
0 0 0 0 0
( , )1
( , )
1 ( , )
( , )
p
w
R
F R
CF
C
p y x p
p y x p
w x w y
w x w y
. 
 40 
We next re-organize the remaining revenue and cost ratios as 
(ix)  =TFPF ∆ ×TEff ∆ ×AEff
 ⋅ ⋅  
 ⋅ ⋅ ⋅ 
1
1 1 1 1 1 2
1 1 1 1 1
1
0 0 0 0 0 2
0 0 0 0 0
( , )
( , )
( , )
( , )
p
w
R
FC
FR
C
p y x p
w y w x
p y x p
w y w x
 
Further, we introduce the profitability functions as 
(x)  =TFPF ∆ ×TEff ∆ ×AEff
ρ ρ
ρ
ρ
ρ ρ
    ⋅        ⋅       ⋅       ⋅    ⋅    ⋅    
1
21 1 1 1 1
1 1 1 1 1
1 1 1 1 1 1
1 1 1
0 0 01
20 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
( , )
( , )
( , ) ( , )
( , )
( , )( , )
( , )
( , ) ( , )
p
w
R
C
F
FR
C
p y x p
w y w x
w p w p
w p
w pp y x p
w y w x
w p w p
. 
The third component is the change in scale efficiency ( ), that is, ∆SEff
(xi) =TFPF ∆ ×TEff ∆ ×AEff ∆SEff ρρ
 ×  
1 1 1
0 0 0
( , )
( , )
p
w
F
F
w p
w p
. 
Finally, we expand the ratio of profitability functions as 
(xii)  =TFPF ∆ ×TEff ∆ ×AEff ∆SEff
ρ ρ ρ ρ
ρ ρ ρ ρ
    × ⋅      
1 1
1 0 0 1 1 1 0 1 1 1 1 12 2
0 0 0 0 1 1 0 0 0 1 0 0
( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
p
w
F
F
w p w p w p w p
w p w p w p w p

 
The first component within the brackets is the technology change measure ( ) and the remaining second 
component and the price deflator form the market strength component ( ). Thus, reorganizing produces our 
decomposition of F  
∆Tech
∆MS
TFP
(xiii) .      =TFPF ∆ ×TEff ∆ ×Tech ∆SEff ×∆AEff ×∆MS ,
 41