1. Faculty of Business, Economics and Law
School of Economics
A Parametric Approach to Analysing
Labour Productivity Growth
An Honours Thesis submitted to the School of Economics, The University of
Queensland, in partial fulfillment of the requirements for the degree of
BEcon(Honours)
by
Daniel Pool
supervised by
Professor Christopher O‘Donnell
&
Professor Prasada Rao
Approximated Word Count : 23,200
2. I declare that the work presented in this Honours thesis is, to the best of my knowledge and
belief, original and my own work, except as acknowledged in the text, and that material has
not been submitted, either in whole or in part, for a degree at this or any other university.
Daniel Pool, November 9, 2015
3. Acknowledgments
I would firstly like to thank my two supervisors. Christopher O’Donnell, for his kind guid-
ance, time and patience. Throughout this year, I have learnt so much from him, and not once
did he hesitate to provide his time and advice. Secondly, my dearest gratitude goes to my
secondary supervisor, Prasada Rao, for all the help he provided, including the wise advice
regarding the structure of my Thesis. I truly believe that I could not of asked for a better
pair supervisors to work with throughout this year, and I will be always grateful to them.
Of course, I would like to thank the School of Economics. In particular, Jeff Kline and
Heidi Ellis for all the effort they put into the Honours program. As well as all the lecturers,
who in one year taught me more than I ever would of dreamed of.
There are several friends who throughout this project, were always there for me. Some
of which who were in the same program, some of which were far, far away. While the list
could go on and on, there a few that deserve a special mention. I would like to thank Lucille,
Mai and Francis for sharing this journey with me. As well as Dominique, who has never
failed to provide her support despite our distance.
Lastly, but surely not least. I would like to thank my mother, for her unconditional love
and support she has always given me. Furthermore, for never giving up on me. I could not
leave out my Uncle Pierre, who once saved me and inspired me to be the man I am today.
4. Abstract
Since its introduction by Solow (1957), Growth Accounting has been the main framework
used to analyse labour productivity change over time. Many of its critics over the years
have flagged the strong assumptions made with the model, as well its interpretation of the
‘Solow Residual’ as a proxy of ‘technological change’ as shortcomings. The main objective
of this research is to undertake an econometric approach to labour productivity, relaxing
the assumptions of constant returns to scale, and that economic units always operate on
the frontier. According to this model constructed in this thesis, environmental changes are
considered as driver of shifts to the production function in addition to technological change.
A significant departure from the standard Growth Accounting framework is to allow for
noise and measurement errors with the use of Stochastic Frontier Analysis. Finally, based on
the model constructed, labour productivity change is decomposed into technological change,
input deepening, as well as changes in technical efficiency, a scale effect, and noise. The
decomposition is the applied to the Agricultural, Hunting, Forestry and Fishing Sector for 15
countries for the period between 1977 and 2007, using data from the EU KLEMS Productivity
and Growth Accounts.
10. Abbreviations
Abbreviations
BLS Bureau of Labour Statistics
COLS Corrected Ordinary Least Squares
DEA Data Envelopment Analysis
DFA Deterministic Frontier Analysis
EI Environmental Index
GMM Generalised Method of Moments
GY Geometric Young
HD1 Homogeneous of Degree 1
ICT Information & Communications Technology
IDI Input Deepening Index
iid independently and identically distributed
KLEMS Capial Labour Energy Materials & Services
L.O.S Level of Significance
LP Labour Productivity
LPI Labour Productivity Index
MLE Maximum Likelihood Estimation
ND Non Decreasing
NN Non Negative
OECD Organisation for Economic Co-operation and Development
11. Abbreviations
OLS Ordinary Least Squares
OTE Output Oriented Technical Efficiency
OTEI Output Oriented Technical Efficiency Index
SFA Stochastic Frontier Analysis
SNI Statistical Noise Index
TE Technical Efficiency
TFP Total Factor Productivity
TI Technological Change Index
UK United Kingdom
US United States
12. Chapter 1
Introduction
1.1 Background
Labour productivity, measured as the output per unit of labour input, can be loosely in-
terpreted as the ability of labour to convert inputs into outputs. Its importance to the
economic literature ranges from its role as a key determinant of trade according to the Ri-
cardian model, all the way to its close links to income per capita, and therefore the wealth
of individual countries (Balk 2014). Its importance has led to various statistical agencies
publishing labour productivity measures and their determinants for the economy as a whole,
as well as for various sectors. These measures are mostly done according to the standard
framework, Growth Accounting, introduced by Solow (1957). For example, the OECD (2001)
Productivity Manual, which sets the standards for national statistical agencies, follows this
framework.
Under the assumptions of constant returns to scale, perfectly competitive markets and full
technical and allocative efficiency, the Solow model allows the input cost shares to act as
weights for respective input contributions to production. Over time, it provided a framework
in which any sources of growth that is not explained by the changes in inputs, also known
as input deepening, can be attributed to what is referred to as ‘Technical Change’. This
measure is calculated as a residual, and therefore represents any factors that explain growth
1
13. CHAPTER 1. INTRODUCTION 2
in output besides this growth in inputs, which led to the component being dubbed the ‘Solow
Residual’. Many authors in the literature interpret this measure as changes in technology,
for example see Balk (2001), Hulten (2000), Del Gatto et al. (2011).
Unfortunately for the Solow model, the assumptions it is based on seldom hold in the real
world, especially over the short-term or when analysing sectoral data. Moreover, the frame-
work is not robust to these assumptions. As (Balk 2001, p. 7), in his seminal article argues
these assumptions “appears to hold only in an economically perfect world”. Should these as-
sumptions fail to hold, the residual will reflect many factors in addition to ‘technical change’.
Furthermore, as various forms of errors are forever present in empirical studies, these will
also be reflected in the measure. This is a fact that has been acknowledged by many authors
in the field such as Abramovitz (1956), OECD (2001), Balk (2003).
Over the years, there have been efforts made to relax these assumptions. For example,
Aigner and Chu (1968) and F¨are et al. (1994) used linear programming methods to allow
for technical inefficiency and analyse its effects on labour productivity. Furthermore, papers
have been produced to study the effect of labour productivity when the assumption of con-
stant returns to scale do not hold (Balk 2001, Diewert et al. 2011, Mizobuchi 2014).
However, of the studies that take these factors into account, none have constructed a model
that allows for errors in the framework or data when analysing labour productivity. The cur-
rent approaches, such as Data Envelopment Analysis (DEA) and the Solow model all assume
there are no errors present. Due to the ever present errors of all sorts in empirical work,
especially when dealing with macroeconomic data series, we argue this is an assumption too
strong to be ignored.
All these factors have motivated the aim of this thesis, which is to introduce a framework
that allows labour productivity change to be explained more comprehensively when the stan-
14. CHAPTER 1. INTRODUCTION 3
dard assumptions of the Solow model do not hold. Furthermore, to do so by moving away
from the deterministic approaches of numerical programming and index numbers, towards a
parametric approach with the use of Stochastic Frontier Analysis (SFA).
1.2 Objective
The objective of this thesis is to construct a production framework that relaxes the assump-
tions of the Solow model, but still provides a comprehensive explanation of productivity
change over time. We will do this by using the Solow model as our base, then progressively
relaxing the its assumptions. The analysis starts with the introduction of the concept of en-
vironmental variables, and identifying how its changes can also cause shifts in the production
function. We then move on to relaxing the assumptions of constant returns to scale (CRS)
and then allowing for technical inefficiency. Lastly, we introduce the possibility of functional
form and measurement errors into the framework (i.e: a noise component). Along the way,
the implications of these factors and how they affect the interpretation of Solow’s ‘technical
change’ will be discussed. We show that if the main assumptions of the Solow model are
violated, the measure of ‘technical change’ is biased.
The final outcome of the economic framework is a labour productivity measure that is driven
by six main factors, these being: (i) technological progress, (ii) input deepening, (iii) envi-
ronmental change, (iii) a scale effect, (v) technical efficiency, and finally (vi) noise change.
Following the development of the framework, we then discuss alternative ways of apply-
ing the model to data. These include the use of Corrected Ordinary Least Squares (COLS)
to estimate a deterministic frontier model and Maximum Likelihood Estimation (MLE) to
estimate a stochastic frontier model.
Lastly, we use the EU KLEMS Labour Productivity and Growth Accounts to provide an
empirical application of the framework. One factor motivating our use of the parametric
15. CHAPTER 1. INTRODUCTION 4
approach is the ability to use statistical tests to assess the main assumptions of the Solow
model. We aim to run several regressions based on COLS and MLE on a panel of 15 countries.
Following these regressions, several statistical tests are made based on the assumptions of the
Solow model, including constant returns to scale and the presence of technical inefficiency.
The MLE coefficients are then used to decompose the labour productivity index (LPI) over
time for each of the 15 countries in the panel. Unfortunately, due to limitations in the data,
we are not able to identify the environmental variables in the decomposition.
For illustrative purposes, we intend to focus on the drivers of labour productivity growth
in the US and Germany. This will start with the full, flexible decomposition of the labour
productivity index. We then subsequently show how tightening these assumptions effect the
LPI decomposition. Furthermore, we compare the full decomposition to the decomposition
resulting from Growth Accounting to analyse the effects of the assumptions on ’technical
change’. We show that if these assumptions do not hold and are not accounted for, then
‘technical change’ represents not only the factors that cause shifts in the production func-
tion, but also represents scale effects, technical efficiency and noise.
1.3 Contribution
This thesis extends the current analysis of Growth Accounting to allow for weaker and less re-
strictive assumptions. In addition to relaxing the assumptions already made in the literature,
we introduce the concept of environmental variables into the Growth Accounting analysis.
Furthermore, the most significant contribution to the literature is allowing for noise in the
labour productivity index decomposition.
According to the final output model produced in this thesis, we identify 6 main drivers
of labour productivity change over time, these being: (i) technological progress, (ii) input
deepening, (iii) environmental change, (iii) a scale effect, (v) technical efficiency, and finally
(vi) a noise effect. To measure the effects on changes in labour productivity, we define an
16. CHAPTER 1. INTRODUCTION 5
index to identify and measure these changes. Decomposing the index into environmental
change, technical efficiency change, noise change and a measure of output scale-mix effi-
ciency change, which combines the effects of changes in input deepening and the scale factor.
While there have been advances in the application of Growth Accounting over the years,
in particular with the implementation of Data Envelopment Analysis, there have been no
attempts to apply the analysis using an econometric approach. This thesis proposes the use
of Stochastic Frontier Analysis in the implementation of such an approach.
With the use of an empirical application, this thesis applies the proposed framework on
the Agriculture, Hunting, Forestry and Fishing sector from the EU KLEMS Productivity
and Growth Accounts. We use statistical tests to show that with the data set used, the
Solow assumptions do not hold. Accordingly, we compare the drivers of labour productivity
to show that, as expected, when the main assumptions are violated, that there will be other
factors driving labour productivity.
1.4 Structure of the Thesis
The remaining structure of this thesis is as follows:
Chapter 2, which is the crux of this thesis, starts with a review of the Solow model, and
then progressively relaxes its assumptions. Along the way, the implications of the assump-
tions on the restricted Solow model when thy do not hold are also discussed. It concludes
with the decomposition of the labour productivity index in a less restrictive framework.
Chapter 3 introduces a generalised production function which accommodates the drivers of
labour productivity outlined in the previous chapter. The Growth Accounting methodology
is briefly discussed, followed by a more detailed outline of the two econometric methodolo-
gies used to estimate the function; Stochastic Frontier Analysis and Deterministic Frontier
17. CHAPTER 1. INTRODUCTION 6
Analysis.
Chapter 4 provides an empirical application of the framework introduced in Chapters 2
and 3. The importance of high quality data is discussed, followed by a description of the
measures used. Several statistical tests are conducted based on the assumptions of the Solow
model. Moreover, the interpretation of the results and a cross comparison between the US
and Germany are then discussed. Finally, we restrict our full model to analyse the effects on
the labour productivity decomposition when the assumptions do not hold.
Finally, Chapter 5 concludes the thesis.
18. Chapter 2
Economic Framework
2.1 The Solow Model
In his seminal paper, Solow (1957) introduced the concept of an aggregate production function
used to explain the relationship between real output and real inputs in an economy or sector.
This paper introduced an economic framework to the former axiomatic approach introduced
by Tinbergen (1942). Tinbergen’s work however, was published in German and not discovered
by the US schools until later on (Griliches 1996). The production function was used to
describe the transformation of a firm’s capital and labour to produce an aggregate output
under the neoclassical framework. This model is expressed below:
Qt = F(Kt, Lt, A(t)) (2.1)
where Qt, Kt and Lt is the aggregated real output, capital and labour, respectively. The
production function F(Kt, Lt, A(t)) is assumed to be homogeneous of degree 1 in capi-
tal and labour. This, therefore implies that the technology exhibits constant returns to
scale. Throughout this thesis, we will be using these two terms interchangeably. Finally, the
component A(t) represents a measure of ‘technical change’. In the literature (Solow 1957,
Abramovitz 1962, Jorgenson 2001), it is common to assume that ‘technical change’ is ‘Hicks-
Neutral’. The implications of such an assumption is that ‘technical change’ causes shifts
7
19. CHAPTER 2. ECONOMIC FRAMEWORK 8
in the production function, leaving the ratios of the inputs’ marginal products unchanged.
Accordingly, the output equation can be expressed as:
Qt = A(t)F(Kt, Lt) (2.2)
Equation (2.2) shows that under the Solow model, there are two factors affecting output.
These are: (i) movements along the production function due to changes in Lt and Kt, and
(ii) exogenous ‘shifts’ in the production function caused by changes in A(t). The former
is a relatively straight forward measure, given that the outputs and factor inputs are ap-
propriately measured (a factor that plays a role of great importance in the analysis). On
the other hand, there has been much controversy concerning A(t). Solow referred to it as “
Technical Change”, without going into details about what may cause these shifts. However,
he suggested “...improvements in the education of the labour force, and all sorts of things
will appear as technical change” (Solow 1957, p. 312). Due to the rather ambiguous nature
of this measure, Domar (1961) later on called it the “residual”, as it may contain other fac-
tors such as external economies, changes in product mix, better management and so forth.
Several other authors and institutions, such as Hulten (2000) and the OECD (2001) have
since shared this view.
This ‘residual’ can be interpreted as a measurement of Total Factor Productivity (TFP)
under the assumptions of the Solow framework. The OECD defines TFP as total aggre-
gated output divided by total aggregated input (OECD 2001, p. 13). Therefore, by dividing
equation (2.2) by F(Kt, Lt), which can be viewed as measure of total input, Xt, we get:
TFPt ≡
Qt
Xt
=
Qt
F(Kt, Lt)
= A(t) (2.3)
It is important to keep in mind that when using the production function as a scalar aggre-
gator function to measure Xt, that it satisfies the following properties: (i) non-negativity
20. CHAPTER 2. ECONOMIC FRAMEWORK 9
(NN), (ii) non-decreasing (ND) and (iii) linearly homogeneous (HD1) (O’Donnell 2012). The
Cobb-Douglas production function proposed by Solow satisfies the NN and ND properties.
Furthermore, through the assumption of constant returns to scale, it is also HD1.
Despite the lack of understanding of what drives these changes in TFP, the Solow Model
nonetheless provides a framework that can calculate it. This is done by using index numbers
to measure changes in the aggregate measures of outputs and inputs. By taking the difference
between the output index and the input index, what is left is a measure of the change in
TFP. It is therefore a residual measure, which is why it is commonly known as the ‘Solow
Residual’ in many modern textbooks. Note that with the labour productivity measure (refer
to equation (2.4) in the next paragraph), output is not divided by a measure of total input.
Therefore, in this respect, the complexity of dealing with the changes of LP are arguably far
less than that of TFP.
Another concept that is critical to Growth Accounting and various strands of economic
literature is labour productivity (LP). As opposed to TFP, which is as multi-factor produc-
tivity measure, labour productivity is a single factor productivity measure (OECD 2001). It
is defined broadly defined as the ratio of a measure of output to a measure of labour input.
It also can be thought of a special case of TFP, with the aggregate measure of Xt assigning
all weight to labour and zero to all other inputs. As a result, Xt = Lt.
By dividing equation (2.2) by Lt, and under the assumption of constant returns to scale,
labour productivity can be expressed as:
LPt ≡
Qt
Lt
= A(t)F
Kt
Lt
, 1 (2.4)
Equation (2.4) shows that under the Solow model, there are two main drivers of labour pro-
ductivity. The first is shifts in the production function driven by ‘technical change’. Secondly,
changes in the capital to labour ratio; this effect is also referred to as ‘capital deepening’.
21. CHAPTER 2. ECONOMIC FRAMEWORK 10
In his initial paper, Solow’s model expressed the changes in labour productivity and TFP as
derivatives. Therefore implying continuous changes of such variables with respect to time.
The Divisia index number was used to approximate these continuous changes. This index
number was propsed by by French economist Divisia (1925) to measure the changes in both
prices and quantities by the weighted average rates of growths of its components (Jorgenson
and Griliches, 1971 ). This index was also used in other studies ( Jorgenson and Griliches,
1967 and Denison, 1972 ). However, it is important to note that in economics, neither prices
nor quantities are measured or recorded continuously. While some prices are recorded more
frequently than others, it is seldom, if not ever done gradually. As a result, the Tornqvist
(1936) index is commonly used to provide an approximation to the Divisia index (Hulten
2000, OECD 2001, Bureau of Labor Statistics 2015).
The Divisia index numbers used in Solow’s analysis use weights assigned to different compo-
nents of the aggregator function. The benefit the economic framework put forward by Solow
is that through the assumptions that the marginal products are paid at their marginal social
costs, and constant returns to scale, the observed cost shares from the market could be used
to approximate these output elasticities. While Solow himself did not explicitly make any
assumptions about the behaviour of firms and the market (Acemoglu 2008), the following
demonstrates that under the assumption of perfectly competitive markets, and the profit
maximizing behaviour of firms, we obtain the same results. These assumptions are also in
line with the OECD Productivity Manual, in which their framework makes similar assump-
tions (OECD 2001, p. 19).
Following the assumption that the aggregate production function represents the economy,
suppose that the representative firm maximizes its profits in a perfectly competitive market.
22. CHAPTER 2. ECONOMIC FRAMEWORK 11
Therefore, their decision problem in period t is
max
Kt≥0, Lt≥0
{PtQt − wtLt − rtKt subject to Qt = A(t)F(Kt, Lt) } (2.5)
Which can also be expressed as:
max
Kt≥0, Lt≥0
{PtA(t)F(Kt, Lt, ) − wtLt − rtKt}
Where:
• Pt is the price of output Qt in time period t
• wt is the wage rate paid for labour services in time period t
• rt is the rental rate paid to capital services in time period t
By differentiating the objective function with respect to labour and capital, the first order
conditions imply
PtA(t)FL(Kt, Lt, A(t)) = wt
PtA(t)FK(Kt, Lt, A(t)) = rt
where FL and FK are the marginal products of labour and capital, respectively. Furthermore,
if the price of output in period t is normalised to 1:
A(t)FL(Kt, Lt, A(t)) = wt (2.6)
A(t)FK(Kt, Lt, A(t)) = rt (2.7)
Thus, we can see that the marginal revenues are equal to their marginal costs (The same
assumptions Solow made) .
Solow further made the assumption that the production function is homogeneous of de-
gree 1 in capital and labour. Following Euler’s Theorem (Acemoglu 2008), we know that if
23. CHAPTER 2. ECONOMIC FRAMEWORK 12
the production function is homogeneous of degree r ∈ R++ in capital (Kt ∈ R) and labour
(Lt ∈ R), then :
rF(Kt, Lt) = FK(Kt, Lt)Kt + FL(Kt, Lt)Lt
In the case of the Solow model, constant returns to scale implies that r = 1. Therefore:
F(Kt, Lt) = FK(Kt, Lt)Kt + FL(Kt, Lt)Lt
Furthermore, by multiplying both sides of the equation by A(t) :
A(t)F(Kt, Lt) = A(t)FK(Kt, Lt)Kt + A(t)FL(Kt, Lt)Lt (2.8)
The solution to the optimization problem has to satisfy the constraint : Qt = A(t)F(Kt, Lt).
By substituting equation (2.8) into the technology constraint, it is evident that output can
be expressed as below:
Qt = A(t)FK(Kt, Lt))Kt + A(t)FL(Kt, Lt)Lt (2.9)
Dividing equation (2.9) by Qt gives:
1 = αKt + αLt (2.10)
Where αKt = A(t)FK Kt
Qt
and αLt = A(t)FLLt
Qt
represent the contribution of capital and labour to
total income, respectively (recall that the price in period t is normalized to 1).
It is evident from equations (2.6) and (2.7), that αKt = rtKt
Qt
and αLt = wtLt
Qt
, while the
marginal products are not observed, their respective marginal costs are part of income shares,
that can be observed in the market. The implications following this result is that Solow pro-
vided a framework in which the value of the factors of input (income shares), obtained by
observed market transactions could be used as the factor shares. These shares could then be
used for weights in the index numbers used to calculated changes in the aggregate measures
24. CHAPTER 2. ECONOMIC FRAMEWORK 13
over time.
The first application of the framework was done by Solow (1957), in which he applied the
model to US data in order to estimate changes in labour productivity of the economy be-
tween 1909 and 1949. In this article, there was an emphasis on the difficulty of obtaining the
appropriate data for the analysis. This was evident as compiling the three data series nec-
essary (capital services, labour services and real output) proved to be problematic. Despite
preferring a measure of real net national product, Solow instead had to use a gross measure
due to the former being hard to come by. Furthermore, the measure of capital services is one
that may be full of errors due to its nature. The capital measure used was net of government,
agricultural and consumer durables.
Several key features can be highlighted from Solow’s application of the US data. One is that
labour productivity appeared to be trending upwards throughout the period, with the figure
doubling over the interval (Solow 1957). Of this growth, an astounding 87.5% was attributed
to ‘technical change’. By construction, the remaining 12.5% of growth was explained by
input deepening. Furthermore, Solow used a scatter plot of the calculated ‘technical change’
against the capital to labour ratio, which suggested no trace of a relationship. This result
was in favour of his assumption of the neutrality of technology throughout the period. This
may therefore have motivated the use of the Hicks-Neutrality assumption in his future studies.
Since then, there have been several adoptions of the approach put forward by Solow and
Tinbergen. The performance of the US growth in the post war period was a topic hotly
debated in the literature. Denison (1972), for example, reported an annual average growth
in ‘technical change’ of 1.37% between 1950 and 1960. On the other hand, Jorgenson and
Griliches (1967) put forward an economic framework in order to obtain a more accurate
measurement of capital and labour (these measures will be discussed in detail later on in
this thesis). According to their framework, they calculated an average growth rate of only
25. CHAPTER 2. ECONOMIC FRAMEWORK 14
0.1% between 1945 and 1965. Therefore stating that after appropriately accounting for the
contribution of the factor inputs, on average, ‘technical change’ contributed only 2.8% to the
growth in output. This result was in line with their hypothesis, that after controlling for the
factor inputs, ‘technical change’ contributed very little to output growth. These results were
therefore of great contrast to the contributions of ‘technical change’ calculated by Solow and
Denison.
In a rebuttal to the results put forward by Jorgenson and Griliches, Denison (1972) ar-
gued that the difference in the figures were due to many factors such as the definition of
scope, output and the time period analysed. But most significantly due to difference in
the measures and inferences of factor inputs. Denison argued that his method provided a
broader measure for capital, allowing for shifts in quality changes. Further, he argued that
the utilization assumptions of capital and land made by Jorgenson and Griliches were not
consistent with the empirical surveys made in the period. This example shows the impor-
tance of the appropriate assumptions made in regards to the underlying economic framework
used to measure inputs, and how they reflect on the measurement of productivity. It should
therefore be evident that the measurement of productivity is no easy task.
Until now, our analysis has focused on cases where the practice of Growth Accounting has
been done on the aggregate economy level. Domar (1961) was the first to dig deeper into this
issue. Motivated by the thought that the ‘residuals’ from different industries that make up
the economy may provide more information to explaining the performance of the economy
as a whole, Domar outlined a framework of various weighting systems according to different
scenarios that led to ‘residual invariant’ aggregation and integration of firms and sectors.
The difference between aggregation and integration in the context of productivity is that ag-
gregation refers to the building up of productivity measures from sectors to the economy as a
whole. This method provides additional explanatory power, through highlighting the sectors
which are driving growth in the economy’s productivity. On the other hand, integration
26. CHAPTER 2. ECONOMIC FRAMEWORK 15
takes into account the existing upstream and downstream relationships between industries
when aggregating these sectors (OECD 2001). As integration does not take into account the
intra-industry product flows, there tends to be ’double counting’ in the process. The study
of productivity measures aggregated from the industry level was later on extended by Hulten
(1978)
In light of the developments put forward by Domar and Hulten, Jorgenson (1988) published
a further study regarding the performance of the post war US economy between 1948 and
1979, using productivity measures at the industry level in the US. Jorgenson also included
contributions of intermediate inputs in his analysis. Some interesting figures found in his pa-
per was that 46 out of the 51 sectors had growth predominantly explained by capital, labour
and intermediate inputs. Jorgenson, as well as Denison (1985), found a decrease in the US
economic growth during some periods. Both papers found that the decrease was primarily
driven by a slowdown in productivity growth. By analysing the data at an industry level,
Jorgenson then found that there had also been slower growth of productivity at the industry
level too. To explain the growth of productivity at the sector level, Jorgenson proposed the
use of econometric methods, through which he argued that without the ‘parametric tool’,
the decline in the US economic growth would of remained unexplained.
While the empirical applications discussed above have focused on the US, the analysis has
been applied to many different countries and sectors. Two country specific examples are an
in depth analysis of the industry level productivity comparisons of Japan by Jorgenson et al.
(1987), and the performance from various sectors in Russia by Timmer and Voskoboynikov
(2014) . Furthermore, Timmer et al. (2008) provides a comprehensive Growth Accounting
analysis to compare the superior performance of the US’ productivity compared to those of
the Eurozone members. Their findings, consistent with Jorgenon’s work (Jorgenson 2001),
found that investment in the Information Technology (IT) sectors have been spurring growth
in the US since the mid 1990’s, as opposed to the Eurozone members. Timmer et al. con-
27. CHAPTER 2. ECONOMIC FRAMEWORK 16
cluded that this factor was the main driver of the productivity differences between the two
regions throughout the period.
The examples given have focused on the studies published in academic circles. Most, if
not all of these have contributed to the standards of Growth Accounting set by the OECD
in their Productivity Manual (OECD 2001). This document provides detailed guidelines
for productivity measurement and analysis, offering empirical examples and interpretations.
Furthermore, there have been collaborations between the European countries, as well as Aus-
tralia, Japan and the US to provide harmonized measurements of data that can be used for
direct comparisons of productivity analysis. Such figures can be found from the EU KLEMS
database (O’Mahony and Timmer 2009). The World KLEMS initiative (World KLEMS 2015)
also provides information in a similar, harmonized manner. This includes KLEMS data for
the countries in the EU KLEMS database, as well as Canada, Korea, India and Argentina.
Due to its importance in growth and income for countries and industries, various national
statistical agencies provide TFP and labour productivity measures for a broad range of in-
dustries and sectors, as they can be crucial for policies resulting based on these measures.
The Bureau of Labor Statistics (2015) for example, posts annual measures of output per unit
of combined inputs for the private business, non farm business, and manufacturing sectors
and industries. The measure is obtained from an aggregate of 18 manufacturing industries,
providing a measure of output per input unit though the KLEMS framework, i.e: capital (K),
labour (L), energy (E), materials (M), and purchased business services (S) (Bureau of Labor
Statistics 2006). The data are aggregated within the sectors through the use of Tornqvist
index numbers, and the TFP measures are obtained from the residuals, consistent with the
Growth Accounting framework.
There have been several extensions to the analysis of Growth Accounting that deviate from
the initial Solow model. Many of these studies have been a motivation for this thesis, and
28. CHAPTER 2. ECONOMIC FRAMEWORK 17
will be mentioned in the subsequent sections where relevant.
In his textbook, Acemoglu argues that the Solow model should be thought of as a “...spring-
board for richer models” (Acemoglu 2008, p. 27). Accordingly, throughout the rest of this
chapter, we will be extending the Solow Model to control for factors such as environmental
changes, through which we argue may also cause shifts in the production function. Further-
more, we allow for variable returns to scale, account for inefficiencies of firms, and allow for
statistical noise in the analysis. We will do so by cumulatively extending the framework set
out, starting with equation (2.2).
2.2 Gross Output and Omitted Inputs
Despite the standard solow model being constructed based on one unit, several further studies
have done multi-country comparisons of productivity. Our model allows for such compar-
isons. From here onward, our model will be based on a panel of i countries. Note in the case
of a single country, the unit descriptor can simply be omitted.
When making comparisons of labour productivity, one should be careful to make the dis-
tinction between gross and value added measures of labour productivity. The difference
between the two comes from the measure of the output used in the numerator. According to
the OECD (2001), gross output is a measure of output in regards to both primary inputs,
being capital and labour, and intermediate inputs. Value added, on the other hand, is a
measure of the value added by primary inputs to the intermediate inputs.
For the purposes of this thesis, our definition of labour productivity is in line with that
of the gross labour productivity measurement from the OECD Productivity Manual (OECD
2001, p. 14). Therefore, the model used from here onwards will be based on such a measure.
To accommodate this, we introduce the gross output measure to the Solow framework by
including intermediate inputs (which consist of an aggregated measure of Energy, Materials
29. CHAPTER 2. ECONOMIC FRAMEWORK 18
and Services) in the production process, acknowledging output Qit as being a gross output
measure. Therefore, keeping in mind that this model accounts for i countries, gross output
can be an be expressed as below, and all subsequent analysis will be based on :
Qit = A(t)F(Kit, Lit, Mit) (2.11)
where :
• Qit is a measure of gross output produced from country i in period t
• Kit is a measure of capital used by country i in period t
• Lit is a measure of labour used by country i in period t
• Mit is a measure of intermediate inputs used by country i in period t
• F(Kit, Lit, Mit) is homogeneous of degree 1 in capital, labour, and intermediate inputs.
In his paper, Solow (1957) did not take into account intermediate inputs, neither did he
explicitly define which output measure he was using. We argue that given the nature of A(t)
being calculated as a residual, such intermediate inputs would be captured in this component
according to our definition of output. Therefore, according to our model, the intermediate
inputs are considered as omitted inputs based on the standard Solow output equation (2.2).
Later papers that focus on the sectoral analysis of labour productivity, account for interme-
diate inputs in the production function (Jorgenson 1988, Timmer et al. 2008).
After explicitly defining our measure of output and introducing intermediate inputs into
the production function, labour productivity is expressed as below :
LPit ≡ A(t)
1
Lit
F(Kit, Lit, Mit) (2.12)
30. CHAPTER 2. ECONOMIC FRAMEWORK 19
2.3 Environmental Factors and Technological Progress
The first extension we make to the standard Solow framework is in line with the concept of
‘technical change’. In his paper, Solow referred to the term as “... a short hand expression
for any kind of shift in the production function” (Solow 1957, p. 312). While in the past, the
interpretation of these shifts have been primarily focused on changes (mostly advances) in
technology, we argue that there are environmental factors that may also lead to such shifts in
the production function. Accounting for these changes in the production environment may
provide a more accurate explanation for changes in labour productivity. As all previous stud-
ies provide extensive analysis on the technological advance component, this section focuses
on the effects of changes in the production environment.
For the purposes of this framework, we follow the definition of an environmental variable
from (O’Donnell 2015, p. 2). According to him, “characteristics of the production environ-
ment are variables that are physically involved in the production process, but never chosen
by firms”. By the nature of this definition, all environmental variables will be considered
exogenous. Changes in the production environment, as mentioned, will cause shifts in the
production function. Therefore, it will not be considered an input in the production process,
but rather a ‘shifter’. Sun et al. (2015) also refer to these factors as exogenous, non tradi-
tional inputs and acknowledges they may shift the production function either neutrally or
non-neutrally. For example, Zhang et al. (2012) provide a good example in which environ-
mental change causes non neutral shifts, while Bhaumik et al. (2015) provide the case where
an environmental variable causes a neutral shift in the production function.
In practice, changes in the production environment are not uncommon. However, when
when conducting a thorough analysis of a production process, it is crucial to identify the de-
cision making unit of interest. By doing so, one can have a clear idea of which variables can
be considered environmental. Or in other words, which factors can be chosen or controlled
by the decision making unit. In the case of Growth Accounting, the analysis is applied at
31. CHAPTER 2. ECONOMIC FRAMEWORK 20
both macro and micro levels (Del Gatto et al. 2011). Therefore, the environmental factors
that are relevant may differ depending on the particular market or economy being considered
at the time.
The El Ni˜no phenomenon provides a broad, contemporary example of an event that may
have an effect on the output of several industries. According to the Australian Government
Bureau of Meteorology (2015), the effects of this phenomena, which is associated with periods
of warming in the central and eastern tropical Pacific, has consequences for many industries.
Its effects on the climate include reduced rainfall, warmer temperatures and decreased snow
depths. These factors may flow into several industries across Australia. For example, reduced
rainfall and a warmer temperature has its clear effects on the agricultural sector. While the
warmer sea temperatures may cause various water species to migrate elsewhere ( National
Oceanic and Atmospheric Administration, 2015) , which may effect the output and inputs
in some fishing industries. Theses effects are not limited to agricultural and fisheries. For
example, the warmer temperatures and decreased snow depths may have damaging effects
on the tourism industry in these particular regions.
The inclusion of environmental effects is an important, yet relatively underutilized concept in
the analysis of productivity. O’Donnell (2014b) for example, outlines a methodology to de-
compose a TFP index into various measures including ‘technical change’, and also included
temperature as an environmental variable in the analysis of the US agricultural industry
(O’Donnell 2014a).
In line with our argument that ‘technical change’ is a component of both technological ad-
vance and environmental change (changes in the production environment), we will introduce
a new variable to ‘technical change’ such that A(t, zit), where zit = (x1it, . . . , xJ∗it) ∈ RJ∗
and represents a vector of J∗
environmental variables in time period t for a given country i.
Note, J∗
is the total number of environmental variables in the production process. Equation
32. CHAPTER 2. ECONOMIC FRAMEWORK 21
(2.13) illustrates this adjustment:
Qit = A(t, zit)F(Kit, Lit, Mit) (2.13)
Equation (2.13) shows that what was referred to as ‘technical change’ by Solow is now a func-
tion of not only time, but also of changes in the production environment. Therefore, either
exogenous changes in the technology available to all participants in the economy or changes
in environmental factors may cause shifts in the production function and effect real output.
Note that as in the case of shifts caused by technology change, environmental changes do not
effect the ratio of the marginal products, but instead cause a shift in the production function
accordingly.
Following our discussion and from equation (2.13), it is clear that with the inclusion of
environmental variables, the labour productivity function is extended. This is shown in
equation (2.14):
LPit ≡
Qit
Lit
= A(t, zit)F
Kit
Lit
,
Mit
Lit
, 1 (2.14)
Figure 2.1 shows the effect of an upward shift caused by environmental change on labour
productivity. With real output on the y-axis and labour services on the x-axis, the diagram
gives the output produced by the production function. By definition, the gradient of the
straight line extending from the origin and passing through a production point gives a mea-
sure of labour productivity. The curve represents the production function, which shows the
relationship between maximum output and labour services, holding the amount of capital
and intermediate inputs fixed.
According to the Solow model, all production points will be on the frontier (an assump-
tion that will be relaxed later in this chapter). Starting from the initial point A in Figure
2.1, for a given set of inputs xt = (xLt, xKt, xMt) ∈ R3
(where xt is a vector of capital, labour
33. CHAPTER 2. ECONOMIC FRAMEWORK 22
and intermediate inputs) in time period t, an upward shift will create an increase in output,
ceteris paribus. This will lead to a movement to production point B. As the diagram shows,
this will lead to a steeper slope, and therefore a higher level labour productivity.
B
A
L
Q
Figure 2.1: Effect of a shift in the production function caused by environmental change on
labour productivity
It is common in the literature of Growth Accounting for authors to dismiss technological
regress. This may happen when the production function shifts inward. Perhaps with the use
of environmental variables, the inward shifts in the production function can be explained.
For example in the case of the agricultural sector, the technologies may remain the same,
but unfavourable environmental factors may cause inward shifts in the production function,
lowering labour productivity.
Due to the exogenous nature of these environmental variables, their implications for pol-
icy measures are limited. It is important, nonetheless, to control for these effects in order to
better understand the factors driving labour productivity, as omissions of these changes may
lead to biased contributions or misinterpretations of technological advance. Furthermore, as
a fundamental economics measure, we want to understand the drivers in labour productiv-
ity, and environmental changes can help provide us with more information in regards to its
changes.
34. CHAPTER 2. ECONOMIC FRAMEWORK 23
2.4 Relaxing the Assumption of Constant Returns to
Scale
Until now, the aggregate production function used in this thesis has been restricted to being
homogeneous of degree 1. This, of course, is a restriction to the general case of which the
production function is homogeneous of degree r. In this section, we allow for the possibility
of increasing or decreasing returns to scale. By doing so, we show that labour productivity
may also be driven by scale effects.
We wish to introduce a framework that allows for variable returns to scale, or in other words,
where the production function is homogeneous of degree r, where r is the scale elasticity. In
the case of Growth Accounting, a single aggregated output measure is considered. According
to several authors (Ray 1999, Diewert et al. 2011, Zelenyuk 2013), the scale elasticity in the
case of a single output is given by:
r ≡
∂lnf(λx)
∂lnλ λ=1
=
M
m=1
∂f(x)
∂xm
xm
f(x)
= xf(x)x/f(x)
where x is a vector of M inputs (i.e : x = (x1, x2, . . . xM ) ∈ RM
+ ) and xf(x) gives
the vector of partial derivatives of the production function with respect the inputs (i.e :
xf(x) ≡ (∂f(x)/δx1, ∂f(x)/δx2, . . . ∂f(x)/δxM ) ∈ RM
+ and λ ∈ R++ .
The standard economic interpretation of constant returns to scale is that an increase in
all inputs of the same proportion will lead to an increase in output by the same proportion
(i.e: r = 1). Therefore under the Solow model, rises in all inputs have to be met by a propor-
tionate rise in output. On the other hand, increasing returns to scale is the case where output
increases in a greater proportion than the increase in inputs (i.e; r > 0). While decreasing
returns to scale is the case where output increases less than the proportion increases in inputs
(i.e: r < 0).
35. CHAPTER 2. ECONOMIC FRAMEWORK 24
To relax the assumption of constant returns to scale, we simply assume that the produc-
tion function is now homogeneous of degree r.
Qit = A(t, zit)F(Kit, Lit, Mit) (2.15)
According to the new output equation (2.15), we want to analyse the effects of variable
returns to scale on labour productivity. To do so, we will start by dividing equation (2.15)
by Lit to get:
LPit ≡ A(t, zit)
1
Lit
F(Kit, Lit, Mit) (2.16)
As the aggregate production function is no longer homogeneous of degree 1, we will proceed
to show that the relationship in equation (2.14) no longer holds.
Starting with the assumption that F(Kit, Lit, Mit) is homogeneous of degree r, we obtain:
λr
F(Kit, Lit, Mit) = F(λKit, λLit, λMit)
where λ > 0 and r ∈ R++. By letting λ = 1
Lit
, we obtain:
1
Lr
it
F(Kit, Lit, Mit) = F
Kit
Lit
,
Mit
Lit
, 1
By multiplying both sides by Lr−1
it we obtain the following expression:
1
Lit
F(Kit, Lit, Mit) = Lr−1
it F
Kit
Lit
,
Mit
Lit
, 1 (2.17)
By substituting equation (2.17) into (2.16), we derive a new labour productivity expression
under the assumption of variable returns to scale in the production function:
LPit ≡ A(t, zit)Lr−1
it F
Kit
Lit
,
Mit
Lit
, 1 (2.18)
Notice that the expression for labour productivity under constant returns to scale is different
36. CHAPTER 2. ECONOMIC FRAMEWORK 25
to that of variable returns to scale. The difference, being Lr−1
it , represents a scale effect. Of
course, in the case of constant returns to scale, r=1, and the term collapses down equation
(2.14). From here on, we will proceed with the more general labour productivity measure
given by equation (2.18).
The analysis of productivity has been moving away from the restricted constant returns
to scale framework. This is evident, as several statistical agencies have begun to avoid mak-
ing assumptions regarding constant returns to scale (Diewert et al. 2011). Furthermore, there
have recently been several studies that allow for effects of scale elasticity in Growth Account-
ing.
According to Balk (2003, p. 36), “Although one could argue that the assumption of con-
stant returns to scale can validly be made on a global level and for the long run, it appears
to be hardly tenable on a sectoral level and for the short run.” This statement highlights the
implications of the failure of such an assumption. As mentioned, statistical agencies produce
these measures annually. Therefore, in the short term, it may be that the assumption of
constant returns to scale does not hold. Furthermore, the state of the art implementation
of Growth Accounting takes into account the aggregation of sectors, and how they drive
productivity.
Following the discussion of the effects of variable returns to scale by Denny et al. (1981),
Kumbhakar et al. (2000) applied a parametric framework using a Translog production func-
tion to decompose TFP change into ‘technical change’, economies of scale and economic
efficiency ( both technical and allocative). The authors found the effects of scale to change
over the time periods of the years analysed, in other words, being ’ U-shaped ’.
Balk (2001), in his analysis of a panel of 18 rubber-processing firms in the Netherlands
throughout 1978-1992, compared a Malmquist index with the encompassing numbers of pro-
37. CHAPTER 2. ECONOMIC FRAMEWORK 26
ductivity and found a significant difference between the two measures. As the Malmquist
index captures only the change in technical efficiency (to be discussed in the next chapter)
and ‘technical change’, the difference suggested that there were scale factors present causing
the deviation. Balk concluded that neglecting the scale effects present and the changes in
input-mix would lead to overstated productivity change in his study.
Diewert et al. (2011) provide a good example of analysing TFP growth in the presence
of scale effects. The article discusses the case of several Japanese industries that underwent
major changes in size and structure from the early 1960’s to the late 1980’s. With this ap-
proach, they were able to analyse the relevance of government policy making and its effects
on the scale elasticity of these industries, and how these scale effects translated into TFP
changes over the period. Their estimates found that for the textile industry, there were alter-
nating periods of decreasing and increasing returns to scale. They argue that the pattern was
due to increased production to meet high demand in earlier periods, and improvements in
technology that led to increasing returns in the later periods. The pulp and paper industry,
on the other hand, exhibited increasing returns throughout the same time interval, besides a
period of supply shortages which led to a brief spell of decreasing returns to scale. The sec-
tor’s increasing returns to scale was mostly attributed to superior shipping for transporting
wooden chips, as well as technologies that later on enabled the industries to incorporate new
input mixes in order to overcome the supply shortages.
O’Donnell (2014b) provides a breakdown of Australian TFP change from 1970 to 2007, based
on 18 different sectors using a Geometric Young (GY) index. Using econometric techniques,
he was able to conduct statistical tests and reject some assumptions of the Growth Account-
ing framework such as constant returns to scale. Furthermore, O’Donnell identified scale-mix
efficiency as the most important driver of TFP change in the Australian economy. The author
pointed out that the mining sector’s decline in TFP was driven by increases in both capital
and labour. This coincides with the rising global commodity prices at the time, motivating
38. CHAPTER 2. ECONOMIC FRAMEWORK 27
producers to increase their output, and therefore use more inputs.
As mentioned by Mizobuchi (2014), economists have focused mainly on the technical progress
and growth in capital inputs when focusing on labour productivity. Much less attention has
been paid to the changes in labour services, ceteris paribus, and how it effects labour pro-
ductivity. Mizobuchi provided a framework which decomposed growth in labour productivity
into technical progress, capital input growth and a ‘returns to scale effect’. He applied this
framework to US industry data for the time period between 1987 and 2009, and found that
returns to scale did have an effect on labour productivity growth in both goods and services.
This section highlights the importance of the constant returns to scale assumption in the
production process and the implications it has for labour productivity. Allowing for an anal-
ysis of movements along the frontier provides us with a more deeper understanding of what
factors may be driving productivity, whether it be TFP or labour productivity. Furthermore,
as Diewert et al. (2011) emphasize, government policies such as the opening of trade and ex-
panding markets may provide opportunities for sectors to expand their output and improve
productivity. O’Donnell (2014b) also mentions the relevance of policy measures to improve
TFP in the case of variable returns to scale. As a result, allowing for variable returns to scale
in the analysis of labour productivity does not only provide us with more precise drivers of
the measure, but also creates a framework relevant for policy making decisions.
2.5 Allowing for Inefficiency
Del Gatto et al. (2011) differentiate between productivity analysis methodologies that allow
for the case of inefficiency, and those that do not, as Frontier and Non-Frontier, respectively.
The frontier, also referred to as the production possibility frontier, can be defined as the
maximum output possible for a given input vector. The Non-Frontier approach is classified
as a category that assumes firms are always operating on the frontier. On the other hand,
the Frontier approach allows for the possibility of firms operating below the frontier, in other
39. CHAPTER 2. ECONOMIC FRAMEWORK 28
words, being inefficient. The behavioural assumptions of firms in the Solow model, that is,
profit maximizing in a competitive market, implies that for each given input vector, the firm
will be producing the maximum output possible. Otherwise, they will eventually be forced
out of the market. Therefore, the Solow model can be classified as a Non-Frontier approach.
In this section, we wish to extend the framework to allow for a broader case, where various
factors may impede the mechanisms of the perfectly competitive benchmark. This, in turn,
may eventually lead to inefficiencies. As a result, we wish to provide a Frontier based frame-
work that allows for the possibility of inefficiency.
At this point, it is important to explicitly express the type of efficiency used in our model.
The efficiency we are considering is somewhat similar to that outlined by Leibenstein (1975),
in which he extends the notion of efficiency to distinguish between allocative and what he
called ’X-efficiency’. The concept of ’X-efficiency’ was referred to as “the difference between
maximal effectiveness of utilization and actual utilization” (Leibenstein 1975, p. 582). Our
perspective of efficiency is similar. We will be referring to the measure of Technical Efficiency
(TE), introduced by Farrell (1957). He described TE as the success in producing as large
as possible output from a given set of inputs. Alternatively, technical efficiency can also be
input oriented, in the sense that TE can be described as using the least amount of inputs to
produce a given level output.
Figure 2.2 provides an illustration of the concept of output oriented technical efficiency
(OTE). This diagram shows the combinations of two outputs, q1 and q2, that can be pro-
duced with a given set of inputs, in given time period, and in a given production environment.
The straight line extending from the origin represents a given output mix, while the curve
shows the combinations of the maximum output that can be produced for a given input vec-
tor. Therefore, it represents the production possibilities frontier. This curve can be thought
of as a benchmark efficient entity that produces the maximum output possible, holding all
40. CHAPTER 2. ECONOMIC FRAMEWORK 29
other factors constant.
A
B
L
Q
Figure 2.2: Illustration of output oriented technical efficiency
To outline TE in this case, note that of the two production points A and B, who are both
using the same mix of inputs, A is producing a greater amount of outputs than B. Further-
more, as it is located on the frontier, it is producing the maximum amount, given the input
vector. To obtain a measure of TE, we can divide the respective distance from the origin of
point A (0A) and of point B (OB), i.e : TE = 0B
0A
. Therefore, this measure gives the extent
to which a firm is maximizing its output vector, for a given input vector, at time period t,
in a given environment zit. Therefore, at point A, the firm will be at full technical efficiency
and the TE ratio will be equal to 1. As the distance between production points A and B
grows, the level of technical efficiency will also decrease.
Figure 2.3 provides an illustration of the effect of technical efficiency on labour productivity.
Corresponding to Figure 2.2, point A represents a production point that is at full technical
efficiency, while point B. From the illustration, it is evident that an increase in the technical
efficiency from point B to A leads an increase in the slope of the gradient, and therefore a
higher level of labour productivity.
41. CHAPTER 2. ECONOMIC FRAMEWORK 30
A
B
L
Q
Figure 2.3: Effect of technical efficiency on labour productivity
From Figure 2.3, it should be clear by now that the Solow model, which implicitly assumes
that all firms produce on the frontier, may lead to upward biases in the contribution of ‘tech-
nical change’ to labour productivity in the case where the assumption of full TE fails. After
all, ‘technical change’ is simply calculated as a residual. The approach of Kumbhakar et al.
(2000) for example, apply their econometric analysis to three separate panel data sets and
found that the contribution of technical efficiency was rather large in each case and stated
that ignoring these drivers could lead to biased or misleading estimates of TFP change. Note
that the same would apply in the case of labour productivity change.
Following Farrell’s paper, there has been several applications to the analysis of productivity
to allow for technical inefficiency. Aigner and Chu (1968) for example, applied the non para-
metric approach of linear programming, also called Data Envelopment Analysis (DEA) to
calculate an efficiency measure as well as its changes over time in the Steel industry. While
F¨are et al. (1994) used a similar methodology to construct a ‘world frontier’, based on data
from 17 OECD countries from 1979 to 1988. There are also several other papers that have
used DEA to analyse the productivity of various sectors and economies over time ( Perelman
(1995), Gouyette and Perelman (1997), Weber and Domazlicky (1999), Kumar (2006)).
As opposed to the traditional approach of Growth Accounting, in which the performance
42. CHAPTER 2. ECONOMIC FRAMEWORK 31
of a country or industry is compared only to itself relative to past periods, the recent exten-
sions of literature focus on panel studies, in which the benchmark entity is according to the
constructed ‘grand frontier’. By constructing such a frontier, the performances of countries
can be analysed relative to the best practices and technologies according to the set of coun-
tries in the panel. This is particularly relevant to the literature of convergence and catch
up. However, it is important to emphasize the difference between catch up, which is the
catch up TFP measures of countries, and convergences in income. This distinction is made
by Dowrick and Nguyen (1989).
In the study of F¨are et al. (1994), the authors were able to isolate catch up and shifts
in the frontier itself. The main findings were that the US was producing efficiently, i.e: on
the frontier for all the periods considered. Furthermore, the largest improvements (growth)
in technical efficiency were from Japan, this is consistent with the findings of Baumol (1986)
and Abramovitz (1986) who estimated an inverse relationship between a countries TFP catch
up rate and the country’s initial level of relative labour productivity. Furthermore, as the
US was the only country operating on the frontier, they were able to identify the shifts in
the world frontier caused by advances in US technology.
These studies provide a good example of how accommodating technical efficiency into Growth
Accounting can be done. However, when making such comparisons across countries, it is im-
portant to keep in mind the interpretation of technological change in these studies, and the
assumptions made. Filippetti and Peyrache (2012) in their paper, highlight this fact, stat-
ing that when making such comparisons of catch up, the assumptions are that all relevant
countries have access to the same technologies, and can therefore use these technologies to
catch up to their counterparts. These authors argue that there may be three reasons why
countries may seem inefficient relative to a panel frontier. These are attributed to: (i) an in-
sufficient development of these technologies, (ii) absorbive capacity, or (iii) a lack of efficiency
using these inputs. There may be difficulties in identifying these components, and while this
43. CHAPTER 2. ECONOMIC FRAMEWORK 32
thesis focuses on the interpretation of the third point, it is useful to keep these factors in mind.
The linear programming method used by the authors in the previous paragraphs allowed
for the decomposition of F¨are’s Malmquist Productivity index under the assumptions of con-
stant returns to scale. The advantages over the Tornqvist index previously discussed, is
that the Tornqvist index fails to allow for the possibility of technical inefficiency. Note that
for both of these indexes, the assumption of constant returns to scale is made. Using the
Malmquist index, these authors were able to decompose the changes of TFP over time into
‘technical change’ (in which they interpreted solely as changes in technology) and changes
in technical efficiency. An important observation made in the study of F¨are et al. (1994)
was that despite using a similar data set, the results from the Tornqvist (consistent with the
traditional Growth Accounting approach) and the Malmquist index were substantially dif-
ferent. The authors suggested two possible reasons for this, one could be the possibility that
the prices did not actually represent the marginal products, leading to allocative inefficiency
of firms. The other is was presence of technical inefficiency. Note that the Tornqvist index
allows for neither, and hence the suggested discrepancies.
More recently, there have been developments in parametric approaches to estimating frontier
functions (for example, see Kumbhakar et al. (2000), Coelli et al. (2008), Kumbhakar et al.
(2015), O’Donnell (2015) ). Such approaches will be discussed in the next section when allow-
ing for the presence of noise, as these studies allow for noise as well as technical inefficiency.
Furthermore, the details of these estimation methods will be outlined in the next chapter,
and will be used in the analysis done in Chapter 4.
While the mentioned applications have focused on the decomposition of TFP changes over
time, the use of linear programming to decompose a labour productivity index was introduced
much later on. Kumar and Russell for example, decomposed the labour productivity index
into changes in technology (‘technical change’), technological catch up (technical efficiency)
44. CHAPTER 2. ECONOMIC FRAMEWORK 33
and capital deepening. This paper motivated several similar studies of labour productivity
growth and convergence in the regions of Italy by Piacentino and Vassallo (2011) , Spanish
regions by Salinas-Jim´enez (2003) and also for a panel of European countries by Filippetti
and Peyrache (2012). While the assumptions made in the paper done by Mizobuchi (2014)
assumed implicitly that all firms were efficient, the author did acknowledge that if their as-
sumptions were relaxed, the effect of technical inefficiency would also have to be included in
the decomposition of labour productivity growth.
Now that we have discussed the concept of technical efficiency and its applications in the
literature, we will incorporate it into our framework. In the previous section, as full technical
efficiency in production is assumed, equation (2.15) holds with equality. This is because
the given level of output was always maximized for a given input vector. Now that we are
allowing for inefficiency, an inequality is introduced into the equation, as it allows for the
possibility of firms not producing at the maximum feasible output. This is shown below:
Qit ≤ A(t, zit)F(Kit, Lit, Mit) (2.19)
or equivalently, this expression can be written as :
Qit = A(t, zit)F(Kit, Lit, Mit) exp(−uit) (2.20)
where uit = ln A(t, zit)F(Kit, Lit, Mit) − ln Qit ≥ 0 is a measure of technical inefficiency.
This term uit will pick up the slack from firms not producing on the frontier. Since uit ≥ 0,
output oriented technical efficiency for country i in period t is OTEit = exp(−uit) ∈ (1, 0].
At full technical efficiency, uit = 0 and therefore, OTEit = exp(−uit) = 1. While as uit
increases, OTE approaches zero.
Following the introduction of these new components, the labour productivity decomposi-
45. CHAPTER 2. ECONOMIC FRAMEWORK 34
tion becomes :
LPit ≡ A(t, zit)Lr−1
it F
Kit
Lit
,
Mit
Lit
, 1 exp(−uit) (2.21)
As shown in Figure 2.2 and 2.3, an increase in inefficiency will lead to a decrease in labour
productivity, consistent with equation (2.21). Therefore, not acknowledging technical effi-
ciency as a driver of labour productivity, as the Solow model does, will attribute changes of
technical efficiency to ‘technical change’, causing a bias in the measure(see Hulten (2000)).
However, in practice, these issues can be adequately addressed by including a component
that can account for and measure technical efficiency in the production process. By doing
so, we can track its changes over time and identify its effect on labour productivity change.
Estimates of technical efficiency provide useful information for policy makers and managers.
For example, government policies can help raise labour productivity by identifying and im-
proving sectors associated with low technical efficiency levels. If the cause of TE is based
on the failure of workers to follow instructions, education programs and workshops can be
arranged to fix such an issue. On the other hand, it is possible that certain laws or forms of
’red tape’ may prohibit firms from effectively using particular technologies. In this regard,
its effect on labour productivity should be considered when discussing the trade-offs associ-
ated with any policy reforms. Therefore, accommodating and calculating technical efficiency
also has policy implications if the aim for government or managers is to improve labour
productivity performance.
2.6 Statistical Noise
The final component is an allowance for measurement and functional form errors (what we
also refer to as ‘noise’ or ‘statistical noise’). While Solow (1957) used a Cobb-Douglas pro-
duction function in his analysis of US productivity, the nature of empirical work will always
lead to errors in the functional form of the production function. It is common to see different
production function models used in the literature, such as Translog production functions
46. CHAPTER 2. ECONOMIC FRAMEWORK 35
(Kumbhakar et al. 2000) and various others restricted or extended forms. Moreover, the
production functional form may change from case to case, depending on the structure of the
production process. Such differences may also depend on the nature of the industry being
analysed. We argue that an allowance for noise may capture any differences between the
actual production process and the production function being used to estimate it.
Furthermore, there are other factors not being taken into account that may be affecting
output. An example would be measuring labour quality through the innate ability of indi-
viduals. Without an appropriate proxy, which is hard to come by, the unobserved component
will be absorbed into such an error term. In the absence of allowing for noise, this will end
up as a factor contributing to the other drivers we have established, in particular to the effi-
ciency component. This is due to the estimation procedures used to estimate such changes,
which will be discussed in the next chapter.
Aigner et al. (1977) provide a prime example of the role of noise in the analysis of production.
This seminal paper states the hypothetical case of a farmer being considered unlucky when a
storm affects production, as it will be controlled for by the error term. However if the error
term is not included, it may be considered inefficient. This example is in particular relevant
to the issues covered in this paper. In the previous section, we acknowledged the effects of
environmental factors mentioned in the example such as droughts or storms. In the case that
these variables can be observed and measured, such distinctions should not have to be made.
The scenario nonetheless highlights the fact that without controlling for unobserved effects
or measurement errors, the estimates of technical efficiency may be biased. This will in turn
lead to an inaccurate representation of the labour productivity drivers.
It is not hard to argue that in practice, there will be certain environmental factors that
are omitted. In that case, the vector of environmental variables included in the model is
zt ∈ RJ
, where J ≤ J∗
. This is also a form of an omitted variable, and if not accounted for
47. CHAPTER 2. ECONOMIC FRAMEWORK 36
(with noise), it may lead to biased measures of the other drivers in the model, as highlighted
by the example in the previous paragraph.
Lastly, as mentioned, the use of index numbers cannot be avoided when aggregating data.
Whether it be on an economy, industry or regional level, there will always be the use of
index numbers. There are several errors that may arise from aggregation, a lot of these,
outlined by Jorgenson and Griliches (1967), will be discussed in the later section regarding
data. The measure given by an index used to approximate an aggregate’s change over time
may create various errors if certain properties are violated. For example, the Tornqvist index
used commonly in the literature ( O’Mahony and Timmer 2009, Bureau of Labour Statistics
2006 ) does not satisfy the transitivity property (see O’Donnell (2014b)). The transitivity
property implies that if in the period between 1 and 2, the index reports a 20% increase,
and between period 2 and 3 the change is 30%, then the change between period 1 and 3 is
56% ( i.e : 1.2 × 1.3 − 1 = 0.56). Of course, this represents an inaccuracy. There are other
index numbers used in the analysis of productivity that contain such errors. For example,
the Malmquist index violates the assumption of circularity (Pastor and Lovell, 2005) . All of
these errors may also be absorbed in the noise component used.
Until now, the analysis of Growth Accounting has focused on the index number approach
(Solow (1957), Denison (1972), Jorgenson and Griliches (1967)) and later on the non-parametric
methods of DEA ( Aigner and Chu 1968, Fare et al 1994, Kumar and Russell 2002). All
these methods are non-stochastic and in the case of DEA, any measurement errors will be
interpreted as inefficiency (Hailu and Tanaka 2015) . These methods, however, make the
strong assumption that there are no errors present.
The most common model used to allow for the presence of noise in frontier analysis is
Stochastic Frontier Analysis (SFA). According to Park et al. (2015), since the introduction
of stochastic frontier models by Aigner et al. (1977), these models, and their implementation
48. CHAPTER 2. ECONOMIC FRAMEWORK 37
using maximum likelihood methods have been among the most popularly used in productiv-
ity and efficiency analysis. Further theoretical advancements have been made since, including
a well known textbook based on the topic by Kumbhakar and Lovell (2000).
While SFA will be discussed further in the estimation chapter, we still proceed to discuss
some of its applications present in the literature. In their introductory paper, Aigner et al.
(1977) provided two examples of an analysis allowing for noise, applying the stochastic frame-
work to the US primary metals industry and the US agricultural sector. Battese and Coelli
(1988,1992) extended the estimation of production frontiers to panel data to dairy farms in
Australia, and paddy farms in an Indian village, respectively. Sun et al. (2015) for example,
used a semi-parametric frontier approach to analyse productivity and technical efficiency,
including a noise component. While Kumbhakar et al. (2015) applied the framework to al-
low for the analysis to returns to scale, efficiency, productivity and noise in the Norwegian
electricity distribution industry between 1998 and 2010. Unfortunately, no such analysis has
been applied to labour productivity.
As mentioned, the noise in our model can come from several sources, one being the func-
tional form assumed in the analysis. To accommodate for this, we introduce the concept
of an approximating function, f(Kit, Lit, Mit), to represent the functional form assumed
by the researcher. In the presence of errors in the functional form, omitted variables,
or measurement errors, the approximating function will deviate from the true functional
form F(Kit, Lit, Mit). To allow for this difference, we introduce an error term vit, where
vit ≡ ln F(Kit, Lit, Mit) − ln f(Kit, Lit, Mit) = ln F(Kit,Lit,Mit)
f(Kit,Lit,Mit)
. By rearranging this ex-
pression to get F(Kit, Lit, Mit) = exp(vit)f(Kit, Lit, Mit), we can substitute it into equation
(2.20) to get equation (2.22). Furthermore, the same is done for labour productivity, shown
in equation (2.23) :
Qit = A(t, zit)f(Kit, Lit, Mit) exp(−uit) exp(vit) (2.22)
49. CHAPTER 2. ECONOMIC FRAMEWORK 38
LPit ≡ A(t, zit)Lr−1
it f(
Kit
Lit
,
Mit
Lit
, 1) exp(−uit) exp(vit) (2.23)
At this point, it is important summarize the implications of applying the standard Solow
output equation (2.2) when the standard assumptions do not hold. Given that A(t) is calcu-
lated a residual, all the components, including environmental effects, a scale effect, technical
efficiency and noise will all be a part of it. After controlling for such effects, the noise com-
ponent which we have introduced will be the residual. However this residual will carry a lot
less of a role in driving and explaining labour productivity growth, as opposed to the ‘Solow
Residual’, given that the main economic drivers (scale effect, TE, environmental variables)
are controlled for. Furthermore, even in the case where other factors are not controlled for,
they will be absorbed by the noise component, and not technical change. Therefore, it will
help reduce biases in the ‘technical change’ component.
This last extension made to the Solow framework incorporates the effect of noise, or in
other words, the various forms of errors and omissions. The presence of errors in a broad
range of forms are ever present in the world of economics. Authors such as Jorgenson and
Griliches (1967) have contributed significantly to the reduction of these errors, however they
may never be able to completely eliminate them. By allowing for functional form errors,
measurement errors and omitted variables, we acknowledge the presence of all these proper-
ties and attempt to control for them such that precision of the other drivers outlined are not
compromised through various biases.
2.7 Complete Framework
This final outcome of the fully extended production framework, is the labour productivity
decomposition below :
LPit ≡ A(t, zit)Lr−1
it f
Kit
Lit
,
Mit
Lit
, 1 exp(−uit) exp(vit)
50. CHAPTER 2. ECONOMIC FRAMEWORK 39
This expression collapses down to that of the Solow model under its original assumptions.
Firstly, the Solow model assumes no errors. Therefore, f(Kit
Lit
, Mit
Lit
, 1) = F(Kit
Lit
, Mit
Lit
, 1) and
vit = 0 or exp(vit) = 1. Next, as the assumptions of profit maximizing firms in a perfectly
competitive market imply that all firms produce at the frontier, the possibility of technical
inefficiency is eliminated. As a result, uit = 0 and exp(−uit) = 1. Lastly, the assumption of
firms exhibiting constant returns implies that the scale elasticity, r, equals to 1. Therefore,
L1−1
it = 1. Accordingly, labour productivity collapses down to that derived from the Solow
Model :
LPit ≡ A(t, zit)f
Kit
Lit
,
Mit
Lit
, 1
Here labour productivity is driven by ‘technical change’ or input deepening. Of course, in
this framework, we argue that shifts caused by Solow’s ‘technical change’ is a result of either
environmental or technological change. Furthermore, if we assume that there are no environ-
mental factors and omit the contribution of intermediate inputs in the production function,
we get back to the standard Solow (1957) equation.
As mentioned, if any of the assumptions made by Solow do not hold, they will all result
in contributions to the ‘technical change’ component of labour productivity growth. This
may have implications for how the researcher interprets this measure in practice. Further-
more, if these components such as scale effects and technical efficiency can be measured and
analysed, there may be opportunities for policy making to improve labour productivity, as
highlighted in the case of technical efficiency and variable returns to scale.
2.8 Labour Productivity Index and its Decomposition
While the labour productivity decomposition outlined in this chapter is based on a theoretical
framework, the remaining part of this thesis focuses on its empirical application. To do so,
we will have to make certain choices and/or assumptions about the components of the model.
51. CHAPTER 2. ECONOMIC FRAMEWORK 40
From here onward, we will be:
1. Choosing the approximating production function, f(Kt, Lt, Mt), to be of the Cobb-
Douglas form (Cobb and Douglas, 1928 ). The implications of this form, as opposed to
a Translog production function (see p. 204-206, Heady and Dillon, 1961 ), is that the
scale elasticity remains constant. While we acknowledge the more flexible properties
of the Translog production function, we will be using the Cobb-Douglas due to its
simplicity providing a more parsimonious labour productivity index decomposition.
2. Approximating A(t, zit) with a linear function. Therefore A(t, zit) = exp(α0 + α1t +
J
j=1 δjzjit)
According to these assumptions, the output equation (2.20) is now expressed as below:
Qit = exp α0 + α1t +
J
j=1
δjzjit ×
M
m=1
xβm
mit × exp(−uit) × exp(vit) (2.24)
Where α0, α1, δj for all, j = 1, . . . J and βm for all M are unknown parameters.
As we wish to identify these labour productivity drivers over time, we want to firstly obtain a
measure of labour productivity according to output equation. We derive this measure in two
steps. Firstly, by multiplying and dividing equation (2.24) by M
m=2 xβm
1it , where x1it = Lit,
we get :
Qit = exp α0 + α1t +
J
j=1
δjzjit ×
M
m=2
xβm
mit ×
1
M
m=2 xβm
1it
× xβ1
1it × (
M
m=2
xβm
1it ) × exp(−uit)
× exp(vit)
Qit = exp α0 + α1t +
J
j=1
δjzjit ×
M
m=2
xmit
x1it
βm
× (x1it)
M
m=1 βm
× exp(−uit) × exp(vit)
(2.25)
Following our definition of labour productivity, we divide output by x1it. Furthermore, by
using the property that M
m=1 βm = r, we get the labour productivity measure for country i
52. CHAPTER 2. ECONOMIC FRAMEWORK 41
in period t, shown below:
LPit = exp α0 + α1t +
J
j=1
δjzjit ×
M
m=2
xmit
x1it
βm
× xr−1
1it × exp(−uit) exp(vit) (2.26)
To make comparisons labour productivity over time and across countries we will use an index,
we use the index that compares the labour productivity of a country i in period t with that
of country k in period s, defined as LPIksit = LPit
LPks
. Of course, in the case of a single country
being analysed, the country descriptors i and k are omitted. This labour productivity index
can be expressed described as:
LPIksit =
exp(α1t)
exp(α1s)
×
J
j=1
exp(zjit)
exp(zjks)
δj
×
M
m=2
xmit/x1it
xmks/x1ks
βm
×
x1it
x1ks
(r−1)
×
exp(−uit)
exp(−uks)
×
exp(vit)
exp(vks)
(2.27)
From equation (2.27), we identify six main drivers of labour productivity. These are ‘tech-
nical change‘, environmental change, input deepening, changes in the scale effect, changes in
technical efficiency and changes in noise.
Equation (2.27) gives a decomposition of the labour productivity index according to the
framework constructed, where the first term represents a technology change index (TI);
the second term is an environmental change index (EI); the third term and fourth term,
representing input deepening and the input scale effect, can combined to obtain an out-
put scale-mix efficiency index (OSMEI) (O’Donnell 2015). The two last terms represent an
output-orientated technical efficiency index (OTEI) and a statistical noise index (SNI).
The scale-mix efficiency component gives a measure of the extent to which firms are capturing
economies of scale and scope. Countries may be expanding the scale of output or adjusting
their input mix, which may improve or worsen such a measure. Frisch (1964) identified the
53. CHAPTER 2. ECONOMIC FRAMEWORK 42
input bundle where the scale elasticity equals to one as the ’optimal scale’ for a given in-
put mix. Therefore, any input bundle that deviates from this ’optimal scale’ bundle will be
deemed inferior or inefficient. Output scale efficiency on the other hand, is the ratio of the
given level of aggregated output to the maximum level of output that can be produced with
the same input mix. Output scale-mix efficiency (OSME) combines these two measures.
Equation (2.27) can therefore be expressed as
LPI = TI × EI × OSMEI × OTEI × SNI
The decomposition above can be implemented through SFA, which will be discussed in the
next section. However, the following shows out this labour productivity index is effected when
we make different assumptions. If we neglect noise in the framework (as all other studies
relating to labour productivity does), then there will be no changes in noise and therefore
SNI = 1. This is line with the DFA assumptions, and therefore :
LPI = TI × EI × OSMEI × OTEI
If we apply a non-frontier approach and do not take into account technical inefficiency, then
the index representing technical efficiency change, OTEI will also disappear, therefore :
LPI = TI × EI × OSMEI
Furthermore, according to O’Donnell (2014b), OSMEI can be broken down into an output
scale efficiency component and a residual mix efficiency component. In the case of constant
returns to scale, the scale elasticity equates to 1. As discussed earlier, this implies full
scale efficiency (Ray 1999). What is left is the residual mix efficiency component that is
also interpreted as input deepening (O’Donnell 2015). This change in input deepening is
measured by an input deepening index (IDI). Also, the Solow model does not take into
account environmental factors and as a result, EI=1. Therefore:
54. CHAPTER 2. ECONOMIC FRAMEWORK 43
LPI = TI
Technical Change
× IDI
Input Deepening
It is important to once again reiterate that according to the basic Solow model, all these
factors that drive productivity change when the standard assumptions do not hold will be
captured by the residual, ‘technical change’. Later on in Chapter 4, we will be comparing
the labour productivity index decomposition based on Growth Accounting to the framework
produced in this chapter in order to highlight these effects.
Throughout this chapter, we have progressively built up a framework that relaxes the strong
assumptions made by the Solow model. We have then made assumptions on the resulting
output equation in order to construct and index representing labour productivity change. In
the remaining of this thesis, we will be discussing the ways in which such a framework can be
applied to data. However, before moving on to the data analysis, the next chapter will be out-
lining the techniques needed to obtain the unknown parameters in equation (2.27) required
to decompose the labour productivity index. United States (US) (Total Factor Productivity
(TFP)) (Non Decreasing (ND)) (Homogeneous of Degree 1 (HD1)) (Labour Productivity
(LP)) Bureau of Labour Statistics (BLS) (Labour Productivity Index (LPI) (Information &
Communications Technology (ICT)) Organisation for Economic Co-operation and Develop-
ment (OECD) Capial Labour Energy Materials & Services (KLEMS) Technical Efficiency
(TE)) (Data Envelopment Analysis (DEA)) (Corrected Ordinary Least Squares (COLS))
(Maximum Likelihood Estimation (MLE)) (Deterministic Frontier Analysis (DFA))(Ordinary
Least Squares (OLS)) (independently and identically distributed (iid)) (Statistical Noise In-
dex (SNI)) (Stochastic Frontier Analysis (SFA)) (Generalised Method of Moments (GMM))
(Geometric Young (GY)) (United Kingdom (UK)) (Level of Significance (L.O.S)) (GMM)
(GY) (UK) (L.O.S) (SNI) (SFA) (GMM) (GY) (UK) (L.O.S) (GMM) (GY) (UK) (L.O.S)
(Output Oriented Technical Efficiency (OTE)) (Non Negative (NN)) (Environmental Index
(EI)) (Technological Change Index (TI)) (Input Deepening Index (IDI)) (Output Oriented
Technical Efficiency Index (OTEI))
55. Chapter 3
Estimation Strategies
The previous chapter concluded with an output equation that can be used to estimate the
unknown parameters needed to decompose our labour productivity into various components.
While the input and output measures used are observable, the rest of the components, in-
cluding the output elasticities, the technical efficiency component as well as the noise com-
ponent are not. As a result, we identify these components by estimating them. Throughout
this chapter, we will be outlining the techniques used to estimate these parameters. These
techniques include Corrected Ordinary Least Squares (COLS) and Maximum Likelihood Es-
timation (MLE). Beforehand, the traditional Growth Accounting approach will briefly be
discussed.
The stochastic frontier model we will be using is shown in equation (3.1). This equation
is obtained by taking the logarithm of equation (2.24) outlined in the previous chapter.
ln(Qit) = α0 + α1t +
J
j=1
δjzjit +
M
m=1
βm ln(xmit) − uit + vit (3.1)
This model, as previously discussed, accounts for environmental factors, the presence of
technical inefficiency and statistical noise. Now that we have introduced the general form of
the equation, all methods discussed will be based on output equation (3.1), restricting it in
various ways depending on the assumptions made.
44
56. CHAPTER 3. ESTIMATION STRATEGIES 45
3.1 Growth Accounting Approach
The Solow model is a Non-Parametric (deterministic) and Frontier based approach, accord-
ing to Del Gatto et al. (2011). The deterministic approach implies that no econometric
estimation is required. Instead, only a few values are needed to decompose growth in labour
productivity into ‘technical change’ and input deepening. The required data are the individ-
ual quantity measures of output and inputs, as well as the respective income shares of the
inputs.
As was shown in Chapter 2, due to the assumptions of the Solow model, the factors of
production can be weighted according to their respective income shares. As a result, refer-
ring to the general equation (3.1), for every input xm, βm = ¯sm, where ¯sm is the average
income share for input xm. Furthermore, as there is no technical inefficiency present in the
Solow model, uit = −ln(1) = 0, and as there is no noise, vit = 0. This leaves us with equation
(3.2) :
ln(LPit) = ln(A(t, zit)) +
M
m=2
ln
xmit
x1it
¯sm
(3.2)
where ‘technical change’, A(t, zit) = exp(α0 + αtt + J
j=1 δjzjit). Note that we have included
the environmental variables in the ‘technical change’ component. This is consistent with the
Solow (1957) framework, as he described ‘technical change’ as anything that causes a shift
in the production function. This component is calculated as a residual. Therefore:
A(t, zit) = exp ln(LPit) −
M
m=2
ln
xmit
x1it
¯sm
(3.3)
As discussed, according to this methodology, if these assumptions do not hold, the component
A(t, zit) will be biased and it will not represent solely advances in technology, but also other
components. For example, if there is noise present (which is very likely), such errors will be
absorbed by ‘technical change’. Therefore, in that case, A(t, zit) = A(t, zjit, vit), or :
57. CHAPTER 3. ESTIMATION STRATEGIES 46
ln A(t, zjit, vit) = α1t +
J
j=1
δjzjit + vit
Furthermore, in the case where there is technical inefficiency present, but not accounted for,
‘technical change’ will also represent such drivers. Thus, A(t, zjit) = A(t, zjit, vit, uit), or :
ln A(t, zjit, vit, uit) = α1t +
J
j=1
δjzjit − uit + vit
Furthermore, if the assumption of CRS is violated, there will be scale effects present. These
will also be captured by the ‘residual’. Therefore, A(t, zjit) = A(t, zjit, vit, uit, r).
This is a particular feature and shortcoming of Growth Accounting. Solow and the sub-
sequent studies that followed assumed that ‘technical change’ caused purely shifts in the
production function. However, according to our model, the only factor besides technological
progress that causes shifts in the production function are environmental changes. If the main
assumptions do not hold, then factors such as the scale effect will be captured by ‘technical
change’. As features such as scale are associated with movements along the frontier, and not
shifts, it follows that under the failure of these assumptions, the Solow model simply does
not work accordingly. Other factors such as noise and technical efficiency changes also are
not directly associated with shifts in the production function, but they may still be captured
and drive ‘technical change’.
In the next chapter, we will be conducting tests based on the assumptions of the Solow
model. After doing so, with the help of the model proposed in Chapter 2, we show how
these effects influence ‘technical change’ if these assumptions do not hold. However, before
we move onto such an analysis, the methodologies used to estimate the parameters required
to decompose the labour productivity index outlined in the previous section will be outlined.
58. CHAPTER 3. ESTIMATION STRATEGIES 47
3.2 Corrected Ordinary Least Squares
The previous section outlined the deterministic approach of the traditional Growth Account-
ing methodology. This procedure was in line with the Solow model, and as a result did not
include any of the extensions made in Chapter 2. We will now move on to an econometric
approach that allows for the estimation of the frontier, which provides us with a benchmark
for technical efficiency. Furthermore, the parameters estimated from the production function
provide the output elasticities of the input parameters which sum up to give an estimate
of the scale elasticity, and therefore, can provide a statistical test for the presence of con-
stant returns to scale. These econometric methods have long been suggested for such use.
Jorgenson and Griliches (1967) for example, emphasized the use of econometrics as a good
“supplementary tool” for the use of Growth Accounting.
We will begin modeling our framework using Deterministic Frontier Analysis. Note that
the following assumptions are made:
DFA1: inputs, environmental variables and outputs are all observed without any errors
DFA2: no errors in the assumed functional form.
While this assumption is unrealistic, it allows us with a base case scenario to measure the
technical efficiency of a production function through uit.
While we do not have any measured environmental effects that can be controlled for, we
make the assumption that there are unobserved, time-invariant effects present in each coun-
try that we can identify with through fixed effects. As a result, the environmental effects
considered in our estimation from here onwards are time-invariant. We will estimate these
effects by including a dummy variable for all countries excluding 1 in order to avoid the
‘dummy variable trap’.
Following assumptions DFA1 - DFA2, if we refer to the model outlined in equation (2.22)
