Origins of productivity measurement can be traced back to seminal papers of Tinbergen (1942), Fabricant (1954), Abramovitz (1956) and Solow (1957) who have decomposed output growth to input growth and productivity residuals. These early studies have mainly focused on aggregate TFP and its role in economic growth. Many textbooks have explained various productivity functions and forms (Besanko et al., 2015). To extend our knowledge of the recent techniques within the productivity estimation, the following section focuses entirely on productivity and how it is estimated in the recent literature. It does not aim to look at established estimators; rather the focus is on the pioneering approaches applied within the field so that either an appropriate extension to the existing method could be chosen or an alternative proposed.
Several articles (Van Biesebroeck, 2007; Eberhardt and Helmers, 2010; Del Gatto et al., 2011; van Beveren, 2012a) are used as a starting point. They provide an overview of the methodological issues when estimating TFP at an establishment level. To look at the most recent techniques applied in productivity estimation, this part of the literature review focuses on publications in the Journal of Productivity Analysis (JPA). This journal publishes theoretical and applied research addressing the measurement, analysis, and improvement of productivity. Given that the aim is to look for pioneering studies departing from the established disciplines, the journal seems to be an appropriate choice. To further our understanding, all articles from this journal published in a period between 2014 and 2017 were coded with NVivo. This analysis is used to estimate statistics and look for forward-looking methodologies. The findings from this analysis are combined with studies from other journals and are also presented in Section 5.1.4.
5.1.1 General Trends in the Journal of Productivity Analysis
First and foremost, an overview of findings is provided to inform about the underlying trends in the JPA. A variation of the parametric Stochastic Frontier approach is the most popular in the Journal of Productivity Analysis. Several empirical and many theoretical papers have discussed many estimation issues like heterogeneity, measurement error in capital and endogeneity, which are detailed in the Research Design Chapter (esp.
Section 6.3.1.1). Also, it is evident that authors are choosing techniques according to the assumptions of technology and technical change. All papers have been looking at diverse areas. Thus, their results are not directly applicable to this analysis, rather they indicate the best practice and recent techniques used to estimate and define productivity.
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The most popular industry being investigated is manufacturing, with 54 published articles during 2014-2017. Other popular industries are agriculture, finance, education and hospitality. Although productivity analysis stays popular in more output-input manufacturing firms, it seems that a large number of authors are considering the service sector. Furthermore, productivity studies come from emerging, developing and developed countries. The vast majority of studies are from European countries; however, the US, Japan and China also remain popular. There were only very few UK studies published in this journal.
5.1.2 Functional Form
To estimate TFP, first and foremost a functional form should be defined. The Cobb-Douglas functional form has been the most popular among the researchers in the Journal of Productivity Estimation. The 94 articles from this literature review explicitly stated that they were using Cobb-Douglas functional form, while most others were using a variation of Cobb-Douglas functional form, such as the constant elasticity of substitution (CES) which is sometimes used to access the elasticity of substitution. For instance, US researchers (Akay and Dogan, 2013) estimate the elasticity of substitution by using CES production function (as described in Appendix 10.3.1.5) with one mobile factor (labour) and 25 industries of the US to suggest how these estimates describe the general equilibrium of production.
On the other hand, the nested production functions could also be applied to question a relationship between some factors and the critical productivity elements (capital and labour) which may experience different substitution effects. For example, Shankar and Quiggin (2013) use the stochastic frontier approach with a CES specification of technology. Various functional forms are defined in Appendix 10.3.1. Given the nature of the data, namely micro level longitudinal data25, only methods that can deal with microdata will be discussed in order to find an appropriate approach for this thesis.
Usually, an establishment-level production function is a mathematical expression that describes a systematic relationship between inputs and output in an economy (Katayama et al., 2009). The fundamental mechanisms in standard theories based on neoclassical school of thought are discussed in the Theory Review Chapter (Section 2.2.1).
In hand with these models, Del Gatto et al. (2010) start by quantifying productivity:
𝑌𝑖𝑡 = 𝐴𝑖𝑡∗ 𝐹(𝑍𝑖𝑡) → 𝑇𝐹𝑃 = 𝐴𝑖𝑡 = 𝑌𝑖𝑡 𝐹(𝑍𝑖𝑡),
25 For further detail, see the Research Design Chapter, Section 6.1.3.
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where Y is the output of a unit (country/industry/firm) i at time t to a (1 × N) vector of inputs Z and the term 𝐴 defines how much output a unit can produce from a certain volume of inputs, given the technological level. The state of technology, embodied by the function F(.), is given and common to all enterprises.
Thus, the TFP index at time t is the ratio of produced output and total inputs employed.
5.1.2.1 Cobb-Douglas
The most commonly applied is the Cobb-Douglas productivity function named after C. W. Cobb and P. H. Douglas (1934). The mathematical form of the Cobb-Douglas production function can be given by
𝑌𝑖𝑡(𝐿, 𝐾) = A𝐾𝛽𝐾𝐿𝛽𝐿 → 𝑌𝑖𝑡(𝑡𝐿, 𝑡𝐾) = A(t𝐾)𝛽𝐾(𝑡𝐿)𝛽𝐿 = 𝑡𝛽𝐾+𝛽𝐿𝑌𝑖𝑡(𝐿, 𝐾)
This is a preferable choice because Cobb-Douglas productivity function can exhibit any degree of returns to scale: constant (𝛽𝐾+𝛽𝐿=1), increasing (𝛽𝐾+𝛽𝐿>1) and decreasing (𝛽𝐾+𝛽𝐿<1).
It is quite straightforward to show that the elasticity of substitution is equal to 1:
𝑀𝑅𝑇𝑆 = 𝛽𝐿A𝐾𝛽𝐾𝐿𝛽𝐿−1 𝛽𝐾A𝐾𝛽𝐾−1𝐿𝛽𝐿 = 𝛽𝐿
𝛽𝐾 𝑘
𝑙 → ln(𝑀𝑅𝑇𝑆) = ln (𝛽𝐿
𝛽𝐾) + ln (𝑘 𝑙)
𝑆𝐸 = %∆ ln (𝑘 𝑙)
%∆ ln(𝑀𝑅𝑇𝑆)= 1
This has influenced scholars to apply the constant returns-to-scale version to estimate the aggregate productivity in numerous countries. The constant 𝛽𝐾 is then the elasticity of output with regards to capital input, and 𝛽𝐿 is the elasticity of output with regards to labour input.
In the case of multiple inputs, the function takes the following form:
𝑌 = ∏ 𝑥𝑖𝛽𝑖
𝑛
𝑖=1
If ∑𝑛𝑖=1𝛽𝑖 = 1, then the equation exhibits constant returns to scale. In the constant-returns-to-scale Cobb-Douglas function, β is the elasticity of Y with respect to input x. If 𝛽 is in the range of 0 and 1, each x should exhibit diminishing marginal productivity. However, any degree of increasing returns to scale can now be incorporated.
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With regards to the elasticity of substitution, this could be expressed similarly as with one input (as shown in Appendix 10.3.1) and result in:
𝑆𝐸 =
%∆ ln (𝑥𝑖 𝑥𝑗)
%∆ ln(𝑀𝑅𝑇𝑆)= 1
This constraint is the primary limitation of Cobb-Douglas productivity function.
5.1.2.1.1 Technical Progress in Cobb-Douglas Productivity Function
Let’s assume constant returns to scale and technological progress occurring at a steady exponential, then a production function with technological progress becomes:
𝑌𝑖𝑡(𝐿, 𝐾) = Ae𝜃𝑡𝐾𝛽 𝐿1−𝛽
The technical change feature is explicitly modelled, and the output elasticities are specified with the exponents in the Cobb-Douglas.
The Cobb-Douglas functional form cannot include annual improvements directly. If the annual improvement in capital (e𝜑𝑡) and labour (e𝜌𝑡) is added, the Cobb Douglas function would still convert to its previous form:
𝑌𝑖𝑡(𝐿, 𝐾) = A(e𝜑𝑡𝐾)𝛽 (e𝜌𝑡𝐿)1−𝛽 = Ae[𝛽𝜑+(1−𝛽)𝜌]𝑡𝐾𝛽 𝐿1−𝛽 = Ae𝜃𝑡𝐾𝛽 𝐿1−𝛽 , 𝑤ℎ𝑒𝑟𝑒 𝜃 = 𝛽𝜑 + (1 − 𝛽)𝜌
However, it is questionable whether the annual improvement in capital and labour could be precisely estimated.
5.1.2.1.2 Trans log Form Cobb-Douglas Productivity Function
Taking logarithmic function from Cobb Douglas productivity function and adding varying intercept as well as further condition (𝛽𝑖𝑗𝑙𝑛𝑥𝑖𝑙𝑛𝑥𝑗) produces translog function:
𝑙𝑛𝑌 = 𝛽0+ ∑ 𝛽𝑖𝑙𝑛𝑥𝑖
The trans log production function incorporates many substitution possibilities among various inputs and can take any degree of returns to scale.
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The logarithmic Cobb-Douglas functional form seemed to be an appropriate method to further our understanding since it includes the key inputs and different variations which may help to achieve better estimates. More specifically, the trans log production function seems to be the most appropriate as it can incorporate many substitution possibilities among various inputs and can take any degree of returns to scale.
5.1.3 Estimation Techniques
Once this thesis selected how the productivity function is likely to look, the focus now turns towards the estimation techniques. Figure 5:2 provides a broad classification of the techniques approached in this literature survey. The shape and underlying assumptions of the productivity function influence the choice of estimation techniques. Authors have differing preferences and arguments for using different techniques, implying that none of the estimation techniques are significantly better than others.
Figure 5:2 Classification of the estimation techniques. Based on publications in the JPA between 2014 and 2017.
Broadly, Del Gatto et al. (2011) highlight the main difference between non-frontier and frontier models. The primary reason leading to the implementation of frontier models is their ability to disentangle the main sources of productivity growth, which are technical efficiency and technological change. Technical progress measures the shift of the frontier over time and the change in technical efficiency measures the movement of an economy away from (or towards) the production frontier. Kumbhakar and Lovell (2000) and Kumbhakar et al. (2015) have provided an overview of stochastic frontier models. The
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primary issue with such models is the degree to which variables are included in the deterministic part of the model (to set the frontier) and to what degree these same variables explain inefficiency and, hence, enter a determinant of the one-sided inefficiency term (Battese and Coelli, 1995). These models do not control for endogeneity and selection biases, which are detailed in the Research Design Chapter (Section 6.3.1.1, p. 158).
Van Biesebroeck (2007) publishes one of the most cited articles in productivity estimation after Cobb Douglas. He uses the Monte Carlo approach to investigate the sensitivity of the most widely used methods, which are data envelopment analysis, index numbers, instrumental variables, stochastic frontiers and semi-parametric estimation. He explores three different scenarios concerning measurement error and simultaneity bias, which is described in the Research Design Chapter (Section 6.3.1.1) and examines their credibility with Monte Carlo.
Van Biesebroeck (2007) suggests using index numbers for cases with small measurement error. GMM estimators are useful for estimating productivity levels. DEA is suitable when returns to scale are not constant and technology is heterogeneous.
Parametric approaches are suitable when optimisation or measurement errors are minor.
The authors ranked these techniques by the persistence of the productivity differentials between firms (in decreasing order). This is the order they proposed: the stochastic frontiers, GMM, or semiparametric estimation methods.
With the data proposed for this study (see Chapter 6), it seems counterintuitive to assume that variables could have little or no measurement error because of the survey nature and some variables like capital as it may be defined and estimated in various ways.
Similarly, some technological heterogeneity should be assumed since all firms may not share the same technology. In this scenario, the author suggested using system GMM estimator.
The technological differences are arguably not too diverse in the UK. Technological change and technical efficiency change in Britain varies. However, the frontier approach may be more applicable to several countries where technology variation within companies is more apparent. Arguably, companies within the same sector can possess similar technology. If the technology is immutable, it does not contribute to productivity improvements. The same effects are present with technical inefficiency. If it does not vary over time, it also does not have any effect on the rate of change of productivity.
The review of techniques showed that frontier approaches are becoming less popular, while the usage of non-frontier approaches seems to be increasing. It appears that
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researchers are looking into new ways to employ non-parametric approaches. This literature review did not identify any other paper that used decision trees to explain productivity. Thus, it seems reasonable to complement standard econometric techniques, which are also presented with other empirical examples in the following section, with such machine learning alternatives as Classification and Regression Trees described in Section 5.3. This could not only contribute to the literature on productivity estimation, but also assure the results estimated by one of the econometric techniques.
More specifically, summary statistics presented in Figure 5:3 (p. 104) shows that there were 87 empirical and 35 theoretical papers published between 2014 and 2017 (up to May). Theoretical papers mainly focused on either comparisons or improvements of methodology, which is the reason why they were excluded from further comparisons.
Overall, more than two-thirds of these papers were looking at frontier analysis. However, this data shows that non-frontier approaches are becoming increasingly more popular, while frontier techniques seem to lose their popularity. With regards to underlying assumptions of data distribution, non-parametric approaches were most often used, while the popularity of parametric techniques decreased. Furthermore, semi-parametric studies seem to be increasingly more popular. However, none of the 2017 studies were published using this methodology. It is worth mentioning that this review indicates the most used and appropriate methods, rather than the exact figures since just one journal is included in the analysis.
Figure 5:3 Count of articles published in different years by their approach between
2014 and 2017. Counts for 2017 were partly imputed for comparability. Source
author’s calculation.
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All studies directly related to UK business rates (BRs) fit within the empirical debate on capitalisation, and are therefore discussed in the Empirical Review Chapter. For this reason, in this section, SBRR is assumed to be merely a cash inflow and greater attention is paid to the approach rather than results. As a result of the data and previous comparison, the focus naturally turns towards the econometric approaches. The underlying mechanisms of these estimators were discussed in Appendix 10.2.2. This section provides several examples of Difference in Difference (Section 5.1.4.1), Ordinary Least Squares (Section 5.1.4.2), Fixed Effects (section 5.1.4.3), Ordinary Least Squares with Matching (section 5.1.4.4) and Instrumental Variable (Section 5.1.4.5) estimation techniques.
The focus is on the approaches rather than findings since no studies have been identified to investigate how non-domestic property taxation relates to either survival or productivity. Other findings are too diverse and do not add much value towards the analysis. For instance, Irwin and Klenow (1996) find no impact on labour productivity of R&D subsidies for U.S. high-tech companies. Whilst, for Japanese forestry, Managi (2010) finds a negative relationship between grants and TFP; Einio (2014) reports no immediate impacts of R&D support programmes in Finland on productivity (although there is evidence of long-term gains). Huang (2015) shows that tax credit use among Taiwanese firms enhances their productivity. Koski and Pajarinen (2015) report that R&D subsidies have no statistically significant impact on labour productivity in Finnish firms during 2003-2010.
Given the differing environments of the firms and various issues within the tax setting and estimation, these papers are not reviewed in this section.
5.1.4.1 Difference-in-Difference (DiD) estimator
One of the most fundamental methods to analyse BRs is the Difference-in-Difference estimator. This has been employed in various studies, such as Bronzini and de Blasio (2006) which evaluates the income of Italian investment incentives. This is a standard case described in Appendix 10.2.2. They use simple DiD to compare subsidised firms with the rejected applications. They find subsidies to be positive for the recipients.
The problem with these analyses is the assumption that the differentials in growth for companies that experienced support would have remained the same to non-supported firms’ differentials.
Earlier studies have consisted of simulations with macro data before and after the implementation of policies. For instance, Moore and Rhodes (1974) use the shift-share approach to find the effect of a policy change on investment and employment between 1960 and 1963. They construct a series to simulate what investment and employment would
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have been after 1963 without any policy change. This is done by comparing the overall British industry growth rates to the investment and employment levels of each industry.
This approach assumes a similar limiting assumption to the DiD estimator. The underlying assumption of this study is that growth rates in the development areas should be the same in the other regions. Canning et al. (1987) control for these. They investigate the differences between expected and actual values by exploiting data from before the change in policy.
They claim that regional policy in Northern Ireland created 33,000 new manufacturing jobs.
However, they also unrealistically assume that the expected and actual values would have continued to rise to the same degree, had there been no change. Thus, the self-selection issue, which will be defined in the Research Design Chapter (Section 6.3.1.1.2) is apparent.
Furthermore, estimates are sensitive to the period used to investigate actual and expected values.
5.1.4.2 Ordinary Least Squares (OLS)
To control for these differences in observed characteristics between control and treatment groups, the DiD may be performed with OLS. Bergstroem (2000) analyses the effect on TFP by selective capital subsidies in Sweden. His treatment group consists of firms that were grant recipients in 1989 and the control group is a random sample. Whilst, the growth in output between 1989 and 1990/1993 is a dependent variable in his primary regression. The treatment variable is recorded as the amount of subsidy received in 1989.
They estimate their equation with bounded OLS estimator to minimise the influence of outliers. They show that subsidies increase TFP in the short run, but are associated with reduced TFP in the long-term. However, they do not discuss the correlation between the error term and grant. The former comes from differences in unobserved characteristics across control and treatment groups. Furthermore, the correlation between the growth of factor inputs and error term is ignored and endogeneity of control variables is not accounted for, as it should be according to Froelich (2008) or the econometric issues in Section 4.6 (p. 89).
Additionally, Irwin and Klenow (1996) investigate whether R&D subsidies affect US high-tech companies’ labour productivity. Their sample consists of 71 firms with annual observations between 1970 and 1993. They compare R&D of companies receiving support with unsupported companies. They acknowledge that businesses may have expected R&D intensity and that is why they received the support. To control for the differences between observations, they include age dummies and estimate their regression with OLS and weighted-least-squares (WLS). It is worth noting that they include the lagged dependent variable. As discussed in Section 4.6, neither OLS nor WLS provides an unbiased estimate
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for an equation with a dependent lagged variable. They should include IVs. They find no effect on productivity.
Various simulations with US macro data indicates that capital subsidies are likely to cause lower consumption and lower employment. For instance, Lee (1996) uses multivariate regressions to investigate the effect of industrial policy on gross value added, capital growth and TFP with the data from South Korea. He allows for time and fixed effects.
His model is estimated with three-stage-least-squares in order to control for the endogeneity of the treatment variables. Furthermore, weighted least squares are also employed to correct for cross-equation heteroscedasticity in this model. Nevertheless, as Bond (2002) shows, if the error term is autoregressive, the once lagged policy variables are not valid instruments since they are likely to be correlated with the error term. Similarly, Harris (1991) uses simulation to look at the effect of capital subsidies on employment within the manufacturing sector in Northern Ireland. This model includes production, factor demand and industry demand equations. The maximum likelihood is used to estimate the parameters. He uses these parameters to generate estimates of output, labour and capital and finds that although production has increased, employment has decreased
His model is estimated with three-stage-least-squares in order to control for the endogeneity of the treatment variables. Furthermore, weighted least squares are also employed to correct for cross-equation heteroscedasticity in this model. Nevertheless, as Bond (2002) shows, if the error term is autoregressive, the once lagged policy variables are not valid instruments since they are likely to be correlated with the error term. Similarly, Harris (1991) uses simulation to look at the effect of capital subsidies on employment within the manufacturing sector in Northern Ireland. This model includes production, factor demand and industry demand equations. The maximum likelihood is used to estimate the parameters. He uses these parameters to generate estimates of output, labour and capital and finds that although production has increased, employment has decreased