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On the productive efficiency of Australian businesses: firm size and age class effects

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Abstract

After 26 years of growth, the Australian economy is beginning to show signs of stress and declining productivity. In this paper, we consider aspects of productive efficiency using an Australian business population data set. Using a production function approach, several key findings are uncovered. Firstly, decreasing returns to scale are identified as a significant feature of the Australian business sector. This implies that not all firm growth will lead to productivity gains. Secondly, there are significant differences in the way value added is created between small and large firms. In the largest 25% of firms, the capital contribution to value added is four times that of the smallest 25% of firms. Thirdly, efficiency follows an inverted ‘U’ shaped in firm age with the youngest (0–2 years) and oldest (> 9 years) firms being less productive than the middle 50% of firms. Fourthly, there are also huge industry sector variations in productivity. In particular, financial services appears to be the most productively efficient sector in the Australian economy and mining the least efficient.

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Correspondence to Marc Cowling.

Appendix

Appendix

The CES is an alternative specification of the production function which does not have this property relating to the marginal rate of substitution. Having an underlying non-linear form, the CES can be expressed as:

$$ {Y}_i=\gamma {\left[\delta {K_i}^{-\rho }+\left(1-\delta \right){L_i}^{-\rho}\right]}^{-\nu /\rho }{e}^{\varepsilon i}, $$

Here, γ > 0, 1 > δ > 0, ν > 0, and ρ > − 1. These four parameters of the function, γ, δ, ν, and ρ, are taken to represent efficiency, distributional shares, returns to scale, and factor substitution. In our modelling, it is fundamental in our choice of production function whether or not the ρ parameter, relating to factor substitution, is statistically not different from zero, which would imply that there is an elasticity of substitution of 1 (from 1 / 1 − ρ). If this was found to be the case, then the CES function collapses to the Cobb-Douglas form.

The CES model can be written thus, log Yi = log α – ν / ρ log [δ Capitali−ρ + (1 − δ) Labouri−ρ+ ei. Testing for the most appropriate functional form, Cobb-Douglas or CES, is important before we begin to develop our full modelling strategy. The focus of interest is on the ρ parameter. If ρ is statistically not different from zero, then this would support rejection of CES in favour of the Cobb-Douglas functional form as it implies an elasticity of substitution of 1. If we consider a Taylor series expansion of the CES function (i.e. expand log Y around ρ = 0), then we can generate an equation that can be estimated thus:

$$ \log \kern0.5em {Y}_i=\log \kern0.5em \gamma +\nu \delta \kern0.5em \log \kern0.5em {K}_i+\nu \left(1-\delta \right)\log \kern0.5em {L}_i\hbox{--} 1/2\rho \nu \delta \left(1-\nu \right){\left[\log \kern0.5em {K}_i\hbox{--} \log \kern0.5em {L}_i\right]}^2+{\varepsilon}_i $$

Here, the first element is the Cobb-Douglas model and the second element allow for ρ to have a non-zero value. Again, if ρ = 0, then Cobb-Douglas is favoured over CES. We can derive an unrestricted linear form:

$$ \log \kern0.5em {Y}_i={\beta}_1+{\beta}_2\log \kern0.5em {K}_i+{\beta}_3{L}_i+{\beta}_4{\left[\log \kern0.5em {K}_i\hbox{--} \log \kern0.5em {L}_i\right]}^2+{\varepsilon}_i $$

The relevant parameter here is the estimate of β4 which determines whether we can reject the CES specification. Here, if β4 is statistically zero, then Cobb-Douglas is the most appropriate functional form. The first finding from the unrestricted CES estimate is that our key parameter of interest, β4, is not statistically significant (Test ([log K – log L2] = 0, F = 0.13, Pr > F0.723). The estimating equation can be used to derive the parameters of Eq. (5) as follows:

$$ \delta =\upbeta 2/\upbeta 2+\upbeta 3=0.095/0.921=0.103 $$

The δ corresponds to the distributional shares between capital and labour; hence, the relative shares are 0.103 for capital and 0.897 for labour.

$$ \nu =\upbeta 2+\upbeta 3=0.921 $$

As ν is significantly less than 1, this implies decreasing returns to scale (Test ln K + ln L = 1, F = 8.49, Prob > F = 0.0039)

$$ \rho =-2{\beta}_4\ \left({\beta}_2+{\beta}_3\right)/\upbeta 2+\upbeta 3=-2\times 0(0.921)/0.078=0 $$

The ρ is the substitution parameter, and the marginal rate of substitution is calculated to be 1 using 1 / 1 − ρ. Even if we allow for it to be non-zero, then we get an estimated MRS of 1 / (1 − 0.006) = 0.994 (Table 6).

Table 6 Cobb-Douglas and CES estimates for value added

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Cowling, M., Tanewski, G. On the productive efficiency of Australian businesses: firm size and age class effects. Small Bus Econ 53, 739–752 (2019). https://doi.org/10.1007/s11187-018-0070-0

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