Estimating input-mix efficiency in a parametric framework: application to state-level agricultural data for the United States

ABSTRACT

This paper contributes to the productivity literature by demonstrating novel econometric methods to estimate input-mix efficiency (IME) in a parametric framework. Input-mix efficiency is defined as the potential improvement in productivity with change in input mix. Any change in input-mix (e.g., land to labou r ratio) will result in change in productivity. The advantage of this approach is that it does not require data on input prices to estimate the mix efficiency levels. A nonlinear input-aggregator function (e.g., Constant Elasticity of Substitution) is used to derive an expression for input-mix efficiency. Bayesian stochastic frontier is estimated for obtaining mix efficiency using US state-level agricultural data for the period 1960–2004. Significant variation in input-mix efficiency is noted across the states and regions, attributable to diverse topographic and geographic conditions. Furthermore, comparisons of allocative and mix efficiencies provide insightful policy implications. The production incentives such as taxes and subsidies could help farmers in adjusting their input mix in response to changes in input prices, which can affect the US agricultural productivity significantly. The proposed methodology can be extended by i) using flexible functional forms; ii) introducing various time- and region-varying input aggregators; and iii) defining more sophisticated weights for input aggregators.

KEYWORDS:

• Mix efficiency
• Bayesian stochastic frontier
• productivity
• aggregator function

JEL CLASSIFICATION:

• D21
• D24
• C43

Acknowledgments

I am grateful to Knox Lovell, Chrsitopher O’Donnell, and George Battese for their valuable comments on earlier drafts of this paper. I am thankful to Janet Hohnen for her editorial help in fian draft of the paper. The earlier versions of this paper have been presented in seminar series at UNE Business School, University of New England, Armidale, Australia and School of Economics, The University of Queensland, Brisbane, Australia and I would also like to thank participants of both seminars for providing their feedback on the paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Data availability statement

The data described in this article are openly available in the Open Science Framework at DOI:10.17605/OSF.IO/TPA6U.

Notes

¹ If $f(x)$ is not everywhere differentiable, then [$f_i(x)/f_j(x)$]⁺ ≤ $w_i/w_j$ ≤ [$f_i(x)/f_j(x){]}^{-}\mathrm{\forall }i,j$, as would be the case if technology were modelled with data envelopment analysis.

² ${\bar{X}}_t=X_t/D_I(q_t,x_t).$

³ ${\hat{x}}_t=\underset{x>0}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}\left\{X(x_t):(q_t,x_t)\in T_t\right\}.$

⁴ Properties of input and output distance functions are given in footnote 5 of O’Donnell and Nguyen (2013).

⁵ Studies using the same data set used the average efficiency equal to 0.90 as an informative prior. For instance, O’Donnell (2012a) assumes ${\zeta }^{\ast }=0.90$ using in his study for the same data set. However, different values for the prior (ranging from 0.8 to 0.95) were tried, but there was no variation was noticed in technical efficiency scores with these different values of priors.

⁶ More details on the construction of the output and input variables can be found in Ball, Hallahan, and Nehring (2004). These details are also available on ERS-USDA website: http://www.ers.usda.gov/data.

⁷ All inputs are adjusted for quality using hedonic prices. A detailed methodology for the quality adjustment of inputs can be found in Kellogg et al. (2002).

⁸ Output quantity indexes have been constructed using the methods of Elteto and Koves (1964) and Szulc (1964), known as the EKS indexes (see Ball, Hallahan, and Nehring 2004).

⁹ In Bayesian set up, the hypothesis test is conducted by the posterior odd ratio as $\frac{P(M_1\left|Y)\right.}{P(M_2\left|Y)\right.}=\frac{P(M_1)}{P(M_2)}\frac{L(Y\left|M_1)\right.}{L(Y\left|M_2)\right.}.$ Where $\frac{P(M_1)}{P(M_2)}$is the odd ratio of priors and $\frac{L(Y\left|M_1)\right.}{L(Y\left|M_2)\right.}$is the likelihood ratio of two models under consideration. Since, priors for the two models are same, the Bayes factor is $B{F}_{12}=\frac{L(Y\left|M_1)\right.}{L(Y\left|M_2)\right.}=\frac{265.46}{40.40}=\mathrm{6.57.}$

¹⁰ It is noted that materials input variable has shown the increasing trend throughout the study period. To obtain, a parisomonious several specifications have been chosen and interaction of time trend with material input (i.e., $x_{4t}\times t$) gives a parsimonious model.

¹¹ However, the geometric mean of technical change (computed by using the expression, $\mathrm{\Delta }\mathrm{T}\mathrm{e}\mathrm{c}\mathrm{h}=\mathrm{e}\mathrm{x}\mathrm{p}\left[\left({\alpha }_t+{\beta }_{4t}ln{x}_{4t}\right)\left(t-1\right)\right]$), shows an annual increase of 1.73%.

¹² I am grateful to Eldon Ball and Knox Lovell for providing me a series of input prices which have been used to compute allocative efficiencies in the US agricultural sector.

¹³ $\mathrm{\Delta }IM{E}_{AL}^{2004}/\mathrm{\Delta }IM{E}_{FL}^{2004}=(IM{E}_{AL}^{2004}/IM{E}_{AL}^{1960})/(IM{E}_{FL}^{2004}/IM{E}_{AL}^{1960})=1.225/1.077=1.172$.

¹⁴ $\mathrm{\Delta }IM{E}_{FL}^{2004}/\mathrm{\Delta }IM{E}_{WY}^{2004}=(IM{E}_{FL}^{2004}/IM{E}_{AL}^{1960})/(IM{E}_{WY}^{2004}/IM{E}_{AL}^{1960})=(1.071/0.683)=1.568$.

¹⁵ $\mathrm{\Delta }IM{E}_{AL}^{2004}/\mathrm{\Delta }IM{E}_{WY}^{2004}=(\mathrm{\Delta }IM{E}_{AL}/\mathrm{\Delta }IM{E}_{FL})\times (\mathrm{\Delta }IM{E}_F/\mathrm{\Delta }IM{E}_{WY})=1.178\times 1.568=1.837$.

¹⁶ National Agricultural Statistics Service United States Department of Agriculture (NASS-USDA) also publishes cropland prices by region and states which differ from real estate prices. For example, cropland prices (per acre) for Alabama, Florida, Iowa and Wyoming in 2004 were $1800, $3900, $2320, and $1010, respectively.

¹⁷ Because input-mix efficiency is monotonically increasing (decreasing) with the increasing (decreasing) values of $\theta$, therefore, choosing different values of the substitution parameter only changes the efficiency score without changing the ranking of states.