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Productivity and Efficiency Analysis pp 281–324Cite as

Deterministic Frontier Analysis

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Abstract

Production frontiers are often represented by distance, revenue, cost and/or profit functions. These functions can sometimes be written in the form of regression models in which the explanatory variables are deterministic (i.e., not random). This chapter explains how to estimate and draw inferences concerning the unknown parameters in so-called deterministic frontier models (DFMs). It then explains how the estimated parameters can be used to predict levels of efficiency and analyse productivity change. The focus is on least squares estimators and predictors.

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Notes

  1. 1.

    See, for example, Greene (1980b, Eq. 1) , Färe et al. (1993, Eq. 14) , Grosskopf et al. (1995, Eq. 11) , Coelli and Perelman (1999, Eq. 5) , Ray (1998, Eq. 21) , Fuentes et al. (2001, Eq. 3) , Orea (2002, Eq. 5) , Reig-Martinez et al. (2001, Eq. 13) , O’Donnell and Coelli (2005, Eq. 5) , Vardanyan and Noh (2006, Eq. 3.1) , Ferrari (2006, Eq. 2) , Zhang and Garvey (2008, Eq. 9) and Diewert and Fox (2010, p. 82) .

  2. 2.

    If outputs and inputs are strongly disposable, then the output distance function is nondecreasing in outputs and nonincreasing in inputs for all feasible input-output combinations. If the output distance function is a translog function, then it is possible to find feasible input-output combinations where these monotonicity properties do not hold. Ergo, the output distance function cannot be a translog function. If there is more than one output and the output distance function is a translog function, then there exists a feasible input-output combination where at least one shadow revenue share lies outside the unit interval.

  3. 3.

    If there is more than one output and the output distance function is a double-log function, then output sets are unbounded (O’Donnell 2016 , p.  330).

  4. 4.

    See, for example, Diewert (1980, p. 462) , Deprins et al. (1984, p. 292) , Althin et al. (1996, Eq. 2.1) , Coelli and Perelman (1999, Eq. 10) , Coelli et al. (2003, p. 44) , Tsekouras et al. (2004, p. 98) , Hajargasht et al. (2008, Eq. 9) , Stern (2010, p. 351) , Das and Kumbhakar (2012, p. 211) and Coelli et al. (2013, Eq. 3) .

  5. 5.

    If outputs and inputs are strongly disposable, then the input distance function is nonincreasing in outputs and nondecreasing in inputs for all feasible input-output combinations. If the input distance function is a translog function, then it is possible to find feasible input-output combinations where these monotonicity properties do not hold. Ergo, the input distance function cannot be a translog function. If there is more than one input and the input distance function is a translog function, then there exists a feasible input-output combination where at least one shadow cost share lies outside the unit interval.

  6. 6.

    If the input distance function is a double-log function, then the output distance function is also a double-log function. If there is more than one output and the output distance function is a double-log function, then output sets are unbounded (O’Donnell 2016 , p. 330).

  7. 7.

    See, for example, Banker et al. (2003, Eq. 5).

  8. 8.

    If firms are price takers in output markets, then the revenue function is nondecreasing in p for all nonnegative x. If the revenue function is a translog function, then there exists a nonnegative p and a nonnegative x where this monotonicity property does not hold. Ergo, the revenue function cannot be a translog function. If (a) firms are price takers in output markets, (b) there is more than one output, and (c) the revenue function is a translog function, then there exists a nonnegative p and a nonnegative x where at least one revenue-maximising revenue share lies outside the unit interval.

  9. 9.

    If firms are price takers in output markets and the revenue function is a double-log function, then the output distance function is also a double-log function. If there is more than one output and the output distance function is a double-log function, then output sets are unbounded (O’Donnell 2016 , p.  330).

  10. 10.

    See, for example, Greene (1980b, Eq. 3) , Kopp and Diewert (1982, p. 328) , Banker et al. (1986, Eq. 1) , Baltagi and Griffin (1988, Eq. 4) , Kumbhakar (1997, Eq. 11) , Nadiri and Nandi (1999, p. 489) , Kumbhakar and Lovell (2000, Eq. 4.2.27) and Zheng and Bloch (2014, p.  207) .

  11. 11.

    If firms are price takers in input markets, then the cost function is nondecreasing in w for all producible q. If the cost function is a translog function, then there exists a nonnegative w and a producible q where this monotonicity property does not hold. Ergo, the cost function cannot be a translog function. If (a) firms are price takers in input markets, (b) there is more than one input, and (c) the cost function is a translog function, then there exists a nonnegative w and a producible q where at least one cost-minimising cost share lies outside the unit interval.

  12. 12.

    If firms are price takers in input markets and the cost function is a double-log function, then the output distance function is also a double-log function. If there is more than one output and the output distance function is a double-log function, then output sets are unbounded (O’Donnell 2016 , p. 330).

  13. 13.

    See, for example, Diewert (1980, Eq. 8) , Kumbhakar and Bhattacharyya (1992, Eq. 8) , Chaudhary et al. (1999, Eq. 1) , Kumbhakar (2001, Eqs. 7, 8) and Kumbhakar (2006, p. 254).

  14. 14.

    If firms are price takers in output and input markets, then profit functions are nondecreasing in p for all nonnegative w. If the profit function is a translog function, then there exists a nonnegative w where this monotonicity property does not hold. Ergo, the profit function cannot be a translog function. If (a) firms are price takers in output and input markets, (b) there is more than one output, and (c) the profit function is a translog function, then there exists a nonnegative w where at least one profit-maximising revenue share lies outside the unit interval.

  15. 15.

    Some authors make these assumptions implicitly. For example, Hsieh and Klenow (2009, Eq. 4) assume that production functions are double-log functions with observation-invariant slope coefficients that are positive and sum to one. This implies GA1 to GA4.

  16. 16.

    GA1 to GA3 imply that \(Q(q_{it})=A^t(z_{it})F(x_{it})D_O^t(x_{it}, q_{it},z_{it})\) where F(.) is a nonnegative,, linearly-homogenous, scalar-valued function (see Proposition 17 in Appendix A.1). GA4 implies that F(.) is also nondecreasing. Thus, it can be viewed as an aggregate input.

  17. 17.

    GA5 to GA7 imply that \(D^{t}_I(x_{it},q_{it},z_{it})=1\) (see Sect. 4.4.2). GA3 implies that \(D^{t}_O(x_{it},q_{it},z_{it})=1/D^{t}_I(x_{it},q_{it},z_{it})\) (see the discussion of O10 and DO10 in Sects. 2.1.1 and 2.4.1).

  18. 18.

    If there is no environmental change, then \(A^t(z_{it})\) reduces to a measure of technical change only. In this book, the term ‘technical change’ refers to the discovery of new technologies. In contrast, Solow (1957) uses the term ‘technical change’ “as a shorthand expression for any kind of shift in the production function. Thus, slowdowns, speedups, improvements in the education of the labor force, and all sorts of things will appear as [technical change]” (p. 312).

  19. 19.

    Elsewhere in the deterministic frontier literature, the term ‘COLS’ is often used to refer to slightly different estimators. These alternative estimators involve adjusting \(\alpha ^*\) upwards by an amount that depends on the probability distributions of the inefficiency effects. In this book, these alternative estimators are referred to as modified ordinary least squares (MOLS) estimators. The idea behind MOLS estimation of DFMs can be traced back at least as far as Richmond (1974) . A problem with MOLS estimators is that some observations may lie above the estimated frontier. For more details, see Førsund et al. (1980, p. 12) .

  20. 20.

    For more details concerning the properties of NLS estimators, see, for example, Hill et al. (2011, p. 362) .

  21. 21.

    This type of heteroskedasticity can arise when the inefficiency effects are gamma random variables and the scale parameters are the reciprocals of the shape parameters. It is also possible to imagine some types of group heteroskedasticity where the inefficiency effects have the same mean.

  22. 22.

    A function f(.) is even if \(f(x) = f(-x)\) for all x and \(-x\) in the domain of f(.). Examples include \(f(x)=|x|\) and \(f(x)=x^2\).

  23. 23.

    See, for example, Hill et al. (2011, p. 414).

  24. 24.

    They are, in fact, super-consistent. This means that, as the sample size increases, the distributions of the OLS estimators collapse around the true parameter values even faster than usual.

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    Christopher J. O’Donnell

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O’Donnell, C.J. (2018). Deterministic Frontier Analysis. In: Productivity and Efficiency Analysis. Springer, Singapore. https://doi.org/10.1007/978-981-13-2984-5_7

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