Developing a decomposable measure of profit efficiency using DEA

Abstract

In for-profit organizations efficiency measurement with reference to the potential for profit augmentation is particularly important as is its decomposition into technical, and allocative components. Different profit efficiency approaches can be found in the literature to measure and decompose overall profit efficiency. In this paper, we highlight some problems within existing approaches and propose a new measure of profit efficiency based on a geometric mean of input/output adjustments needed for maximizing profits. Overall profit efficiency is calculated through this efficiency measure and is decomposed into its technical and allocative components. Technical efficiency is calculated based on a non-oriented geometric distance function (GDF) that is able to incorporate all the sources of inefficiency, while allocative efficiency is retrieved residually. We also define a measure of profitability efficiency which complements profit efficiency in that it makes it possible to retrieve the scale efficiency of a unit as a component of its profitability efficiency. In addition, the measure of profitability efficiency allows for a dual profitability interpretation of the GDF measure of technical efficiency. The concepts introduced in the paper are illustrated using a numerical example.

Appendix

Properties of the geometric distance function defined in model (3)

G1.:

0⩽G(x, y)⩽1

G2.:

G(αx, α⁻¹y)⩽(1/α²) G(x, y), α⩾1 and G(αx, α⁻¹y)⩾(1/α²)G(x, y), α⩽1

G3.:

G(αx, y)⩽(1/α) G(x, y)⩽G(x, y), α⩾1

G4.:

G(x, αy)⩽αG(x, y)⩽G(x, y), 0⩽α⩽1

G1 Proof

• The GDF cannot be greater than 1. In order for this to happen the numerator in (3) should be greater than the denominator. However, as every θ _i in the numerator is ⩽1, and every β _r in the denominator is ⩾1, GDF>1 results in an impossibility. This means that the maximum value of G(x, y) is 1, happening when the numerator equals the denominator. As every θ _i in the numerator is ⩽1, and every β _r in the denominator is ⩾1, the equality between the numerator and denominator can only happen when all θ _i and all β _r are 1.

  The GDF may be zero when some inputs (but not all, as we assume that it is not possible to produce outputs with zero inputs) are zero. For zero outputs the model cannot find a feasible solution as it would be possible to find an infinitely large β _ro associated with the zero output.

G2 Proof

• This property states that G(x, y), satisfies sub-homogeneity (eg Russell, 1985) of −2 degree. Indeed,

  

G3 and G4 Proof

• These properties relate with the weak monotonicity properties of the geometric distance function. The input monotonicity implies that

  

  The output monotonicity implies that

  

A unit of maximum profitability is always scale efficient

Consider for the single input output case a unit A (x _A , y _A ) for which y _A /x _A is maximum (being therefore a CRS efficient unit). If this unit is assessed at prices (p _A , w _A ) then clearly for this unit's prices p _A y _A /w _A x _A >p _A y _j /w _A x _j , for every j≠A. Assessing unit B (x _B , y _B ), for which y _B /x _B is not maximum, at prices (p _B , w _B ) we cannot find for this unit's prices p _B y _B /w _B x _B >p _B y _A /w _B x _A , since y _A /x _A is maximum. Therefore, the above model (7) renders maximum profitability units that are also scale efficient. This reasoning can be extended to the multiple input/output case since the way we aggregate output revenues and input costs is through the geometric mean. Therefore replacing above p _j y _j /w _j x _j by (Π _r p _rj y _rj )^1/s/(Π _i w _ij x _ij )^1/m maintains the reasoning valid.