Abstract
In this paper we develop radial and directional measures of the rate of technical change for the class of directional distance functions. For both types, we distinguish between primal and dual measures while the former are further divided into oriented (input- and output-based) and non-oriented. We highlight the pivotal role of translation elasticity in examining the interrelationships among the alternative directional measures and that of scale elasticity in the case of radial measures. We also show that the radial and directional measures are related one another through the normalized (with the value of the direction vector) dual functions.
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Notes
Notice that a there is similar relationship between the input- and the output-based measures of technical efficiency; see Färe and Lovell (1978).
Notice that the Shephard-type output (input) distance function can be viewed as a transformation function that is homogenous of degree one in outputs (inputs).
Ohta (1974) initiated this line of research by establishing the relation between the cost function- and the production function-based measures of the rate of technical change by means of scale elasticity. Note that the production function-based is indeed an output-based measure of the rate of technical change.
The notion of translation elasticity, introduced by Färe and Karagiannis (2011), is presented in the next section.
Notice that \(g_x \) is subtracted from the input quantity vector x while \(g_y \) is added to the output quantity vector y.
The last two equalities in (2) are obtained by using the translation property, i.e., \(( {\nabla _x \vec {D}^T(x,y;g_x ,g_y )})^\prime g_x -( {\nabla _y \vec {D}^T(x,y;g_x ,g_y )})^\prime g_y =1\).
Notice that total differential hereon will involve directional derivatives for inputs and outputs and conventional derivatives for the technical change index. For the definition of directional derivatives see (Rockafellar 1970, p. 241).
Notice that in the single-input, single-output case, \(\vec {D}_{g_x }^{TI} \) coincides with the input-based radial measure of technical change as defined in (D9) and \(\vec {D}_{g_y }^{TO} \) coincides with the output-based radial measure as given in (D10).
This result is analogous to that of Boussemart et al. (2003) where the logarithm of the Malmquist index (i.e., the oriented TFP measure) is roughly twice the Luenberger indicator (i.e., the non-oriented TFP measure) under graph translation homotheticity, which is equivalent to constant returns to translation.
An alternative proof is as follows: Chambers (2002) Lemma 2 shows that graph translation homotheticity, which is equivalent to constant returns to translation (Färe and Karagiannis 2011), implies that \(\vec {D}^I=\vec {D}^O\). Then \(\vec {D}_t^I =\vec {D}_t^O \) and using (D5) and (D7) the first equality follows. The second equality follows by combining (D3), (D5), (D7) and Briec and Kerstens (2004) Proposition 2 which shows that \(\vec {D}^I=\vec {D}^O=2\vec {D}^T\) under graph translation homotheticity and thus \(\vec {D}_t^I =\vec {D}_t^O =2\vec {D}_t^T \).
Even though we have focused exclusively on directional functions, the proposed directional measures of the rate of technical change are readily applicable to all non- directional function representations of production technology, such as the Shephard-type distance functions, production and transformation functions.
Notice that the first six rows and columns make a matrix that is symmetric and, as in Table 1, the elements of the lower triangular block are the inverse of those in the upper block.
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Acknowledgments
We would like to thank two anonymous referees and Walter Briec for helpful comments on an earlier draft of this paper.
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Färe, R., Karagiannis, G. Radial and directional measures of the rate of technical change. J Econ 112, 183–199 (2014). https://doi.org/10.1007/s00712-013-0344-6
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DOI: https://doi.org/10.1007/s00712-013-0344-6