A generalized flexible functional form for α-returns to scale

Abstract

This study proposes a new flexible functional form for distance functions. This generalizes the existing functional form of Diewert (1992b) by allowing for α-returns to scale technology and employing the quadratic mean of order r aggregator function. Because we allow parameter α to be any strictly positive real number and r to be any non-zero real number, it includes a variety of functional forms as special cases, and all of them can now be considered as flexible functional forms.

Appendix

In this appendix, we provide the proof of Propositions 1. For this purpose, we introduce the following lemma. They are the restrictions on the derivatives of the output distance function, implied by homogeneity conditions (5) and (8).

Lemma 1: Suppose that the technology satisfies α-returns to scale and the output distance function D_o is twice differentiable at (x^*, y^*). Then, D_o satisfies the following equations¹⁵:

$$\mathop {\sum}\limits_{n = 1}^N {\frac{{\partial D_o\left( {x^ \ast ,y^ \ast } \right)}}{{\partial x_n}}} x_n^ \ast = - \alpha D_o\left( {x^ \ast ,y^ \ast } \right)$$

(26)

$$\mathop {\sum}\limits_{v = 1}^N {\frac{{\partial ^2D_o\left( {x^ \ast ,y^ \ast } \right)}}{{\partial x_u\,\partial x_v}}} x_v^ \ast = - \left( {1 + \alpha } \right)\frac{{\partial D_o\left( {x^ \ast ,y^ \ast } \right)}}{{\partial x_u}},\,\forall u \in \{ {1, \ldots ,N} \}$$

(27)

$$\mathop {\sum}\limits_{n = 1}^N {\frac{{\partial ^2D_o\left( {x^ \ast ,y^ \ast } \right)}}{{\partial x_n\,\partial y_m}}x_n^ \ast = - \alpha \frac{{\partial D_o\left( {x^ \ast ,y^ \ast } \right)}}{{\partial y_m}},\,\forall m \in \{ {1, \ldots ,M} \}}$$

(28)

$$\mathop {\sum}\limits_{m = 1}^M {\frac{{\partial D_o\left( {x^ \ast ,y^ \ast } \right)}}{{\partial y_m}}y_m^ \ast = D_o\left( {x^ \ast ,y^ \ast } \right)}$$

(29)

$$\mathop {\sum}\limits_{j = 1}^M {\frac{{\partial ^2D_o\left( {x^ \ast ,y^ \ast } \right)}}{{\partial y_j\,\partial y_k}}y_j^ \ast = 0,\,\forall k \in \{ {1, \ldots ,M} \}}$$

(30)

$$\mathop {\sum}\limits_{m = 1}^M {\frac{{\partial ^2D_o\left( {x^ \ast ,y^ \ast } \right)}}{{\partial x_n\,\partial y_m}}y_m^ \ast = \frac{{\partial D_o\left( {x^ \ast ,y^ \ast } \right)}}{{\partial x_n}},\,\forall n \in \{ {1, \ldots ,N} \}}$$

(31)

Proof of Proposition 1: This proposition claims that even under the restrictions of (13)–(17), g_α,r can approximate any arbitrary output distance function $D_o^ \ast$ to the second-order at (x^*, y^*). Stated differently, the following equations are satisfied.¹⁶

$$g_{\alpha ,r}\left( {x^ \ast ,y^ \ast } \right) = D_o^ \ast \left( {x^ \ast ,y^ \ast } \right)$$

(32)

$$\frac{{\partial g_{\alpha ,r}\left( {x^ \ast ,y^ \ast } \right)}}{{\partial x_n}} = \frac{{\partial D_o^ \ast \left( {x^ \ast ,y^ \ast } \right)}}{{\partial x_n}},\,\forall n \in \{ {1, \ldots ,N} \}$$

(33)

$$\frac{{\partial g_{\alpha ,r}\left( {x^ \ast ,y^ \ast } \right)}}{{\partial y_m}} = \frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m}},\,m \in \{ {1, \ldots ,M} \}$$

(34)

$$\frac{{\partial ^2g_{\alpha ,r}\left( {x^ \ast ,y^ \ast } \right)}}{{\partial x_u\,\partial x_v}} = \frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u\,\partial x_v}},\,\forall u,v \in \{ {1, \ldots ,N} \}$$

(35)

$$\frac{{\partial ^2g_{\alpha ,r}\left( {x^ \ast ,y^ \ast } \right)}}{{\partial y_j\,\partial y_k}} = \frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j\,\partial y_k}},\,\forall j,\,k \in \{ {1, \ldots ,M} \}$$

(36)

$$\frac{{\partial ^2g_{\alpha ,r}\left( {x^ \ast ,y^ \ast } \right)}}{{\partial x_n\,\partial y_m}} = \frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_n\,\partial y_m}},\,\forall n \in \{ {1, \ldots ,N} \},\,\forall m \in \{ {1, \ldots ,M} \}$$

(37)

We define the parameters in g_α,r as follows:¹⁷

$$\sigma = D_o^ \ast \left( {x^ \ast ,y^ \ast } \right)$$

(38)

$$\begin{array}{l}a_{j,k} = \left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( \left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j\partial y_k}}} \right) - \left( {1 - r} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\right.\\ \qquad \quad \left. \times\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_k}}} \right) \right)y_j^{ \ast 1 - r/2}y_k^{ \ast 1 - r/2}\\ \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \forall j \ne k \in \{ {1, \ldots ,M} \}\end{array}$$

(39)

$$\begin{array}{l}a_{m,m} = \left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left(\left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m^2}}} \right) - \left( {1 - r} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\right.\\ \quad \quad \quad \left. \times \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m}}} \right)^2 + \left( {1 - \frac{r}{2}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m}}} \right)\left( {\frac{1}{{y_m^ \ast }}} \right)\right)y_m^{ \ast 2 - r}\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \forall m \in \{ {1, \ldots ,M} \}\end{array}$$

(40)

$$\begin{array}{l}b_{u,v} = \left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left(- \left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u\partial x_v}}} \right) + \left( {1 + r} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) \right.\\ \quad \quad \left. \times \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_v}}} \right) \right)x_u^{ \ast 1 - \alpha r/2}x_v^{^\ast 1 - \alpha r/2}\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \qquad \qquad \forall u \ne v \in \{ {1, \ldots ,N} \}\end{array}$$

(41)

$$\begin{array}{l}b_{n,n} = \left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( - \left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_n^2}}} \right) + \left( {1 + r} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\right.\\ \quad \quad \quad \left. \times \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_n}}} \right)^2 - \left( {1 - \frac{{\alpha r}}{2}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_n}}} \right)\left( {\frac{1}{{x_n^ \ast }}} \right)\right)x_n^{ \ast 2 - \alpha r}\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \qquad \quad \forall n \in \{ {1, \ldots ,N} \}\end{array}$$

(42)

$$\begin{array}{l}c_{m,n} = \left( {\frac{4}{{\alpha r}}} \right) \left( {\frac{{y_m^{ \ast 1 - r/2}x_n^{ \ast 1 + \alpha r/2}}}{{\left( {\mathop {\sum}\nolimits_{j = 1}^M {a_jy_j^{ \ast r/2}} } \right)\left( {\mathop {\sum}\nolimits_{u = 1}^N {b_ux_u^{ \ast - \alpha r/2}} } \right)}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) \left(\left( { - \frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m\partial x_n}}} \right)\vphantom{\left( {\frac{{y_m^{ \ast 1 - r/2}x_n^{ \ast 1 + \alpha r/2}}}{{\left( {\mathop {\sum}\nolimits_{j = 1}^M {a_jy_j^{ \ast r/2}} } \right)\left( {\mathop {\sum}\nolimits_{u = 1}^N {b_ux_u^{ \ast - \alpha r/2}} } \right)}}} \right)}\right.\\ \quad \quad \quad \left.\vphantom{\left( {\frac{{y_m^{ \ast 1 - r/2}x_n^{ \ast 1 + \alpha r/2}}}{{\left( {\mathop {\sum}\nolimits_{j = 1}^M {a_jy_j^{ \ast r/2}} } \right)\left( {\mathop {\sum}\nolimits_{u = 1}^N {b_ux_u^{ \ast - \alpha r/2}} } \right)}}} \right)}+ \left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m}}} \right) \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_n}}} \right)\right)\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \qquad \forall n \in \{ {1, \ldots ,N} \},\,\forall m \in \{{1, \ldots ,M} \}\end{array}$$

(43)

First, we show that these parameters specified by (38)–(43) satisfy the restrictions (14)–(17). By summing $a_{j,k}y_k^{ \ast r/2}$ over k, we can derive the following equation:

$$\begin{array}{l}\mathop {\sum}\limits_{k = 1}^M {a_{j,k}y_k^{ \ast r/2}} = \mathop {\sum}\limits_{k \ne j} {a_{j,k}y_k^{ \ast r/2} + a_{j,j}y_j^{ \ast r/2}}\\= \mathop {\sum}\limits_{k \ne j} \left( {\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left(\left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j\partial y_k}}} \right) - \left( {1 - r} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_k}}} \right)\right)y_j^{ \ast 1 - r/2}y_k^ \ast } \right)\\\quad \quad+ \left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) \left(\left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j^2}}} \right) - \left( {1 - r} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j}}} \right)^2 \right. \\ \quad \quad \left. + \left( {1 - \frac{r}{2}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j}}} \right)\left( {\frac{1}{{y_j^ \ast }}} \right)\right)y_j^{ \ast 2 - r/2}\end{array}$$

from (39) and (40),

$$\begin{array}{l} = \left( \mathop {\sum}\limits_{k = 1}^M \left(\left( \frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j\partial y_k}} \right)y_k^ \ast\right.\right.\\\left.\left. - \left( {1 - r} \right)\left(\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_k}}} \right)y_k^ \ast\right) \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)y_j^{ \ast 1 - r/2}\\+ \left( {1 - \frac{r}{2}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j}}} \right)y_j^{ \ast 1 - r/2}\\ = \left( {\mathop {\sum}\limits_{k = 1}^M {\left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j\partial y_k}}y_k^ \ast } \right)} } \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)y_j^{ \ast 1 - r/2}+ \left( - \left( {1 - r} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( {\mathop {\sum}\limits_{k = 1}^M {\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_k}}y_k^ \ast } \right)} } \right)\right.\\\left.+ \left( {1 - \frac{r}{2}} \right) \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j}}} \right)y_j^{ \ast 1 - r/2}\\ = \left( { - \left( {1 - r} \right)\left( {\frac{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) + \left( {1 - \frac{r}{2}} \right)} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j}}} \right)y_j^{ \ast 1 - r/2}\end{array}$$

from (29) and (30),

$$= \left( {\frac{r}{2}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,y^ \ast } \right)}}{{\partial y_j}}} \right)y_j^{ \ast 1 - r/2}$$

(44)

Then, it implies (14) as follows:

$$\begin{array}{l}\mathop {\sum}\limits_{j = 1}^M {\mathop {\sum}\limits_{k = 1}^M {a_{j,k}y_j^{ \ast r/2}y_k^{ \ast r/2}} } \\= \mathop {\sum}\limits_{j = 1}^M {\left( {\left( {\frac{r}{2}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j}}} \right)y_j^{ \ast 1 - r/2}} \right)y_j^{^\ast r/2}} \end{array}$$

$$\begin{array}{l} = \left( {\frac{r}{2}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\mathop {\sum}\limits_{j = 1}^M {\left( {\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j}}} \right)y_j^ \ast } \right)} \\= \left( {\frac{r}{2}} \right)\left( {\frac{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) = \frac{r}{2}\end{array}$$

from (29)

By summing $b_{u,v}x_v^{ \ast \alpha r/2}$ over v, we can derive the following equation.

$$\begin{array}{l}\mathop {\sum}\limits_{v = 1}^N {b_{u,v}x_v^{ \ast \alpha r/2}} = \mathop {\sum}\limits_{v \ne u} {b_{u,v}x_v^{ \ast \alpha r/2} + b_{u,u}x_u^{ \ast \alpha r/2}} \\ = \mathop {\sum}\limits_{v \ne u} {\left(\left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) \left(- \left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u\partial x_v}}} \right) + \left( {1 + r} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_v}}} \right)\right)x_u^{ \ast 1 - \alpha r/2}x_v^ \ast \right)} + \left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\\ \qquad\times \left(- \left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u^2}}} \right) + \left( {1 + r} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u}}} \right)^2 - \left( {1 - \frac{{\alpha r}}{2}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u}}} \right)\left( {\frac{1}{{x_u^ \ast }}} \right)\right)x_u^{ \ast 2 - \alpha r/2}\end{array}$$

from (41) and (42),

$$\begin{array}{l} = \left( {\mathop {\sum}\limits_{v = 1}^N {\left(- \left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u\partial x_v}}} \right)x_v^ \ast + \left( {1 + r} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_v}}} \right)x_v^ \ast \right)} } \right)\\ \quad\times\left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)x_u^{ \ast 1 - \alpha r/2} - \left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( {1 - \frac{{\alpha r}}{2}} \right)\\ \times\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u}}} \right)x_u^{ \ast 1 - \alpha r/2}\end{array}$$

$$\begin{array}{l} = \left(- \mathop {\sum}\limits_{v = 1}^N {\left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u\partial x_v}}x_v^ \ast } \right)} + \left( {1 + r} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u}}} \right)\mathop {\sum}\limits_{v = 1}^N {\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_v}}x_v^ \ast } \right)}\right)\\ \quad\times\left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)x_u^{ \ast 1 - \alpha r/2} - \left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) \\\quad\times\left( {1 - \frac{{\alpha r}}{2}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u}}} \right)x_u^{ \ast 1 - \alpha r/2}\end{array}$$

$$\begin{array}{l} = \left(- \left( { - \left( {1 + \alpha } \right)} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u}}} \right)\right.\\ \left. + \left( {1 + r} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u}}} \right)\left( { - \alpha } \right)D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)\right)\\ \quad\times\left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)x_u^{ \ast 1 - \alpha r/2} - \left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\\\times\left( {1 - \frac{{\alpha r}}{2}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u}}} \right)x_u^{ \ast 1 - \alpha r/2}\end{array}$$

from (26) and (27),

$$\begin{array}{l} = \left( {\left( {1 + \alpha } \right) - \left( {1 + r} \right)\alpha - \left( {1 - \frac{{\alpha r}}{2}} \right)} \right)\left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\\ \qquad\times \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u}}} \right)x_u^{ \ast 1 - \alpha r/2}\end{array}$$

$$= - \left( {\frac{{\alpha r}}{2}} \right)\left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,y^ \ast } \right)}}{{\partial x_u}}} \right)x_u^{ \ast 1 - \alpha r/2}$$

(45)

Then, it implies (15) as follows:

$$\begin{array}{l}\mathop {\sum}\limits_{u = 1}^N {\mathop {\sum}\limits_{v = 1}^N {b_{u,v}x_u^{ \ast \alpha r/2}x_v^{ \ast \alpha r/2}} } = \\ \mathop {\sum}\limits_{u = 1}^N {\left( { - \left( {\frac{{\alpha r}}{2}} \right)\left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u}}} \right)} \right)x_u^ \ast } \end{array}$$

$$\begin{array}{l} = - \left( {\frac{{\alpha r}}{2}} \right)\left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\mathop {\sum}\limits_{u = 1}^N {\left( {\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u}}} \right)x_u^ \ast } \right)} \\ = - \left( {\frac{{\alpha r}}{2}} \right)\left( {\frac{1}{{\alpha ^2}}} \right)\left( {\frac{{ - \alpha D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) = \frac{r}{2}\end{array}$$

from (26).

By summing $c_{m,n}y_m^{ \ast r/2}$ over m, we obtain (16) so as

$$\begin{array}{l}\mathop {\sum}\limits_{m = 1}^M {c_{m,n}y_m^{ \ast r/2}} = \\\left( {\frac{4}{{\alpha r}}} \right)\left( {\frac{{x_n^{ \ast 1 + \alpha r/2}}}{{\left( {\mathop {\sum }\nolimits_{j = 1}^M a_jy_j^{ \ast r/2}} \right)\left( {\mathop {\sum }\nolimits_{u = 1}^N b_ux_u^{ \ast - \alpha r/2}} \right)}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\\\times\left( {\mathop {\sum}\limits_{m = 1}^M {\left(- \left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m\partial x_n}}} \right)y_m^ \ast + \left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m}}} \right)y_m^ \ast \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_n}}} \right)\right)} } \right)\end{array}$$

from (43),

$$\begin{array}{l} = \left( {\frac{4}{{\alpha r}}} \right)\left( {\frac{{x_n^{ \ast 1 + \alpha r/2}}}{{\left( {\mathop {\sum }\nolimits_{j = 1}^M a_jy_j^{ \ast r/2}} \right)\left( {\mathop {\sum }\nolimits_{u = 1}^N b_ux_u^{ \ast - \alpha r/2}} \right)}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\\\times\left(- \left( {\mathop {\sum}\limits_{m = 1}^M {\left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m\partial x_n}}} \right)} y_m^ \ast } \right)\right.\\\left. + \left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_n}}} \right)\left( {\mathop {\sum}\limits_{m = 1}^M {\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m}}} \right)} y_m^ \ast } \right)\right)\end{array}$$

$$\begin{array}{l} = \left( {\frac{4}{{\alpha r}}} \right)\left( {\frac{{x_n^{ \ast 1 + \alpha r/2}}}{{\left( {\mathop {\sum }\nolimits_{j = 1}^M a_jy_j^{ \ast r/2}} \right)\left( {\mathop {\sum }\nolimits_{u = 1}^N b_ux_u^{ \ast - \alpha r/2}} \right)}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\\ \times \left( { - \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_n}}} \right) + \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_n}}} \right)\left( {\frac{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)} \right) = 0\end{array}$$

from (29) and (31).

By summing $c_{m,n}x_n^{ \ast - \alpha r/2}$ over n, we obtain (17) so as

$$\begin{array}{l}\mathop {\sum}\limits_{n = 1}^N {c_{m,n}x_n^{ \ast - \alpha r/2}} \\= \left( {\frac{4}{{\alpha r}}} \right)\left( {\frac{{y_m^{ \ast 1 - r/2}}}{{\left( {\mathop {\sum}\nolimits_{j = 1}^M {a_jy_j^{ \ast r/2}} } \right)\left( {\mathop {\sum}\nolimits_{u = 1}^N {b_ux_u^{ \ast - \alpha r/2}} } \right)}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\\\times\left( \mathop {\sum}\limits_{n = 1}^N \left(- \left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m\partial x_n}}} \right)x_n^ \ast + \left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m}}} \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_n}}} \right)x_n^ \ast \right) \right)\end{array}$$

from (43),

$$\begin{array}{l} = \left( {\frac{4}{{\alpha r}}} \right)\left( {\frac{{y_m^{ \ast 1 - r/2}}}{{\left( {\mathop {\sum}\nolimits_{j = 1}^M {a_jy_j^{ \ast r/2}} } \right)\left( {\mathop {\sum}\nolimits_{u = 1}^N {b_ux_u^{ \ast - \alpha r/2}} } \right)}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\\\times\left(- \left( {\mathop {\sum}\limits_{n = 1}^N {\left( {\frac{{\partial ^2D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m\partial x_n}}} \right)x_n^ \ast } } \right)\right.\\ +\left. \left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right) \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m}}} \right)\left( {\mathop {\sum}\limits_{n = 1}^N {\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_n}}} \right)x_n^ \ast } } \right)\right)\end{array}$$

$$\begin{array}{l} = \left( {\frac{4}{{\alpha r}}} \right)\left( {\frac{{y_m^{ \ast 1 - r/2}}}{{\left( {\mathop {\sum}\nolimits_{j = 1}^M {a_jy_j^{ \ast r/2}} } \right)\left( {\mathop {\sum}\nolimits_{u = 1}^N {b_ux_u^{ \ast - \alpha r/2}} } \right)}}} \right)\left( {\frac{1}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\left(- \left( { - \alpha } \right)\left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m}}} \right)\right.\\ \left. + \left( {\frac{{\partial D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m}}} \right) \left( {\frac{{ - \alpha D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}{{D_o^ \ast \left( {x^ \ast ,\,y^ \ast } \right)}}} \right)\right) = 0\end{array}$$

from (26) and (28).

Second, we show that these parameters specified by (38)–(43) satisfy the equations for the second-order approximation (32)–(37).

The value of g_α,r evaluated at (x^*, y^*) is given by the following equation, using (14)–(17):

$$g_{\alpha ,r}\left( {x^ \ast ,y^ \ast } \right) = \sigma$$

(46)

Substituting (38) into (46), we obtain (32).

The first-order derivatives of g_α,r with respect to inputs x evaluated at (x^*, y^*) are given by the following equations using (14)–(17):

$$\begin{array}{l}\frac{{\partial g_{\alpha ,r}\left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_n}} = - \alpha \sigma \left( {\frac{r}{2}} \right)^{ - 1}x_n^{ \ast \alpha r/2 - 1}\left( {\mathop {\sum}\limits_{v = 1}^N {b_{n,v}x_v^{ \ast \alpha r/2}} } \right)\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\forall n \in \{ {1, \ldots ,N} \}\end{array}$$

(47)

Substituting (38) and (45) into (47), we obtain (33):

The first-order derivatives of g_α,r with respect to outputs y evaluated at (x^*, y^*) are given by the following equations using (14)–(17):

$$\begin{array}{l}\frac{{\partial g_{\alpha ,r}\left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m}} = \sigma \left( {\frac{r}{2}} \right)^{ - 1}y_m^{ \ast r/2 - 1}\left( {\mathop {\sum}\limits_{k = 1}^M {a_{m,k}y_k^{ \ast r/2}} } \right)\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \forall m \in \{ {1, \ldots ,M} \}\end{array}$$

(48)

Substituting (38) and (44) into (48), we obtain (34):

The second-order derivatives of g_α,r with respect to inputs x evaluated at (x^*, y^*) are given by the following equations using (14)–(17):

$$\begin{array}{l}\frac{{\partial ^2g_{\alpha ,r}\left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_u\partial x_v}} = \alpha \sigma \left( {\frac{r}{2}} \right)^{ - 1}\left(\alpha \left( {1 + r} \right)\left( {\frac{r}{2}} \right)^{ - 1}\left( {\mathop {\sum}\limits_{n = 1}^N {b_{u,n}x_n^{ \ast \alpha r/2}} } \right)\right.\\\left. \times \left( {\mathop {\sum}\limits_{n = 1}^N {b_{v,n}x_n^{ \ast \alpha r/2}} } \right)x_u^{ \ast \alpha r/2 - 1}x_v^{ \ast \alpha r/2 - 1}-\left( {\frac{{\alpha r}}{2}} \right)b_{u,v}x_u^{ \ast \alpha r/2 - 1}x_v^{ \ast \alpha r/2 - 1}\right)\\\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \forall u \ne v \in \{ {1, \ldots ,N} \}\end{array}$$

(49)

$$\begin{array}{l}\frac{{\partial ^2g_{\alpha ,r}\left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial x_n^2}} = \alpha \sigma \left( {\frac{r}{2}} \right)^{ - 1}\left(\alpha \left( {1 + r} \right)\left( {\frac{r}{2}} \right)^{ - 1}\left( {\mathop {\sum}\limits_{u = 1}^N {b_{n,u}x_u^{ \ast \alpha r/2}} } \right)^2 x_n^{ \ast \alpha r - 2}\right.\\\qquad\qquad\quad\left.- \left( {\frac{{\alpha r}}{2}} \right)b_{n,n}x_n^{ \ast \alpha r - 2}- \left( {\frac{{\alpha r}}{2} - 1} \right)\left( {\mathop {\sum}\limits_{u = 1}^N {b_{n,u}x_u^{ \ast \alpha r/2}} } \right)x_n^{ \ast \alpha r/2 - 2}\right)\\\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \forall n \in \{ {1, \ldots ,N} \}\end{array}$$

(50)

Either substituting (38), (41), and (45) into (49) or substituting (38), (42), and (45) into (50), we obtain (35).

The second-order derivatives of g_α,r with respect to outputs y evaluated at (x^*, y^*) are given by the following equations, using (14)–(17):

$$\begin{array}{l}\frac{{\partial ^2g_{\alpha ,r}\left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_j\partial y_k}} =\\ \sigma \left( {\frac{r}{2}} \right)^{ - 1}\left(\left( {1 - r} \right)\left( {\frac{r}{2}} \right)^{ - 1}\left( {\mathop {\sum}\limits_{m = 1}^M {a_{j,m}y_m^{ \ast r/2}} } \right)\right. \\ \left. \times \left( {\mathop {\sum}\limits_{m = 1}^M {a_{k,m}y_m^{ \ast r/2}} } \right)y_j^{ \ast r/2 - 1}y_k^{ \ast r/2 - 1} + \left( {\frac{r}{2}} \right)a_{j,k}y_j^{ \ast r/2 - 1}y_k^{ \ast r/2 - 1}\right)\\\qquad \qquad\qquad\qquad\qquad\qquad\qquad \forall j \ne k \in \{ {1, \ldots ,M} \}\end{array}$$

(51)

$$\begin{array}{l}\frac{{\partial ^2g_{\alpha ,r}\left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m^2}} = \sigma \left( {\frac{r}{2}} \right)^{ - 1}\left(\left( {1 - r} \right)\left( {\frac{r}{2}} \right)^{ - 1}\left( {\mathop {\sum}\limits_{j = 1}^M {a_{m,j}y_j^{ \ast r/2}} } \right)^2 y_m^{ \ast r - 2}\right.\\ \left. \qquad \qquad \qquad + \left( {\frac{r}{2}} \right)a_{m,m}y_m^{ \ast r - 2} + \left( {\frac{r}{2} - 1} \right)y_m^{ \ast r/2 - 2}\left( {\mathop {\sum}\limits_{j = 1}^M {a_{m,j}y_j^{ \ast r/2}} } \right) \right)\\\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad \forall m \in \{ {1, \ldots ,M} \}\end{array}$$

(52)

Either substituting (38), (39), and (44) into (51) or substituting (38), (40), and (44) into (52), we obtain (36).

The second-order cross-derivatives of g_α,r with respect to inputs x and outputs y evaluated at (x^*, y^*) are given by the following equation, using (14)–(17):

$$\begin{array}{l}\frac{{\partial ^2g_{\alpha ,r}\left( {x^ \ast ,\,y^ \ast } \right)}}{{\partial y_m\partial x_n}} =\\\alpha \sigma \left(\left( {\frac{r}{2}} \right)^{ - 2}\left( {\mathop {\sum}\limits_{j = 1}^M {a_{m,j}y_j^{ \ast r/2}} } \right)\left( {\mathop {\sum}\limits_{u = 1}^N {b_{n,u}x_u^{ \ast \alpha r/2}} } \right)y_m^{ \ast r/2 - 1}x_n^{ \ast \alpha r/2 - 1}\right. \\ \left. - \left( {\frac{r}{4}} \right)\left( {\mathop {\sum}\limits_{j = 1}^M {a_jy_j^{ \ast r/2}} } \right)\left( {\mathop {\sum}\limits_{u = 1}^N {b_ux_u^{ \ast - \alpha r/2}} } \right)c_{m,n}y_m^{ \ast r/2 - 1}x_n^{ \ast - \alpha r/2 - 1}\right)\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\forall m \in \{ {1, \ldots ,M} \},\,\forall n \in \{ {1, \ldots ,N} \}\end{array}$$

(53)

Substituting (38), (43)–(45) into (53), we obtain (37). QED.