(65)

*<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> *β*

*β*3 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup>

*<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> *β*

�*β*1�*β*2 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup>

*<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> *β*

*β*3 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup>

*<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> *β*

*β*3 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup>

*<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> *β*

*β*3 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup>

� � <sup>¼</sup> *<sup>p</sup>*1*<sup>K</sup>* <sup>þ</sup> *<sup>p</sup>*<sup>2</sup> *<sup>L</sup>* <sup>þ</sup> *<sup>p</sup>*<sup>3</sup> *<sup>I</sup>*

*<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup>

*β*3 � � *<sup>β</sup>*<sup>3</sup> *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup>

real output of earlier period t-1, and C is the total cost [26].

*<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> *β*<sup>2</sup>

*β*2 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> <sup>2</sup> *p*

*L* ¼ *μ*

*I* ¼ *μ*

appears as follows:

*p*1*K* ¼ *μ*

*p*<sup>2</sup> *L* ¼ *μ*

*p*<sup>3</sup> *I* ¼ *μ*

*p*

*p*

*p*

*C p*1, *p*2, *p*3, *yt*

�1 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> *<sup>t</sup> <sup>α</sup>* �<sup>1</sup>

¼ *B p*

*β*2 *β*1 � � *<sup>β</sup>*<sup>1</sup>

*B* ¼ *μ*

Where B:

**7. Conclusion**

**209**

*p*

*p*

�1 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> *<sup>t</sup> <sup>α</sup>* �<sup>1</sup>

*β*2 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> <sup>2</sup> *p*

�1 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> *<sup>t</sup> <sup>α</sup>* �<sup>1</sup>

*β*2 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> <sup>2</sup> *p*

�1 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> *<sup>t</sup> <sup>α</sup>* �<sup>1</sup>

*β*2 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> <sup>2</sup> *p*

*β*1 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> <sup>1</sup> *p*

�1 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> *<sup>t</sup> <sup>α</sup>* �<sup>1</sup>

*Information Technology Value Engineering (ITVE) DOI: http://dx.doi.org/10.5772/intechopen.95855*

> �*β*1�*β*3 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> <sup>2</sup> *p*

�1 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> *<sup>t</sup> <sup>α</sup>* �<sup>1</sup>

*β*2 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> <sup>2</sup> *p*

Whereas y\*it = the desired output with i = subsystem and t = period, Kit = the regular capital, Lit = the labor expense, Iit = the IT capital, α = total factor productivity, and β1, β2, β<sup>3</sup> = the output elasticity of the regular capital, the labor expense, and the IT capital. Therefore, the partial adjustment for each subsystem is as follows (to simplify, i is disappearing):

$$\boldsymbol{y}\_{t} = \boldsymbol{\mu}\_{t}\boldsymbol{y}\_{t}^{\*} + (\mathbf{1} - \boldsymbol{\mu}\_{t})\boldsymbol{y}\_{t-1} = \boldsymbol{\mu}\_{t}a\boldsymbol{K}\_{t}^{\beta\_{1}}L\_{t}^{\beta\_{2}}I\_{t}^{\beta\_{3}} + (\mathbf{1} - \boldsymbol{\mu}\_{t})\boldsymbol{y}\_{t-1} \tag{49}$$

Whereas μ<sup>t</sup> is the static speed of adjustment and yt-1 is the revenue in the earlier period. Additionally, for cost minimization, the partial derivatives of the Eq. (48) should fulfill these conditions [58, 59]:

$$\frac{\partial \mathbf{y}\_t}{\partial \mathbf{K}\_t} = \mathbf{0}, \quad \frac{\partial \mathbf{y}\_t}{\partial \mathbf{L}\_t} = \mathbf{0}, \quad \frac{\partial \mathbf{y}\_t}{\partial \mathbf{I}\_t} = \mathbf{0} \tag{50}$$

If the Eq. (49) is mathematically derived to K, L, and I, it respectively results in the following equations (whereas p1, p2, and p3 are added to the equations as unit prices of the regular capital (K), the labor expense (L), and the IT capital (I):

$$\frac{\partial \mathcal{y}\_t}{\partial K} = \mu\_t a \beta\_1 p\_1 K^{\beta\_1 - 1} p\_2 L^{\beta\_2} p\_3 I^{\beta\_3} \tag{51}$$

$$\frac{\partial \mathcal{y}\_t}{\partial \mathcal{L}} = \mu\_t a \beta\_2 p\_1 K^{\beta\_1} p\_2 L^{\beta\_2 - 1} p\_3 I^{\beta\_3} \tag{52}$$

$$\frac{\partial \boldsymbol{y}\_t}{\partial \boldsymbol{I}} = \mu\_t a \beta\_3 p\_1 \mathbf{K}^{\beta\_1} p\_2 \mathbf{L}^{\beta\_2} p\_3 \mathbf{I}^{\beta\_3 - 1} \tag{53}$$

Using the Eq. (50) prerequisites, the Eq. (51) = the Eq. (52) = the Eq. (53), further equations arise as follows:

$$K = \frac{p\_3}{p\_1} \frac{\beta\_1}{\beta\_3} I; \quad L = \frac{p\_1}{p\_2} \frac{\beta\_2}{\beta\_1} K; \quad \text{and} \ I = \frac{p\_2}{p\_3} \frac{\beta\_3}{\beta\_2} L \tag{54}$$

If the Eq. (49) is substituted by the Eq. (54) such that the new equation appears in the regular capital (K) variable, the equation is as Eq. (55) and afterwards simplified to become Eq. (56).

$$y\_t = \mu\_t a K^{\beta\_1} \left[ \frac{p\_1}{p\_2} \frac{\beta\_2}{\beta\_1} K \right]^{\beta\_2} \left[ \frac{p\_1}{p\_3} \frac{\beta\_3}{\beta\_1} K \right]^{\beta\_3} + (1 - \mu\_t) y\_{t-1} \tag{55}$$

$$y\_t = \mu\_t a \beta\_1^{-\beta\_2 - \beta\_3} \beta\_2^{\beta\_2} \beta\_3^{\beta\_3} p\_1^{\beta\_1 + \beta\_3} p\_2^{-\beta\_2} p\_3^{-\beta\_3} \mathcal{K}^{\beta\_1 + \beta\_2 + \beta\_3} + (1 - \mu\_t) y\_{t-1} \tag{56}$$

Furthermore, the Eq. (56) becomes K variable as in Eq. (57) and afterwards simplified as in Eq. (58) as follows:

$$K^{\theta\_1 + \theta\_2 + \theta\_3} = \mu\_t^{-1} a^{-1} \rho\_1^{\theta\_1 + \theta\_3} \rho\_2^{-\theta\_2} \rho\_3^{-\theta\_3} p\_1^{-\theta\_2 - \theta\_3} p\_2^{\theta\_2} p\_3^{\theta\_3} \left[ y\_t - (1 - \mu\_t) y\_{t-1} \right] \tag{57}$$

$$\begin{split} K &= \mu\_{t}^{\overline{\left(\rho\_{1}+\rho\_{2}+\rho\_{3}\right)}} a^{\overline{\left(\rho\_{1}+\rho\_{2}+\rho\_{3}\right)}} \rho\_{1}^{\overline{\left(\rho\_{1}+\rho\_{2}+\rho\_{3}\right)}} \rho\_{2}^{\overline{\left(\rho\_{1}+\rho\_{2}+\rho\_{3}\right)}} \rho\_{3}^{\overline{\left(\rho\_{1}+\rho\_{2}+\rho\_{3}\right)}} \rho\_{1}^{\overline{\left(\rho\_{1}+\rho\_{2}+\rho\_{3}\right)}} \\ &\overline{p\_{2}^{\overline{\left(\rho\_{1}+\rho\_{2}+\rho\_{3}\right)}}} \overline{p\_{3}^{\overline{\left(\rho\_{1}+\rho\_{2}+\rho\_{3}\right)}}} \left[ \overline{\rho\_{t} - (1-\mu\_{t})\overline{\nu\_{t-1}}} \right]^{\frac{1}{\left(\rho\_{1}+\rho\_{2}+\rho\_{3}\right)}} \end{split} \tag{58}$$

*y* ∗ *it* <sup>¼</sup> *<sup>α</sup>K<sup>β</sup>*<sup>1</sup>

(to simplify, i is disappearing):

*yt* <sup>¼</sup> *<sup>μ</sup><sup>t</sup> <sup>y</sup>* <sup>∗</sup>

should fulfill these conditions [58, 59]:

further equations arise as follows:

simplified to become Eq. (56).

*yt* <sup>¼</sup> *<sup>μ</sup>tα β*�*β*2�*β*<sup>3</sup>

simplified as in Eq. (58) as follows:

*K* ¼ *μ*

*p*

*<sup>K</sup><sup>β</sup>*1þ*β*2þ*β*<sup>3</sup> <sup>¼</sup> *<sup>μ</sup>*�<sup>1</sup>

**208**

*it <sup>L</sup><sup>β</sup>*<sup>2</sup> *it I β*3

*Computational Optimization Techniques and Applications*

Whereas y\*it = the desired output with i = subsystem and t = period, Kit = the regular capital, Lit = the labor expense, Iit = the IT capital, α = total factor productivity, and β1, β2, β<sup>3</sup> = the output elasticity of the regular capital, the labor expense, and the IT capital. Therefore, the partial adjustment for each subsystem is as follows

> *<sup>β</sup>*<sup>1</sup> *Lt <sup>β</sup>*<sup>2</sup> *It*

> > *∂yt ∂It*

*<sup>K</sup>*; *and I* <sup>¼</sup> *<sup>p</sup>*<sup>2</sup>

<sup>2</sup> *<sup>p</sup><sup>β</sup>*<sup>3</sup>

�*β*2 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> <sup>2</sup> *β*

<sup>1</sup>

*p*3 *β*3 *β*2

Whereas μ<sup>t</sup> is the static speed of adjustment and yt-1 is the revenue in the earlier period. Additionally, for cost minimization, the partial derivatives of the Eq. (48)

If the Eq. (49) is mathematically derived to K, L, and I, it respectively results in the following equations (whereas p1, p2, and p3 are added to the equations as unit prices of the regular capital (K), the labor expense (L), and the IT capital (I):

*<sup>∂</sup><sup>K</sup>* <sup>¼</sup> *<sup>μ</sup>tαβ*<sup>1</sup> *<sup>p</sup>*1*Kβ*1�<sup>1</sup> *<sup>p</sup>*2*Lβ*<sup>2</sup> *<sup>p</sup>*3*<sup>I</sup>*

*<sup>∂</sup><sup>L</sup>* <sup>¼</sup> *<sup>μ</sup>tαβ*<sup>2</sup> *<sup>p</sup>*1*Kβ*<sup>1</sup> *<sup>p</sup>*2*Lβ*2�<sup>1</sup> *<sup>p</sup>*3*<sup>I</sup>*

*<sup>∂</sup><sup>I</sup>* <sup>¼</sup> *<sup>μ</sup>tαβ*<sup>3</sup> *<sup>p</sup>*1*Kβ*<sup>1</sup> *<sup>p</sup>*2*Lβ*<sup>2</sup> *<sup>p</sup>*3*<sup>I</sup>*

Using the Eq. (50) prerequisites, the Eq. (51) = the Eq. (52) = the Eq. (53),

If the Eq. (49) is substituted by the Eq. (54) such that the new equation appears

*p*3 *β*3 *β*1 *K <sup>β</sup>*<sup>3</sup>

<sup>2</sup> *<sup>p</sup>*�*β*<sup>3</sup>

Furthermore, the Eq. (56) becomes K variable as in Eq. (57) and afterwards

*β*2þ*β*3 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> <sup>1</sup> *β*

<sup>3</sup> *yt* � <sup>1</sup> � *<sup>μ</sup><sup>t</sup>* ð Þ*yt*�<sup>1</sup>

*<sup>I</sup>*; *<sup>L</sup>* <sup>¼</sup> *<sup>p</sup>*<sup>1</sup> *p*2 *β*2 *β*1

in the regular capital (K) variable, the equation is as Eq. (55) and afterwards

*∂yt ∂Lt* ¼ 0,

*<sup>t</sup>* <sup>þ</sup> <sup>1</sup> � *<sup>μ</sup><sup>t</sup>* ð Þ *yt*�<sup>1</sup> <sup>¼</sup> *<sup>μ</sup><sup>t</sup> <sup>α</sup>Kt*

¼ 0,

*∂yt ∂Kt*

*∂yt*

*∂yt*

*∂yt*

*<sup>K</sup>* <sup>¼</sup> *<sup>p</sup>*<sup>3</sup> *p*1 *β*1 *β*3

*yt* <sup>¼</sup> *<sup>μ</sup>tαK<sup>β</sup>*<sup>1</sup> *<sup>p</sup>*<sup>1</sup>

<sup>1</sup> *<sup>β</sup><sup>β</sup>*<sup>2</sup> <sup>2</sup> *β β*3 <sup>3</sup> *<sup>p</sup><sup>β</sup>*2þ*β*<sup>3</sup> <sup>1</sup> *<sup>p</sup>*�*β*<sup>2</sup>

*<sup>t</sup> α*�<sup>1</sup> *β*

�1 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> *<sup>t</sup> <sup>α</sup>* �<sup>1</sup>

*β*2 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> <sup>2</sup> *p*

*β*2þ*β*<sup>3</sup> <sup>1</sup> *<sup>β</sup>*�*β*<sup>2</sup> <sup>2</sup> *β* �*β*<sup>3</sup> <sup>3</sup> *<sup>p</sup>*�*β*2�*β*<sup>3</sup> <sup>1</sup> *<sup>p</sup><sup>β</sup>*<sup>2</sup>

*<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> *β*

*β*3 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup>

*p*2 *β*2 *β*1 *K <sup>β</sup>*<sup>2</sup> *<sup>p</sup>*<sup>1</sup>

*it* ð Þ *i* ¼ 1, … , 4 *and t* ¼ 1, 2, … , 11 (48)

*<sup>β</sup>*<sup>3</sup> <sup>þ</sup> <sup>1</sup> � *<sup>μ</sup><sup>t</sup>* ð Þ *yt*�<sup>1</sup> (49)

¼ 0 (50)

*<sup>β</sup>*<sup>3</sup> (51)

*<sup>β</sup>*<sup>3</sup> (52)

*<sup>β</sup>*3�<sup>1</sup> (53)

<sup>þ</sup> <sup>1</sup> � *<sup>μ</sup><sup>t</sup>* ð Þ*yt*�<sup>1</sup> (55)

(57)

(58)

�*β*2�*β*3 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> 1

<sup>3</sup> *<sup>K</sup><sup>β</sup>*1þ*β*2þ*β*<sup>3</sup> <sup>þ</sup> <sup>1</sup> � *<sup>μ</sup><sup>t</sup>* ð Þ*yt*�<sup>1</sup> (56)

<sup>3</sup> *yt* � <sup>1</sup> � *<sup>μ</sup><sup>t</sup>* ð Þ*yt*�<sup>1</sup>

�*β*3 *<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup> <sup>3</sup> *p*

*<sup>β</sup>*1þ*β*2þ*<sup>β</sup>* ð Þ<sup>3</sup>

*L* (54)

Using the equivalent way, the variable L and I can become as follows:

$$L = \mu\_{t}^{\frac{-1}{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}}{\mu\_{1}^{\frac{-\rho\_{1}-\rho\_{3}}{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}}}{\mu\_{2}^{\frac{-\rho\_{1}-\rho\_{3}}{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}}}{\mu\_{3}^{\frac{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}}}{\mu\_{3}^{\frac{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}}}\left[y\_{t}-(1-\mu\_{t})y\_{t-1}\right]^{\frac{1}{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}}\tag{59}$$

$$I = \mu\_{t}^{\frac{-1}{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}}{\mu\_{2}^{\frac{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}}\rho\_{1}^{\frac{-\rho\_{1}}{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}}\rho\_{2}^{\frac{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}}\rho\_{3}^{\frac{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}{\left(\rho\_{1}+\rho\_{2}+\beta\_{3}\right)}}$$

If K, L, and I are multiplying p1, p2, and p3 as unit prices respectively, then it appears as follows:

$$\begin{split} p\_1 \mathbf{K} &= \mu\_t^{\frac{-1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} a^{\frac{-1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \boldsymbol{\theta}\_1^{\frac{\rho\_2 + \rho\_3}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \boldsymbol{\theta}\_2^{\frac{-\rho\_1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \boldsymbol{\theta}\_3^{\frac{-\rho\_3}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \boldsymbol{\theta}\_1^{\frac{\rho\_1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \\ p\_2^{\frac{\rho\_1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \boldsymbol{\theta}\_3^{\frac{\rho\_3}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \left[y\_t - (\mathbf{1} - \boldsymbol{\mu}\_t) \boldsymbol{y}\_{t-1}\right]^{\frac{1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \end{split} \tag{61}$$

$$\begin{split} p\_2 L &= \mu\_t^{\frac{-1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} a^{\frac{-1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \rho\_1^{\frac{-\rho\_1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \rho\_2^{\frac{\rho\_1 + \rho\_3}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \rho\_3^{\frac{-\rho\_1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \mathbf{p}\_1^{\frac{\rho\_1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \\ p\_2^{\frac{\rho\_2}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \mathbf{p}\_3^{\frac{\rho\_3}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \left[y\_t - (1 - \mu\_t) y\_{t-1}\right]^{\frac{1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \end{split} \tag{62}$$

$$\begin{split} p\_3 I = \mu\_t^{\frac{-1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} a^{\frac{-1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \rho\_1^{\frac{-\rho\_1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \rho\_2^{\frac{-\rho\_1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \rho\_3^{\frac{\rho\_1 + \rho\_2}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \mathbf{p}\_1^{\frac{\rho\_1 + \rho\_2}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \\ p\_2^{\frac{\rho\_1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \mathbf{p}\_3^{\frac{\rho\_1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \left[y\_t - (\mathbf{1} - \mu\_t) y\_{t-1}\right]^{\frac{1}{\left(\rho\_1 + \rho\_2 + \rho\_3\right)}} \end{split} \tag{63}$$

Moreover, the Eqs. (61), (62), and (63) substituted into Eq. (64), the total cost of yielding y units in the low-cost technique manifest as the Eq. (64) and (65).

$$\begin{split} \mathcal{C}(p\_1, p\_2, p\_3, \mathbf{y}\_t) &= p\_1 K + p\_2 L + p\_3 I \\ &= B \, p\_1^{\frac{\rho\_1}{(\rho\_1 + \rho\_2 + \rho\_3)}} \, p\_2^{\frac{\rho\_2}{(\rho\_1 + \rho\_2 + \rho\_3)}} \, p\_3^{\frac{\rho\_3}{(\rho\_1 + \rho\_2 + \rho\_3)}} \, \big[ \mathbf{y}\_t - (\mathbf{1} - \boldsymbol{\mu}\_t) \mathbf{y}\_{t-1} \big]^{\frac{1}{(\rho\_1 + \rho\_2 + \rho\_3)}} \end{split} \tag{64}$$

Where B:

$$\begin{split} B &= \mu\_{1}^{\frac{-1}{(\rho\_{1}+\rho\_{2}+\rho\_{3})}} a^{\frac{-1}{(\rho\_{1}+\rho\_{2}+\rho\_{3})}} \left[ \left(\frac{\beta\_{1}}{\beta\_{2}}\right)^{\frac{\rho\_{2}}{(\rho\_{1}+\rho\_{2}+\rho\_{3})}} \left(\frac{\beta\_{1}}{\beta\_{3}}\right)^{\frac{\rho\_{3}}{(\rho\_{1}+\rho\_{2}+\rho\_{3})}} + \\ & \left(\frac{\beta\_{2}}{\beta\_{1}}\right)^{\frac{\rho\_{1}}{(\rho\_{1}+\rho\_{2}+\rho\_{3})}} \left(\frac{\beta\_{2}}{\beta\_{3}}\right)^{\frac{\rho\_{3}}{(\rho\_{1}+\rho\_{2}+\rho\_{3})}} + \left(\frac{\beta\_{3}}{\beta\_{1}}\right)^{\frac{\rho\_{1}}{(\rho\_{1}+\rho\_{2}+\rho\_{3})}} \left(\frac{\beta\_{3}}{\beta\_{2}}\right)^{\frac{\rho\_{2}}{(\rho\_{1}+\rho\_{2}+\rho\_{3})}} \right] \end{split} \tag{65}$$

Whereas p1, p2, and p3 is unit prices of the regular capital (Kt), the labor expense (Lt), and the IT capital (It) respectively, yt is the real output of period t, yt-1 is the real output of earlier period t-1, and C is the total cost [26].

#### **7. Conclusion**

The significant problem surrounding this study is to sustain superior firm performance as desired at optimal costs due to the IT presence, which has inevitably become a need for running the business world. Numerous studies on the

relationship of the firm performance of the IT resource were more focused on a statistical method that links between components using survey data. In essence, this study undertakes an analogous study, but with a different approach, namely, the systems engineering approach combined with RBV theory, systems engineering, the theory of partial adjustment, including the CD production function, which, in turn, lead to creating the ITVE. Furthermore, to create the ITVE, the followed stages are to build the conceptual model of the IT value based on the RBV theory, model experiment using PAV, validate PAV, model the ITVE, confirm the ITVE and study managerial impacts of the model.

**References**

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The conceptual model of IT value has logically exemplified the relationship between ITR, FC, FCC, and FP in terms of competitive advantages. The theory of partial adjustment links logically the model, which formulates it in two types of models. Explicitly, the first model addresses PAV with the IT capital presence (with It) inside of its production function, and the second model with the IT capital absence (without It). The applied production function is the CD function while the dynamic factor component of the speed of adjustment is the ROE. However, it may be replaced by other dynamic factors.

The principal problem of this chapter is how to achieve the optimal resources, for instance, IT resource costs, for required business performance. By benefiting the earlier studies, namely the systems engineering methodology, the conceptual model of IT value, the RBV theory, and the PAV theory can solve this problem so that the solution results in the IT value engineering. Furthermore, using the analysis results, a synthesis work leads to composing a block diagram, which depicts a model in terms of the systems engineering of IT value engineering framework, which ultimately results in serial, parallel, and hybrid configurations. Likewise, by benefiting CD production function involved within PAV, the optimal cost of the required firm performance occurs. For that reason, it should surely be an experiment as a simulation on work mechanisms of the model. Consequently, the ITVE technically appears as a framework to study IT value models. However, in practice, this model contributes to managerial implications, which should reinforce the match between techniques and practices.
