An optimization model for investment in technology and government regulation

Abstract

Companies struggle every day to estimate the adequate level of investment in new technologies, and governments lack the tools to determine the impact of their regulations on industry including telecommunications networks. Despite these facts, few studies discuss ways to assess appropriate levels of investment for technological initiatives and government regulations. To fill this gap, this study provides an optimization model for the investment of technology and government regulation, based on efficiencies. Results obtained from surveying northern European companies support the importance of estimating investment in technology and government regulation levels. The survey identified the four most relevant factors for practitioners: quality, cost, technology adoption, and government regulations. Based on the survey’s results, the model evaluates the level of investment for technology adoption and government regulations using cost and quality as target variables. Additional data from a German carrier served to test the model. Results show that technology investment delivers more benefits in cost and quality by increasing technology adoption. However, the model also suggests that diminishing returns make efficiencies stall at a certain level of technology adoption, and shows an investment threshold dependent on the type of benefit, cost, or quality the company seeks to maximize. Regarding government regulation, the model shows a counterintuitive behavior at higher levels of investment for the cost coefficients and at all levels of investment for the quality coefficient. This suggests that government regulation effects could be shifting from fixed-order cost to other types of costs.

Appendix

1.1 Technology adoption cost-oriented model

The derivative of $TC_{c}$ with respect to technology (T) is given by:

$$\frac{{\partial ({\text{TC}}_{\text{c}} )}}{{\partial {\text{T}}}} = \frac{{{\text{O}}\left( {\frac{{\partial {\text{R}}}}{{\partial {\text{T}}}}} \right){\text{D}}}}{\text{Q}} + \frac{{\partial {\text{J}}}}{{\partial {\text{T}}}}{\text{CD}} + 1$$

(10)

From Eqs. 3 and 8:

$$\frac{{\partial {\text{R}}}}{{\partial {\text{T}}}} =\upbeta_{1} \left( {{\text{M}} - {\text{N}}} \right){\text{e}}^{{\upbeta_{1} {\text{T}}}}$$

(11)

From Eqs. 10 and 7, setting Eq. 10 = 0:

$$\frac{{\partial {\text{R}}}}{{\partial {\text{T}}}} = - \left( {\frac{{1 + {\text{CD}}\left( {\frac{{\partial {\text{J}}}}{{\partial {\text{T}}}}} \right)}}{\text{OD}}} \right){\text{Q}}$$

(12)

Given that:

$$\frac{{\partial {\text{J}}}}{{\partial {\text{T}}}} =\upbeta_{5} \left( {{\text{A}} - {\text{E}}} \right){\text{e}}^{{\upbeta_{5} {\text{T}}}}$$

Then from Eqs. 11 and 12:

$$\upbeta_{1} \left( {{\text{M}} - {\text{N}}} \right){\text{e}}^{{\upbeta_{1} {\text{T}}}} = - \left( {\frac{{1 + {\text{CD}}\left( {\upbeta_{5} \left( {{\text{A}} - {\text{E}}} \right){\text{e}}^{{\upbeta_{5} {\text{T}}}} } \right)}}{\text{OD}}} \right){\text{Q}}$$

(13)

Substituting Eq. 2 in Eq. 13’s Q and solving for $R^{*}$:

$${\text{R}}^{ *} = \left[ {\frac{{ -\upbeta_{1} \left( {{\text{M}} - {\text{N}}} \right){\text{e}}^{{\upbeta_{1} {\text{T}}}} }}{{1 + {\text{CD}}\left( {\upbeta_{5} \left( {{\text{A}} - {\text{E}}} \right){\text{e}}^{{\upbeta_{5} {\text{T}}}} } \right)}}} \right]^{2} \frac{\text{ODHI}}{2}$$

(14)

From Eqs. 3 and 14:

$${\text{T}}^{ *} = \frac{{\ln \left( {\frac{{{\text{R}}^{ *} - {\text{N}} + {\text{M}}}}{{{\text{M}} - {\text{N}}}}} \right)}}{{\upbeta_{1} }}$$

(15)

1.2 Technology adoption quality-oriented model

Following the same process, from Eqs. 16 and 17:

$$\frac{{\partial {\text{TC}}_{\text{Q}} }}{{\partial {\text{T}}}} = 1 + \frac{{{\text{HQ}}\left( {\frac{{\partial {\text{I}}}}{{\partial {\text{T}}}}} \right)}}{2}$$

(19)

Setting Eq. 20 = 0

$$\frac{{\partial {\text{I}}}}{{\partial {\text{T}}}} = - \frac{2}{\text{HQ}}$$

(20)

From Eqs. 4 and 18:

$$\frac{{\partial {\text{I}}}}{{\partial {\text{T}}}} =\upbeta_{3} \left( {{\text{L}} - {\text{U}}} \right){\text{e}}^{{\upbeta_{3} {\text{T}}}}$$

(21)

Then:

$$\upbeta_{3} \left( {{\text{L}} - {\text{U}}} \right){\text{e}}^{{\upbeta_{3} {\text{T}}}} = - \frac{2}{\text{HQ}}$$

(22)

Substituting Eq. 2 in Q:

$${\text{I}}^{ *} = \frac{{{\text{ORDH}}\left[ {\upbeta_{3} \left( {{\text{L}} - {\text{U}}} \right){\text{e}}^{{\upbeta_{3} {\text{T}}}} } \right]^{2} }}{2}$$

(23)

$J^{*}$ is obtained from Eqs. 5 and 15:

$${\text{J}}^{ *} = \left( {{\text{E}} - {\text{A}}} \right) + \left( {{\text{A}} - {\text{E}}} \right){\text{e}}^{{\upbeta_{5} {\text{T}}^{ *} }}$$

(24)

Summarizing, the following optimal equations were obtained:

$${\text{R}}^{ *} = \left[ {\frac{{ -\upbeta_{1} \left( {{\text{M}} - {\text{N}}} \right){\text{e}}^{{\upbeta_{1} {\text{T}}}} }}{{1 + {\text{CD}}\left( {\upbeta_{5} \left( {{\text{A}} - {\text{E}}} \right){\text{e}}^{{\upbeta_{5} {\text{T}}}} } \right)}}} \right]^{2} \frac{\text{ODHI}}{2}$$

(14)

$${\text{T}}^{ *} = \frac{{\ln \left( {\frac{{{\text{R}}^{ *} - {\text{N}} + {\text{M}}}}{{\left( {{\text{M}} - {\text{N}}} \right)}}} \right)}}{{\upbeta_{1} }}$$

(15)

$${\text{I}}^{ *} = \frac{{{\text{ORDH}}\left[ {\upbeta_{3} \left( {{\text{L}} - {\text{U}}} \right){\text{e}}^{{\upbeta_{3} {\text{T}}}} } \right]^{2} }}{2}$$

(23)

$${\text{J}}^{ *} = \left( {{\text{E}} - {\text{A}}} \right) + \left( {{\text{A}} - {\text{E}}} \right){\text{e}}^{{\upbeta_{5} {\text{T}}^{ *} }}$$

(24)

1.3 Government regulation optimization

The derivative of $TC_{c}$ with respect to government regulations (G) is given by:

$$\frac{{\partial ({\text{TC}}_{\text{c}} )}}{{\partial {\text{G}}}} = \frac{{{\text{O}}\left( {\frac{{\partial {\text{R}}}}{{\partial {\text{G}}}}} \right){\text{D}}}}{\text{Q}} + \frac{{\partial {\text{J}}}}{{\partial {\text{G}}}}{\text{CD}} + 1$$

(32)

From Eqs. 26 and 31:

$$\frac{{\partial {\text{R}}}}{{\partial {\text{G}}}} = - \frac{{\left( {{\text{N}} - {\text{M}}} \right){\text{e}}^{{\frac{1}{{\upbeta_{2} {\text{G}}}}}} }}{{\upbeta_{2} {\text{G}}^{2} }}$$

(33)

From Eqs. 33 and 30, setting Eq. 33 = 0:

$$\frac{{\partial {\text{R}}}}{{\partial {\text{G}}}} = - \left( {\frac{{1 + {\text{CD}}\left( {\frac{{\partial {\text{J}}}}{{\partial {\text{G}}}}} \right)}}{\text{OD}}} \right){\text{Q}}$$

(34)

Then, from Eqs. 33 and 34:

$$\frac{{\left( {{\text{N}} - {\text{M}}} \right){\text{e}}^{{\frac{1}{{\upbeta_{2} {\text{G}}}}}} }}{{\upbeta_{2} {\text{G}}^{2} }} = \left( {\frac{{1 + {\text{CD}}\left( {\frac{{\partial {\text{J}}}}{{\partial {\text{G}}}}} \right)}}{\text{OD}}} \right){\text{Q}}$$

(35)

Substituting Eq. 2 in Eq. 35’s Q and solving for $R^{*}$:

$${\text{R}}^{ *} = \frac{{\left[ {\left( {{\text{N}} - {\text{M}}} \right){\text{e}}^{{\frac{1}{{\upbeta_{2} {\text{G}}}}}} } \right]^{2} {\text{ODHI}}}}{{2\left[ {\upbeta_{2} {\text{G}}\left( {1 - {\text{CD}}\left( {\frac{{\left( {{\text{E}} - {\text{A}}} \right){\text{e}}^{{\frac{1}{{\upbeta_{6} {\text{G}}}}}} }}{{\upbeta_{6} {\text{G}}^{2} }}} \right)} \right)} \right]^{2} }}$$

(36)

From Eq. 26:

$${\text{G}}^{ *} = \frac{1}{{\upbeta_{2} \ln \left( {\frac{{{\text{R}}^{ *} - {\text{N}} + {\text{M}}}}{{\left( {{\text{N}} - {\text{M}}} \right)}}} \right)}}$$

(37)

1.4 Government regulation quality-oriented model

Following the same process, from Eqs. 39 and 40:

$$\frac{{\partial {\text{TC}}_{\text{Q}} }}{{\partial {\text{G}}}} = 1 + \frac{{{\text{HQ}}\left( {\frac{{\partial {\text{I}}}}{{\partial {\text{G}}}}} \right)}}{2}$$

(41)

Setting Eq. 41 = 0:

$$\frac{{\partial {\text{I}}}}{{\partial {\text{G}}}} = - \frac{2}{\text{HQ}}$$

(42)

From Eqs. 27 and 41:

$$\frac{{\partial {\text{I}}}}{{\partial {\text{G}}}} = - \frac{{\left( {{\text{U}} - {\text{L}}} \right){\text{e}}^{{\frac{1}{{\upbeta_{4} {\text{G}}}}}} }}{{\upbeta_{4} {\text{G}}^{2} }}$$

(43)

Then:

$$- \frac{{\left( {{\text{U}} - {\text{L}}} \right){\text{e}}^{{\frac{1}{{\upbeta_{4} {\text{G}}}}}} }}{{\upbeta_{4} {\text{G}}^{2} }} = - \frac{2}{\text{HQ}}$$

(44)

Substituting Eq. 2 in Eq. 44’s Q:

$${\text{I}}^{ *} = \frac{{{\text{ORDH}}\left[ {\left( {{\text{U}} - {\text{L}}} \right){\text{e}}^{{\frac{1}{{\upbeta_{4} {\text{G}}}}}} } \right]^{2} }}{{2\left[ {\upbeta_{4} {\text{G}}^{2} } \right]^{2} }}$$

(45)

$J^{*}$ is obtained from Eqs. 28 and 38:

$${\text{J}}^{ *} = \left( {{\text{E}} - {\text{A}}} \right) + \left( {{\text{E}} - {\text{A}}} \right){\text{e}}^{{\frac{1}{{\upbeta_{6} {\text{G}}^{ *} }}}}$$

(46)