Investments in information systems: A contribution towards sustainability

Abstract

Empirical research has determined that information systems (IS) can abate far more emissions than they produce. By using its transformative power, Green IS can build energy efficiency along the entire business value chain and thus contribute to sustainable development that goes well beyond that of Green Information Technology (Green IT). However, from a business perspective there is still prevailing uncertainty with regard to the economic viability and optimal extent of Green IS investments. In this paper, we conceptualize a decision model for an IS investment that increases a company’s energy efficiency. We analyze and compare the costs associated with the investment and the realized energy cost savings. Furthermore, we examine the influence of fluctuating energy prices on investment decisions. By integrating risk and return into one decision calculus, we determine an optimal degree of investment, which avoids over-investment while promoting energy efficiency, and therefore establishes the long-term coherence of economic and environmental sustainability. Finally, we demonstrate that reduced exposure to risky energy prices results in comparatively larger investments, thereby implying a higher optimal investment degree, assuming the involvement of risk-averse decision-makers.

Appendix

1.1 Appendix A: Analysis under Certainty

The investment’s NPV (see (1)) is composed of the following elements:

$$ NPV(q)={R_T}(q)-{C_T}(q) $$

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$$ {C_T}(q)=\mathop{\sum}\limits_{t=0}^T\frac{{{c_t}(q)}}{{{{{\left( {1+i} \right)}}^t}}}=\mathop{\sum}\limits_{t=0}^T\frac{{{q^{\beta }}\cdot {c_{{t,\max }}}}}{{{{{\left( {1+i} \right)}}^t}}} $$

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$$ {R_T}(q)=\mathop{\sum}\limits_{t=0}^T\frac{{{r_t}(q)}}{{{{{\left( {1+i} \right)}}^t}}}=\mathop{\sum}\limits_{t=0}^T\frac{{{q^{\gamma }}\cdot {r_{{t,\max }}}}}{{{{{\left( {1+i} \right)}}^t}}} $$

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$$ {r_t}_{,max }={v_t}_{,max }+{e_t}_{,max } $$

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$$ {e_t}_{,max }={s_t}_{,max}\cdot {P_t} $$

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$$ {R_T}(q)=\mathop{\sum}\limits_{t=0}^T\frac{{{q^{\gamma }}\cdot \left( {{v_{t,max }}+{e_{t,max }}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}=\mathop{\sum}\limits_{t=0}^T\frac{{{q^{\gamma }}\cdot \left( {{v_{t,max }}+{s_{t,max }}\cdot {P_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}} $$

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1.2 Appendix B: Derivation of q′ for the optimization under certainty

In order to determine the project size that maximizes the NPV (see (1)) of the IS investment, the first derivative test (δNPV(q)/δq = 0) and the second derivative test (δ ² NPV(q)/δq ² < 0) are used.

$$ \max NPV(q)={q^{\gamma }}\mathop{\sum}\limits_{t=0}^T\frac{{{v_{t,max }}+{s_{t,max }}\cdot {P_t}}}{{{{{\left( {1+i} \right)}}^t}}}-{q^{\beta }}\mathop{\sum}\limits_{t=0}^T\frac{{{c_{t,max }}}}{{{{{\left( {1+i} \right)}}^t}}} $$

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First derivative of (1):

$$ \frac{{\partial NPV(q)}}{{\partial q}}=\gamma {q^{{\left( {\gamma -1} \right)}}}\mathop{\sum}\limits_{t=0}^T\frac{{{v_{t,max }}+{s_{t,max }}\cdot {P_t}}}{{{{{\left( {1+i} \right)}}^t}}}-\beta {q^{{\left( {\beta -1} \right)}}}\mathop{\sum}\limits_{t=0}^T\frac{{{c_{t,max }}}}{{{{{\left( {1+i} \right)}}^t}}} $$

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The first order condition requires $\frac{{\partial \mathrm{NPV}(q)}}{{\partial q}}\mathop{=}\limits^{\mathrm{def}}0$:

$$ \begin{array}{*{20}{c}} {\gamma {{q}^{{\left( {\gamma - 1} \right)}}}\sum\limits_{{t = 0}}^{T} {\frac{{{{v}_{{t,\max }}} + {{s}_{{t,\max }}}\cdot {{P}_{t}}}}{{{{{\left( {1 + i} \right)}}^{t}}}}} - \beta {{q}^{{\left( {\beta - 1} \right)}}}\sum\limits_{{t = 0}}^{T} {\frac{{{{c}_{{t,\max }}}}}{{{{{\left( {1 + i} \right)}}^{t}}}}} = 0} \\ { \Leftrightarrow \gamma {{q}^{{\left( {\gamma - 1} \right)}}}\sum\limits_{{t = 0}}^{T} {\frac{{{{v}_{{t,\max }}} + {{s}_{{t,\max }}}\cdot {{P}_{t}}}}{{{{{\left( {1 + i} \right)}}^{t}}}}} = \beta {{q}^{{\left( {\beta - 1} \right)}}}\sum\limits_{{t = 0}}^{T} {\frac{{{{c}_{{t,\max }}}}}{{{{{\left( {1 + i} \right)}}^{t}}}}} } \\ {\mathop{ \Leftrightarrow }\limits^{{for\,q \ne 0}} \frac{{{{q}^{{\left( {\beta - 1} \right)}}}}}{{{{q}^{{\left( {\gamma - 1} \right)}}}}} = {{q}^{{\left( {\beta - \gamma } \right)}}} = \frac{{\gamma \sum\nolimits_{{t = 0}}^{T} {\frac{{{{v}_{{t,\max }}} + {{s}_{{t,\max }}}\cdot {{P}_{t}}}}{{{{{\left( {1 + i} \right)}}^{t}}}}} }}{{\beta \sum\nolimits_{{t = 0}}^{T} {\frac{{{{c}_{{t,\max }}}}}{{{{{\left( {1 + i} \right)}}^{t}}}}} }}} \\ { \Leftrightarrow = {{{\left[ {\frac{{\gamma \sum\nolimits_{{t = 0}}^{T} {\frac{{{{v}_{{t,\max }}} + {{s}_{{t,\max }}}\cdot {{P}_{t}}}}{{{{{\left( {1 + i} \right)}}^{t}}}}} }}{{\beta \sum\nolimits_{{t = 0}}^{T} {\frac{{{{c}_{{t,\max }}}}}{{{{{\left( {1 + i} \right)}}^{t}}}}} }}} \right]}}^{{\frac{1}{{\beta - \gamma }}}}}} \\\end{array} $$

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Second derivative of (1):

$$ \frac{{{\partial^2}NPV(q)}}{{\partial {q^2}}}=\gamma \left( {\gamma -1} \right){q^{{\left( {\gamma -2} \right)}}}\mathop{\sum}\limits_{t=0}^T\frac{{{v_{t,max }}+{s_{t,max }}\cdot {P_t}}}{{{{{\left( {1+i} \right)}}^t}}}-\beta \left( {\beta -1} \right){q^{{\left( {\beta -1} \right)}}}\mathop{\sum}\limits_{t=0}^T\frac{{{c_{t,max }}}}{{{{{\left( {1+i} \right)}}^t}}} $$

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The second order condition for a maximum requires $\frac{{{\partial^2}NPV(q)}}{{\partial {q^2}}} < 0$:

When analyzing the second derivate (see (14)) of the objective function, we recall that q ∊ [0; 1] with q ≠ 0, γ ∈]0; 1[,β > 1, c _{t, max} > 0, (v _{t, max} +s _{t, max} ·P _t ) = r _{t, max} > 0. We conclude:

$$ \underbrace{{\underbrace{{\gamma \left( {\gamma -1} \right){q^{{\left( {\gamma -2} \right)}}}}}_{<0}\underbrace{{\mathop{\sum}\limits_{t=0}^T\frac{{{v_{{t,\max }}}+{s_{{t,\max }}}\cdot {P_t}}}{{{{{\left( {1+i} \right)}}^t}}}}}_{>0}}}_{<0}\underbrace{{-\underbrace{{\beta \left( {\beta -1} \right){q^{{\left( {\beta -1} \right)}}}}}_{>0}\underbrace{{\mathop{\sum}\limits_{t=0}^T\frac{{{c_{{t,\max }}}}}{{{{{\left( {1+i} \right)}}^t}}}}}_{>0}}}_{<0 } $$

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As both summands are negative, we can conclude that the sum, i.e. the second derivative, is negative. Hence, the NPV (see (1)) has a local maximum at q′ for q ≠ 0:

$$ q\prime :=\min \left\{ {{{{\left[ {\gamma \cdot {{{\left( {\mathop{{\mathop{{\mathop{\sum}\nolimits}}\limits_{t=0 }}}\limits^T\frac{{{v_{{t,\max }}}+{s_{{t,\max }}}\cdot {P_t}}}{{{{{\left( {1+i} \right)}}^t}}}} \right)}} \left/ {\beta } \right.}\cdot \left( {\mathop{{\mathop{{\mathop{\sum}\nolimits}}\limits_{t=0 }}}\limits^T\frac{{{c_{{t,\max }}}}}{{{{{\left( {1+i} \right)}}^t}}}} \right)} \right]}}^{{\frac{1}{{\beta -\gamma }}}}};1} \right\} $$

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1.3 Appendix C: Analysis under uncertainty

Formalization of the risk component:

$$ {{\widetilde{RC}}_T}=-\sqrt{{\mathop{\sum}\limits_{t=0}^T\frac{{{q^{{2\gamma }}}\cdot s_{{t,\max}}^2\cdot {\sigma^2}\left( {{{{\widetilde{X}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}}}=-{q^{\gamma }}\sqrt{{\mathop{\sum}\limits_{t=0}^T\frac{{s_{{t,\max}}^2\cdot {\sigma^2}\left( {{{{\widetilde{X}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}}} $$

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1.4 Appendix D: Derivation of q ^* for the optimization under uncertainty

In order to determine the project size that maximizes the raNPV (see (4)) of the IS investment, the first derivative test (δraNPV(q)/δq = 0) and the second derivative test (δ ² raNPV(q)/δq ² < 0) are used.

$$ \begin{array}{*{20}c} {\max raNPV(q)={q^{\gamma }}\mathop{\sum}\limits_{t=0}^T\frac{{{v_{{t,\max }}}+{s_{{t,\max }}}\cdot E\left( {{{{\widetilde{P}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}-{q^{\beta }}\mathop{\sum}\limits_{t=0}^T\frac{{{c_{{t,\max }}}}}{{{{{\left( {1+i} \right)}}^t}}}} \\ {-\alpha \left( {-{q^{\gamma }}\sqrt{{\mathop{\sum}\limits_{t=0}^T\frac{{s_{{t,\max}}^2\cdot \sigma_t^2\left( {{{{\widetilde{X}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}}}} \right)} \\\end{array}$$

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First derivative of (4):

$$ \begin{array}{*{20}c} {\frac{{\partial raNPV(q)}}{{\partial q}}=\gamma {q^{{\left( {\gamma -1} \right)}}}\mathop{\sum}\limits_{t=0}^T\frac{{{v_{{t,\max }}}+{s_{{t,\max }}}\cdot E\left( {{{{\widetilde{P}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}-\beta {q^{{\left( {\beta -1} \right)}}}\mathop{\sum}\limits_{t=0}^T\frac{{{c_{{t,\max }}}}}{{{{{\left( {1+i} \right)}}^t}}}} \\ {+\alpha \gamma {q^{{\left( {\gamma -1} \right)}}}\sqrt{{\mathop{\sum}\limits_{t=0}^T\frac{{s_{{t,\max}}^2\cdot \sigma_t^2\left( {{{{\widetilde{X}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}}}} \\\end{array}$$

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The first order condition requires $\frac{{\partial \mathrm{raNPV}(q)}}{{\partial \mathrm{q}}}\mathop{=}\limits^{\mathrm{def}}0$:

$$ \begin{array}{*{20}c} {\gamma {q^{{\left( {\gamma -1} \right)}}}\mathop{\sum}\limits_{t=0}^T\frac{{{v_{{t,\max }}}+{s_{{t,\max }}}\cdot E\left( {{{{\widetilde{P}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}-\beta {q^{{\left( {\beta -1} \right)}}}\mathop{\sum}\limits_{t=0}^T\frac{{{c_{{t,\max }}}}}{{{{{\left( {1+i} \right)}}^t}}}+\alpha \gamma {q^{{\left( {\gamma -1} \right)}}}\sqrt{{\mathop{\sum}\limits_{t=0}^T\frac{{s_{{t,\max}}^2\cdot \sigma_t^2\left( {{{{\widetilde{X}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}}}=0} \\ {\Leftrightarrow {q^{{\left( {\gamma -1} \right)}}}\left[ {\gamma \mathop{\sum}\limits_{t=0}^T\frac{{{v_{{t,\max }}}+{s_{{t,\max }}}\cdot E\left( {{{{\widetilde{P}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}+\alpha \gamma \sqrt{{\mathop{\sum}\limits_{t=0}^T\frac{{s_{{t,\max}}^2\cdot \sigma_t^2\left( {{{{\widetilde{X}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}}}} \right]=\beta {q^{{\left( {\beta -1} \right)}}}\mathop{\sum}\limits_{t=0}^T\frac{{{c_{{t,\max }}}}}{{{{{\left( {1+i} \right)}}^t}}}} \\ {\mathop{\Leftrightarrow}\limits^{{for\,q\ne 0}}\frac{{{q^{{\left( {\beta -1} \right)}}}}}{{{q^{{(\gamma -1)}}}}}={q^{{\left( {\beta -\gamma } \right)}}}=\frac{{\gamma \mathop{\sum}\nolimits_{t=0}^T\frac{{{v_{{t,\max }}}+{s_{{t,\max }}}\cdot E\left( {{{{\widetilde{P}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}+\alpha \gamma \sqrt{{\mathop{\sum}\nolimits_{t=0}^T\frac{{s_{{t,\max}}^2\cdot \sigma_t^2\left( {{{{\widetilde{X}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}}}}}{{\beta \mathop{\sum}\nolimits_{t=0}^T\frac{{{c_{{t,\max }}}}}{{{{{\left( {1+i} \right)}}^t}}}}}} \\ {\Leftrightarrow q={{{\left[ {\frac{{\gamma \mathop{\sum}\nolimits_{t=0}^T\frac{{{v_{{t,\max }}}+{s_{{t,\max }}}\cdot E\left( {{{{\widetilde{P}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}+\alpha \gamma \sqrt{{\mathop{\sum}\nolimits_{t=0}^T\frac{{s_{{t,\max}}^2\cdot \sigma_t^2\left( {{{{\widetilde{X}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}}}}}{{\beta \mathop{\sum}\nolimits_{t=0}^T\frac{{{c_{{t,\max }}}}}{{{{{\left( {1+i} \right)}}^t}}}}}} \right]}}^{{\frac{1}{{\beta -\gamma }}}}}} \\\end{array}$$

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Second derivative of (4):

$$ \frac{{{\partial^2}raNPV(q)}}{{\partial {q^2}}}=\gamma \left( {\gamma -1} \right){q^{{\left( {\gamma -2} \right)}}}\sum\limits_{t=0}^T {\frac{{{v_{{t,\max }}}+{s_{{t,\max }}}\cdot E\left( {{{{\widetilde{P}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^2}}}} -\beta \left( {\beta -1} \right){q^{{\left( {\beta -1} \right)}}}\sum\limits_{t=0}^T {\frac{{{c_{{t,\max }}}}}{{{{{\left( {1+i} \right)}}^t}}}+\alpha \gamma \left( {\gamma -1} \right){q^{{\left( {\gamma -2} \right)}}}\sqrt{{\sum\limits_{t=0}^T {\frac{{s_{{t,\max}}^2\cdot \sigma_t^2\left( {\widetilde{{{X_t}}}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}} }}} $$

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The second order condition for a maximum requires $\frac{{{\partial^2}\mathrm{raNPV}\left( \mathrm{q} \right)}}{{\partial {{\mathrm{q}}^2}}} < 0$:

When analyzing the second derivate (see (22)) of the objective function, we recall that q ∊ [0; 1] $with\ q\ne 0,\ \gamma \in \left] {0;1} \right[,\ \beta >1,\ {c_{t,max }} > 0,\ \left( {{v_{t,max }}+{s_{t,max }}\cdot {P_t}} \right)={r_{t,max }} > 0,\ \alpha >0$. We conclude:

$$ \underbrace{{\underbrace{{\gamma \left( {\gamma -1} \right){q^{{\left( {\gamma -2} \right)}}}}}_{<0}\underbrace{{\sum\limits_{t=0}^T {\frac{{{v_{{t,\max }}}+{s_{{t,\max }}}\cdot E\left( {{{{\widetilde{P}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}} -}}_{>0}}}_{<0}\underbrace{{\underbrace{{\beta \left( {\beta -1} \right){q^{{\left( {\beta -1} \right)}}}}}_{>0}\underbrace{{\sum\limits_{t=0}^T {\frac{{{c_{{t,\max }}}}}{{{{{\left( {1+i} \right)}}^t}}}}}}_{>0}}}_{<0 }+\underbrace{{\underbrace{{\alpha \gamma \left( {\gamma -1} \right){q^{{\left( {\gamma -2} \right)}}}}}_{<0}\underbrace{{\sqrt{{\sum\limits_{t=0}^T {\frac{{s_{{t,\max}}^2\cdot \sigma_t^2\left( {\widetilde{{{X_t}}}} \right)}}{{{{{\left( {1+i} \right)}}^2}}}}}}}}_{>0}}}_{<0 } $$

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As all three summands are negative, we can conclude that the sum, i.e. the second derivative, is negative. Hence, the raNPV has a local maximum at q ^* for q ≠ 0:

$$ {q^{*}}:=\min \left\{ {{{{\left[ {{{{\gamma \cdot \left( {\sum\limits_{t=0}^T {\frac{{{v_{{t,\max +{S_{{t,\max }}}}}}\cdot E\left( {\widetilde{{{P_t}}}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}} } \right)+\alpha \sqrt{{\sum\limits_{t=0}^T {\frac{{s_{{t,\max}}^2\cdot \sigma_t^2\left( {\widetilde{{{X_t}}}} \right)}}{{{{{\left( {1+i} \right)}}^2}}}} }}}} \left/ {{\beta \cdot \left( {\sum\limits_{t=0}^T {\frac{{{c_{{t,\max }}}}}{{{{{\left( {1+i} \right)}}^t}}}} } \right)}} \right.}} \right]}}^{{\frac{1}{{\beta -\gamma }}}}};1} \right\} $$

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1.5 Appendix E: Uncertain investment costs

Formalization of the risk component including uncertain investment costs:

$$ \widetilde{R}{C_T}=-{q^{\gamma }}\sqrt{{\mathop{\sum}\limits_{t=0}^T\frac{{s_{t,max}^2\cdot {\sigma^2}\left( {{{{\widetilde{X}}}_t}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}}}+{q^{\beta }}\sqrt{{\mathop{\sum}\limits_{t=0}^T\frac{{{\sigma^2}\left( {{{{\widetilde{c}}}_{t,max }}} \right)}}{{{{{\left( {1+i} \right)}}^t}}}}} $$

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