**A. Appendix**

*Proof.* The optimal profit function for power plant I satisfies the following HJB equation such that *V*ð Þ<sup>I</sup> *<sup>N</sup>* ð Þ *K* :

$$\begin{split} \rho\_1 V\_N^{(l)}(K) &= \max\_{\left\{L\_N^R, L\_N^F\right\} \ge 0} \left\{ \left[ \theta \left( a\_1 L\_N^R(t) + a\_2 L\_N^F + \beta\_1 \check{L}\_N^R + \beta\_2 \check{L}\_N^F + (\Gamma + \delta)K \right) \right] \\ &- \frac{1}{2} \theta^R \left( L\_N^R \right)^2 - \frac{1}{2} \theta^F \left( L\_N^F \right)^2 \\ &+ \frac{\partial V\_N^{(l)}(K)}{\partial K} \left[ \theta\_1 \left( L\_N^R, L\_N^F \right) + \theta\_2 \left( \check{L}\_N^R, \check{L}\_N^F \right) - \xi K \right] + \frac{1}{2} \frac{\partial^2 V\_N^{(l)}(K)}{\partial K^2} \rho^2(K) \right\} \end{split} \tag{A.1}$$

Via the first order conditions, first obtain the optimal values *LR <sup>N</sup>*, *L<sup>R</sup>* ð ÞI � � for power plant I as:

$$L\_N^R = \frac{\theta a\_1 + V\_N^{(\rm I)}(K)\theta\_1^R}{\beta^R} = \frac{\theta a\_1 + a\_1 \theta\_1^R}{\beta^R},\tag{A.2}$$

$$L\_N^F = \frac{\theta a\_2 + \mathcal{V}\_N^{(\mathcal{I})}(K) \theta\_1^F}{\beta^F} = \frac{\theta a\_2 + a\_1 \theta\_1^F}{\beta^F} \tag{A.3}$$

where *<sup>∂</sup>V*ð Þ<sup>I</sup> *<sup>N</sup>* ð Þ *K <sup>∂</sup><sup>K</sup>* � *<sup>V</sup>*0ð Þ<sup>I</sup> *<sup>N</sup>* ð Þ *<sup>K</sup> :* The HJB for power plant (II), using *<sup>∂</sup>V*ð Þ II *<sup>N</sup>* ð Þ *K <sup>∂</sup><sup>K</sup>* <sup>¼</sup> *<sup>V</sup>*0ð Þ II *<sup>N</sup>* ð Þ *K* , then obtain

$$\begin{split} \rho\_{2}V\_{N}^{(\mathrm{I})}(\mathrm{K}) &= \max\_{\left\{\begin{subarray}{c}\hat{\boldsymbol{\Gamma}}\_{N}^{\mathrm{R}},\hat{\boldsymbol{\Gamma}}\_{N}^{\mathrm{F}}\\ \end{subarray}\right\}} \left\{ \left[ (\boldsymbol{\Pi}-\boldsymbol{\theta}) \Big(a\_{1}\boldsymbol{L}\_{N}^{\mathrm{R}}(\boldsymbol{t}) + a\_{2}\boldsymbol{L}\_{N}^{\mathrm{F}} + \boldsymbol{\beta}\_{1}\boldsymbol{\hat{\boldsymbol{\Gamma}}\_{N}^{\mathrm{R}} + \boldsymbol{\beta}\_{2}\boldsymbol{\hat{\boldsymbol{\Gamma}}\_{N}^{\mathrm{F}} + (\boldsymbol{\Gamma}+\boldsymbol{\delta})\boldsymbol{K} \Big) \right]} \right. \\ & \left. - \frac{1}{2}\boldsymbol{\tilde{\beta}}^{\mathrm{F}} \left(\boldsymbol{\tilde{\boldsymbol{\Gamma}}\_{N}^{\mathrm{R}}\right)^{2} - \frac{1}{2}\boldsymbol{\tilde{\beta}}^{\mathrm{F}} \left(\boldsymbol{\tilde{\boldsymbol{\Gamma}}\_{N}^{\mathrm{F}}\right)^{2}} \right. \\ & \left. + \frac{\partial V\_{N}^{(\mathrm{I})}(\boldsymbol{K})}{\partial \boldsymbol{K}} \left[ \boldsymbol{\theta}\_{1} \{\boldsymbol{L}\_{N}^{\mathrm{R}},\boldsymbol{L}\_{N}^{\mathrm{F}}\} + \boldsymbol{\theta}\_{2} \{\boldsymbol{\tilde{\boldsymbol{\Gamma}}\_{N}^{\mathrm{R}},\boldsymbol{\tilde{\boldsymbol{\Gamma}}\_{N}^{\mathrm{F}}\} - \boldsymbol{\xi}\boldsymbol{K} \Big] + \frac{1}{2}\frac{\partial^{2} V\_{N}^{(\mathrm{I})}(\boldsymbol{K})}{\partial \boldsymbol{K}^{2}} \rho^{2}(\boldsymbol{K}) \right]. \end{split} \tag{A.4}$$

Substituting Eqs. (A.2) and (A.3) results to Eq. (A.4) and via the first order conditions, we obtain the optimal values *L*~*<sup>R</sup> <sup>N</sup>*, *<sup>L</sup>*~*<sup>F</sup> N* � � for power plant II as:

*Impact of Hybrid-Enabling Technology on Bertrand-Nash Equilibrium Subject to Energy Sources DOI: http://dx.doi.org/10.5772/intechopen.94016*

$$\tilde{L}\_N^R = \frac{(\mathbf{1} - \boldsymbol{\theta})\boldsymbol{\beta}\_1 + \boldsymbol{V}\_N^{(\text{II})}(\mathbf{K})\boldsymbol{\theta}\_2^R}{\tilde{\boldsymbol{\beta}}^R} = \frac{(\mathbf{1} - \boldsymbol{\theta})\boldsymbol{\beta}\_1 + \boldsymbol{a}\_2\boldsymbol{\theta}\_2^R}{\tilde{\boldsymbol{\beta}}^R},\tag{A.5}$$

$$\tilde{L}\_N^F = \frac{(\mathbf{1} - \boldsymbol{\theta})\boldsymbol{\beta}\_2 + V'^{(\text{II})}\_N(K)\boldsymbol{\theta}\_2^F}{\tilde{\boldsymbol{\beta}}^F} = \frac{(\mathbf{1} - \boldsymbol{\theta})\boldsymbol{\beta}\_2 + \boldsymbol{a}\_2\boldsymbol{\theta}\_2^F}{\tilde{\boldsymbol{\beta}}^F}. \tag{A.6}$$

Hence, the solution of the HJB equation is an unary function with *K*, such that *V*ð Þ<sup>I</sup> *<sup>N</sup>* <sup>¼</sup> *<sup>a</sup>*^1*<sup>K</sup>* <sup>þ</sup> ^ *b*1, *V*ð Þ II *<sup>N</sup>* <sup>¼</sup> *<sup>a</sup>*^2*<sup>K</sup>* <sup>þ</sup> ^ *b*2. Hence, finally *a*^1, ^ *b*1, *a*^2, and^ *b*<sup>2</sup> as:

$$
\hat{a}\_1 = \frac{\theta(\Gamma + \delta)}{(\rho\_1 + \xi)}, \quad \hat{b}\_1 = \frac{\hat{\Phi}\_1}{\rho\_1}, \quad \hat{a}\_2 = \frac{(\mathbb{1} - \theta)(\Gamma + \delta)}{(\rho\_2 + \xi)}, \quad \hat{b}\_2 = \frac{\hat{\Phi}\_2}{\rho\_2}, \tag{A.7}
$$

where

Furthermore, we have shown that due to optimal price reaction functions being upward sloping, the subsidy level plays a decisive role on the payoff function of power plant II as the leader in a Stackelberg game. This model shows that cost reducing R&D investments with efficient hybrid-enabling technology innovation/s strengthens one's competitive bargaining position via the level of subsidy for Power Plant I to become a follower in the Stackelberg game. By analyzing this stochastic differential game model, we capture the government subsidy incentive as well as the subsidy that the leader (Power Plant II) pays the follower (Power Plant I) to

The proposed quantitative framework could assist policymakers when determining the appropriate R&D incentives for the development of hybrid-enabling technology within the energy market to achieve desired short and long-term environmental objectives with respect to budget limitations and environmental consid-

*Proof.* The optimal profit function for power plant I satisfies the following HJB

*<sup>N</sup>*, *<sup>L</sup>*~*<sup>F</sup> N* � �

*<sup>N</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>R</sup>* 1 *<sup>β</sup><sup>R</sup>* <sup>¼</sup> *θα*<sup>1</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>R</sup>*

*<sup>N</sup>* ð Þ *<sup>K</sup> <sup>ϑ</sup><sup>F</sup>* 1 *<sup>β</sup><sup>F</sup>* <sup>¼</sup> *θα*<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*1*ϑ<sup>F</sup>*

*<sup>N</sup>*ðÞþ*<sup>t</sup> <sup>α</sup>*2*L<sup>F</sup>*

*<sup>N</sup>*, *<sup>L</sup>*~*<sup>F</sup> N* � �

Substituting Eqs. (A.2) and (A.3) results to Eq. (A.4) and via the first order

h i

*<sup>N</sup>* ð Þ *<sup>K</sup> :* The HJB for power plant (II), using *<sup>∂</sup>V*ð Þ II

*<sup>N</sup>* <sup>þ</sup> *<sup>β</sup>*1*L*~*<sup>R</sup>*

� *ξK*

*<sup>N</sup>*, *<sup>L</sup>*~*<sup>F</sup> N* � � þ 1 2 *∂*2 *V*ð Þ<sup>I</sup> *<sup>N</sup>* ð Þ *K <sup>∂</sup>K*<sup>2</sup> *<sup>φ</sup>*<sup>2</sup>

n h i �

*<sup>N</sup>* <sup>þ</sup> *<sup>β</sup>*1*L*~*<sup>R</sup>*

nh i �

� *ξK*

*<sup>N</sup>* <sup>þ</sup> *<sup>β</sup>*2*L*~*<sup>F</sup>*

þ 1 2 *∂*2 *V*ð Þ<sup>I</sup> *<sup>N</sup>* ð Þ *K <sup>∂</sup>K*<sup>2</sup> *<sup>φ</sup>*<sup>2</sup>

1

1

*<sup>N</sup>* <sup>þ</sup> *<sup>β</sup>*2*L*~*<sup>F</sup>*

*<sup>N</sup>* þ ð Þ Γ þ *δ K*Þ

ð Þ *K* )

*<sup>N</sup>*, *L<sup>R</sup>* ð ÞI � �

*<sup>β</sup><sup>R</sup>* , (A.2)

*<sup>β</sup><sup>F</sup>* (A.3)

*<sup>N</sup>* ð Þ *K <sup>∂</sup><sup>K</sup>* <sup>¼</sup> *<sup>V</sup>*0ð Þ II

> ð Þ *K* ) *:*

*<sup>N</sup>* þ ð Þ Γ þ *δ K*Þ

for power plant II as:

(A.1)

for

*<sup>N</sup>* ð Þ *K* ,

(A.4)

*<sup>N</sup>*ðÞþ*<sup>t</sup> <sup>α</sup>*2*L<sup>F</sup>*

h i

Via the first order conditions, first obtain the optimal values *LR*

*<sup>N</sup>* <sup>¼</sup> *θα*<sup>1</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup>

*<sup>N</sup>* <sup>¼</sup> *θα*<sup>2</sup> <sup>þ</sup> *<sup>V</sup>*0ð Þ<sup>I</sup>

share hybrid-enabling technology.

erations.

**A. Appendix**

*Carbon Capture*

*<sup>ρ</sup>*1*V*ð Þ<sup>I</sup>

equation such that *V*ð Þ<sup>I</sup>

*<sup>N</sup>* ð Þ¼ *K* max *LR <sup>N</sup>*, *<sup>L</sup><sup>F</sup>* f g*<sup>N</sup>* <sup>≥</sup> <sup>0</sup>

> � 1 2 *β<sup>R</sup> L<sup>R</sup> N* � �<sup>2</sup>

þ *∂V*ð Þ<sup>I</sup> *<sup>N</sup>* ð Þ *K <sup>∂</sup><sup>K</sup> <sup>ϑ</sup>*<sup>1</sup> *<sup>L</sup><sup>R</sup>*

power plant I as:

where *<sup>∂</sup>V*ð Þ<sup>I</sup>

then obtain

*<sup>ρ</sup>*2*V*ð Þ II

**78**

*<sup>N</sup>* ð Þ *K <sup>∂</sup><sup>K</sup>* � *<sup>V</sup>*0ð Þ<sup>I</sup>

*<sup>N</sup>* ð Þ¼ *K* max *L*~*R <sup>N</sup>*, *<sup>L</sup>*~*<sup>F</sup> N*

> � 1 2 *β*~*<sup>R</sup> L*~*<sup>R</sup> N* � �<sup>2</sup>

þ *∂V*ð Þ<sup>I</sup> *<sup>N</sup>* ð Þ *K <sup>∂</sup><sup>K</sup> <sup>ϑ</sup>*<sup>1</sup> *<sup>L</sup><sup>R</sup>*

*<sup>N</sup>* ð Þ *K* :

� 1 2 *β<sup>F</sup> L<sup>F</sup> N* � �<sup>2</sup>

*LR*

*LF*

� � ð Þ <sup>1</sup> � *<sup>θ</sup> <sup>α</sup>*1*L<sup>R</sup>*

� 1 2 *β*~*<sup>F</sup> L*~*<sup>F</sup> N* � �<sup>2</sup>

conditions, we obtain the optimal values *L*~*<sup>R</sup>*

*<sup>N</sup>*, *L<sup>F</sup> N* � � <sup>þ</sup> *<sup>ϑ</sup>*<sup>2</sup> *<sup>L</sup>*~*<sup>R</sup>*

*θ α*1*L<sup>R</sup>*

*<sup>N</sup>*, *L<sup>F</sup> N* � � <sup>þ</sup> *<sup>ϑ</sup>*<sup>2</sup> *<sup>L</sup>*~*<sup>R</sup>*

$$\begin{split} \dot{\Phi}\_{1} &= \left( a\_{1}\theta - \frac{(\theta a\_{1} + a\_{1}\theta\_{1}^{R})}{2} + \theta\_{1}^{R}a\_{1} \right) \left( \frac{\theta a\_{1} + a\_{1}\theta\_{1}^{R}}{\hat{\rho}^{R}} \right) \\ &+ \left( a\_{2}\theta - \frac{(\theta a\_{2} + a\_{1}\theta\_{1}^{F})}{2} + \theta\_{1}^{F}a\_{1} \right) \left( \frac{\theta a\_{2} + a\_{1}\theta\_{1}^{F}}{\hat{\rho}^{F}(1 - a\_{2})} \right) \\ &+ \left( \beta\_{1}\theta + \theta\_{2}^{R}a\_{1} \right) \left( \frac{(1 - \theta)\beta\_{1} + a\_{2}\theta\_{2}^{R}}{\hat{\rho}^{R}} \right) + \left( \theta\_{2}\theta + \theta\_{2}^{F}a\_{1} \right) \left( \frac{(1 - \theta)\beta\_{2} + a\_{2}\theta\_{2}^{F}}{\hat{\rho}^{F}} \right) > 0. \end{split} \tag{A.8}$$

and

$$\begin{split} \dot{\Phi}\_{2} &= \left( (1-\theta)a\_{1} + \theta\_{1}^{R}a\_{2} \right) \frac{\left( \theta a\_{1} + a\_{1}\theta\_{1}^{R} \right)}{\rho^{R}(1-a\_{1})} \\ &+ \left( (1-\theta)a\_{2} + \theta\_{1}^{F}a\_{2} \right) \frac{\left( \theta a\_{2} + a\_{1}\theta\_{1}^{F} \right)}{\rho^{F}(1-a\_{2})} \\ &+ \left( (1-\theta)\beta\_{1} - \frac{\left( (1-\theta)\beta\_{1} + a\_{2}\theta\_{2}^{R} \right)}{2} + \theta\_{2}^{R}a\_{2} \right) \left( \frac{(1-\theta)\beta\_{1} + a\_{2}\theta\_{2}^{R}}{\bar{\rho}^{R}} \right) \\ &+ \left( (1-\theta)\beta\_{2} - \frac{\left( (1-\theta)\beta\_{2} + a\_{2}\theta\_{2}^{F} \right)}{2} + \theta\_{2}^{F}a\_{2} \right) \left( \frac{(1-\theta)\beta\_{2} + a\_{2}\theta\_{2}^{F}}{\bar{\rho}^{F}} \right) > 0. \end{split} \tag{A.9}$$

Substituting *a*1&*a*<sup>2</sup> into Eqs. (A.2), (A.3), (A.5) and (A.6) and simplifying, obtain the optima effort level of technological improvements. By substituting the above results into Eqs. (A.1) and (A.4) we obtain the optimal sharing payoff functions, for power plant (I) and power plant (II).

*Carbon Capture*
