Production decisions of power plants under endogenous hybrid/enabling technological advances

2. Model setup

2.2 Production decisions of power plants under endogenous hybrid/enabling technological advances

Both players will undertake R&D measures on hybrid-enabling technology to ensure immediate reliability and affordability in energy production whilst reducing GHG-emissions. We assume that the strategic effects implemented by power plant I (Player I), has improved hybrid-enabling technology to generate energy and utilize energy sources in a much efficient way. This gives a superior advantage to power plant I overpower plant II (Player II) and both power plants are rational to maxi- mize their profits. Although Power plant II has heterogeneous resources to hybrid- enabling technology, from a practical point of view it is logical for power plant I to share this technology with power plant II, because the price competition is typically characterized by a second-mover advantage. Many researchers have investigated the effects of these commitments in Cournot, Bertrand and Stackelberg setups. See [29–31]. Due to the government incentives, tariff-rate quota, feed-in-tariff and R&D incentive measures, the power companies will be competitive to improve their efficiency. Let L^Rð Þt denotes the R&D effort level of technological improve- ments on renewable sources at time t, and L^Fð Þt denotes the R&D effort level of technological improvements on fossil fuel at time t, of Player I.L~^Rð Þt denotes the R&D effort level of technological improvements on renewable sources at time t, and L~^Fð Þt denotes the R&D effort level of technological improvements on fossil fuel at time t, of Player II. For, further consideration, the sharing cost of advanced hybrid- enabling technology (Player I) and inferior hybrid-enabling technology (Player II) is denoted as C_Ið Þt and C_IIð Þ, which are the quadratic functions of the effect level oft Player I and Player II at time t, respectively. Consider

CI�L^Rð Þ, Lt ^Fð Þ, tt �

¼1

2 β^Rð Þt�L^Rð Þt �2

þβ^Fð Þt�L^Fð Þt �2

� �

, (17)

and

CII�~L^Rð Þ,t ~L^Fð Þ, tt �

¼1

2 ~β^Rð Þt �~L^Rð Þt�₂

þ~β^Fð Þt �~L^Fð Þt �₂

� �

, (18)

where 0< β� ^Rð Þ,t β^Fð Þ,t ~β^Rð Þ,t ~β^Fð Þt�

≤1 and lower the�β^Rð Þ,t β^Fð Þ,t ~β^Rð Þ,t ~β^Fð Þt� , more effective is the technological development.

Let K tð Þdenote the evolution of the hybrid-enabling technology at time t, due to R&D collaborative innovation system of Player I and Player II at time t. The dynam- ics of hybrid-technology is governed by the stochastic differential equation (SDE):

dK tð Þ ¼ ϑ₁ð Þt�L^Rð Þ, Lt ^Fð Þt�

þϑ₂ð Þt�L~^Rð Þ,t L~^Fð Þt�

�ξK tð Þ

h i

dtþφ ffiffiffiffi pK

dW tð Þ K 0ð Þ ¼K₀>0:

8< :

(19)

Then via equationΠ_ij^∗ ¼β_ij�Λ_ij^∗�2

�Fi¼β_ij�θ_ijþω_ijτ_iþχ_ijτ_j�2

�Fi, in Proposition 1, and the payoff matrix of the power plant I is given by:

A¼ a₁₁ a₁₂ a21 a22

� �

¼ ΠF,F ΠF,R

ΠR,F ΠR,R

� �

¼ β_F,FΛ²_F,F�FF β_F,RΛ²_F,R�FF

β_R,FΛ²_R,F�F_R β_R,RΛ²_R,R�F_R

" #

: (9)

Obviously the bimatrix of the power plant II, is given by:

A¼ a11 a21

a₁₂ a₂₂

� �

¼ ΠF,F ΠR,F

ΠF,R ΠR,R

� �

¼ β_F,FΛ²_F,F�FF β_R,FΛ²_R,F�FR

β_F,RΛ²_F,R�F_F β_R,RΛ²_R,R�F_R

" #

: (10)

Proposition 3. The Nash equilibrium for the Bi-matrix game G, is given as ΠR,R�ΠF,R

ð Þ

ΠF,F�ΠR,F�ΠFRþΠR,R

ð Þ, ðΠR,R�ΠF,RÞ

ΠF,F�ΠF,R�ΠR,FþΠR,R

ð Þ

� �

: (11)

Proof. Suppose players I and II use mixed strategies (x,1-x) and (y,1-y), respectively, where

i. The probability that player I choosing row 1 is x and the probability that player I choosing row 2 is 1-x.

ii. The probability that player II choosing row 1 is y and the probability that player II choosing row 2 is 1-y.

Then the value of the game for Player I is

v1ðx, yÞ ¼xyðΠF,FÞ þx 1ð �yÞðΠF,RÞ þð1�xÞyðΠR,FÞ þð1�xÞð1�yÞðΠR,RÞ

¼ððΠF,F�ΠF,R�ΠR,FþΠR,RÞyþðΠF,R�ΠR,RÞÞxþððΠR,F�ΠR,RÞyþΠR,RÞ, (12) and the value of the game for Player II is

v2ðx, yÞ ¼xyðΠF,FÞ þx 1ð �yÞðΠR,FÞ þð1�xÞyðΠF,RÞ þð1�xÞð1�yÞðΠR,RÞ

¼ððΠF,F�ΠR,F�ΠF,RþΠR,RÞxþðΠF,R�ΠR,RÞÞyþððΠR,FR�ΠR,RÞxþΠR,RÞ: (13) Suppose (X, Y) yields a Nash equilibrium. Then for the given payoffs having 0<x<1 implies that

v1¼ðΠF,F�ΠF,R�ΠR,FþΠR,RÞyþðΠF,R�ΠR,RÞ ¼0: (14) Otherwise Player I can change x slightly and do better.

Similarly, for 0<y<1,

v₂¼ðΠF,F�ΠR,F�ΠFRþΠR,RÞxþðΠF,R�ΠR,RÞ ¼0: (15) Otherwise Player II can change y slightly and do better. It follows that the unique Nash equilibrium (x,y), has

ΠR,R�ΠF,R

ð Þ

ΠF,F�ΠR,F�ΠFRþΠR,R

ð Þ, ðΠR,R�ΠR,FÞ

ΠF,F�ΠF,R�ΠR,FþΠR,R

ð Þ

� �

: (16)

Remark 1. Since the power plants plays a symmetric two person bimatrix game G, having two pure strategiesΠF,F 6¼ΠR,F, ΠR,R6¼ΠFR, imply that G, has an evolutionary stable strategy. Then the Nash equilibrium is an outcome in which the strategy chosen by each player is the best reply to the strategy chosen by the other.

This best reply strategy yields the highest payoff to the player choosing it, given the strategy chosen by the co-player, [27, 28].

2.2 Production decisions of power plants under endogenous hybrid/enabling technological advances

Both players will undertake R&D measures on hybrid-enabling technology to ensure immediate reliability and affordability in energy production whilst reducing GHG-emissions. We assume that the strategic effects implemented by power plant I (Player I), has improved hybrid-enabling technology to generate energy and utilize energy sources in a much efficient way. This gives a superior advantage to power plant I overpower plant II (Player II) and both power plants are rational to maxi- mize their profits. Although Power plant II has heterogeneous resources to hybrid- enabling technology, from a practical point of view it is logical for power plant I to share this technology with power plant II, because the price competition is typically characterized by a second-mover advantage. Many researchers have investigated the effects of these commitments in Cournot, Bertrand and Stackelberg setups. See [29–31]. Due to the government incentives, tariff-rate quota, feed-in-tariff and R&D incentive measures, the power companies will be competitive to improve their efficiency. Let L^Rð Þt denotes the R&D effort level of technological improve- ments on renewable sources at time t, and L^Fð Þt denotes the R&D effort level of technological improvements on fossil fuel at time t, of Player I.L~^Rð Þt denotes the R&D effort level of technological improvements on renewable sources at time t, and

~L^Fð Þt denotes the R&D effort level of technological improvements on fossil fuel at time t, of Player II. For, further consideration, the sharing cost of advanced hybrid- enabling technology (Player I) and inferior hybrid-enabling technology (Player II) is denoted as C_Ið Þt and C_IIð Þ, which are the quadratic functions of the effect level oft Player I and Player II at time t, respectively. Consider

CI�L^Rð Þ, Lt ^Fð Þ, tt �

¼1

2 β^Rð Þt�L^Rð Þt�2

þβ^Fð Þt �L^Fð Þt�2

� �

, (17)

and

CII�L~^Rð Þ,t L~^Fð Þ, tt �

¼1

2 ~β^Rð Þt�L~^Rð Þt �₂

þβ~^Fð Þt�L~^Fð Þt�₂

� �

, (18)

where 0< β� ^Rð Þ,t β^Fð Þ,t ~β^Rð Þ,t ~β^Fð Þt �

≤1 and lower the�β^Rð Þ,t β^Fð Þ,t ~β^Rð Þ,t ~β^Fð Þt � , more effective is the technological development.

Let K tð Þdenote the evolution of the hybrid-enabling technology at time t, due to R&D collaborative innovation system of Player I and Player II at time t. The dynam- ics of hybrid-technology is governed by the stochastic differential equation (SDE):

dK tð Þ ¼ ϑ₁ð Þt�L^Rð Þ, Lt ^Fð Þt �

þϑ₂ð Þt �~L^Rð Þ,t L~^Fð Þt�

�ξK tð Þ

h i

dtþφ ffiffiffiffi pK

dW tð Þ K 0ð Þ ¼K₀>0:

8<

:

(19)

ξ∈ð0, 1�, is the attenuation coefficient of hybrid-enabling technology. Let ϑ₁ð Þ ¼t �ϑ^R₁ð Þ þt ϑ^F₁ð Þt�

andϑ₂ð Þ ¼t �ϑ^R₂ð Þ þt ϑ^F₂ð Þt�

denote the influence of the effort level of hybrid-enabling technology sharing on collaboration innovation between Player I and Player II, at time t. W tð Þis a standard Brownian motion andφ ffiffiffiffi

pK ð Þt

� �

random interference factor on hybrid-enabling technology.

LetΠð Þt denotes the total profit under the hybrid-enabling technology system at time t. Letðα₁ð Þt ,α₂ð ÞtÞandðβ₁ð Þt,β₂ð Þt Þdenote the influence of the effort level hybrid-enabling technology on the total profit of Player I and player II, respectively, at time t, namely, the marginal return coefficient of hybrid-enabling technology.

Total profit function can be expressed as:

Πð Þ ¼t �α₁ð ÞLt ^Rð Þ þt α₂ð ÞLt ^Fð Þt �

þ�β₁ð Þt L~^Rð Þ þt β₂ð ÞtL~^Fð Þt�

þðΓþδÞK tð Þ, (20) where

α₁ð Þ ¼t ΠR,Rð Þt

ΠF,Fð Þ �t ΠR,Fð Þ �t ΠF,Rð Þ þt ΠR,Rð Þt , (21) α₂ð Þ ¼t �ΠF,Rð Þt

ΠF,Fð Þ �t ΠR,Fð Þ �t ΠF,Rð Þ þt ΠR,Rð Þt , Γ ¼Γð ÞI þΓð ÞII,δ¼δ_{ð ÞI} þδ_{ð ÞII}, and

(22)

β₁ð Þ ¼t ΠR,Rð Þt

ΠF,Fð Þ �t ΠR,Fð Þ �t ΠF,Rð Þ þt ΠR,Rð Þt , (23) β₂ð Þ ¼t �ΠR,Fð Þt

ΠF,Fð Þ �t ΠR,Fð Þ �t ΠF,Rð Þ þt ΠR,Rð Þt : (24) Γis the influence of the hybrid-enabling technology innovation on total revenue δ∈ð0, 1�;δis the total government subsidy coefficient of hybrid-enabling technol- ogy based on increments of advances in hybrid-enabling technology.

Proposition 4. At least one of the Power Plants has a second mover advantage.

Proof. Demand function D_ij�p_ij, p_ji�

>0, given by Eq. (4), is twice continuously differentiable and

∂Dij�p_ij, p_ji�

∂p_ij ¼ �β_ij<0, and

∂Dij�p_ij, p_ji�

∂p_ji ¼γ_ij>0∀�p_ij, p_ji�

∈P_I�P_II: (25)

The first inequality says that each demand is downward sloping in own price, and the second that goods are substitutes (each demand increases with the price of the other good). [32] shows that in case of symmetric firms, there is a second-mover (first-mover) advantage for both players when each profit function is strictly con- cave in own action and strictly increasing (decreasing) in rival’s action, and reaction curves are upward (downward) sloping.

Then a sufficient condition on the super-modularity of the profit function is obtained via the profit functionΠij, given by Eq. (4):

∂Dij�p_ij, p_ji�

∂p_ji þ p_ij�Ci�vi

� �∂²Dij�p_ij, p_ji�

∂p_jip_ij 2

4

3

5E K tð ð ÞÞ>0, (26)

where E is the expectations. The main implication of this is that it leads to reaction correspondences that are non-decreasing (in the sense that each selection is non-decreasing) but need not be single-valued or continuous. This has a very appealing and precise interpretation: The price elasticity of Power Plant i’s demand increases in the rival’s price, [33]. This is a very intuitive and general condition, though clearly not a universal one. It is satisfied in particular if^∂2Dijð^{pij, pji}Þ

∂p_jip_ij >0, if a higher price by a Power Plant’s rival does not lower the responsiveness of the Power plant’s demand to a change in own price.

We further assume that the total revenue is allocated between two players and θð Þt is the payoff distribution coefficient of player I at time t andθð Þt ∈½0, 1�.

Although Player II has heterogeneous resources of hybrid-enabling technology, Player I can produce electricity more efficiently with lower GHG-emission, ensure immediate reliability and affordability in energy production. Then Player II, can acquire practical outcomes of this hybrid-enabling technological advances. To pro- mote the hybrid-enabling technology, Player II (leader) determine an optimal shar- ing effort level and an optimal subsidy. Then Player I (follower) choose their optimal sharing effort level according to the optimal sharing effort level and sub- sidy. This leads to a Stackelberg equilibrium. Letωð Þ ¼t ðω₁ð Þ,t ω₂ð ÞtÞ, denote the subsidy for hybrid-enabling technology, with Player II willing to pay to Payer I under collaboration. The objective functions of power plant I and power plant II satisfy the following partial differential equations

J_{ð ÞI}ðK0Þ

¼ max

L^R_S, L^F_S

f g^≥0E ð_∞

0e^�ρ1thθð Þt �α₁ð Þt L^Rð Þ þt α₂ð ÞtL^Fð Þ þt β₁ð Þt L~^Rð Þ þt β₂ð ÞtL~^Fð Þt

�

þðΓþδÞK tð ÞÞÞ �1

2β^Rð Þtð1�ω₁Þ�L^Rð Þt �2

�1

2β^Fð Þt ð1�ω₂Þ�L^Fð Þt�2� dt

� ,

(27) and

J_{ð ÞII}ðK₀Þ ¼ max

~L^RS,L~^FS,ωð Þt

� �

≥0

E ð_∞

0e^�ρ2thð1�θð Þt Þ�α₁ð ÞtL^Rð Þ þt α₂ð Þt L^Fð Þ þt β₁ð ÞtL~^Rð Þt

�

þβ₂ð Þt ~L^Fð Þ þt ðΓþδÞK tð Þ�

�1

2~β^Rð Þt�L^Rð Þt�2

�1

2β~^Fð Þt�L~^Fð Þt�₂

�1

2ω₁β^Rð Þt�L^Rð Þt �2

�1

2ω₂β^Fð Þt �L^Fð Þt�2� dt

� ,

(28) whereρ₁andρ₂are the discount rates of Player I and Player II, respectively. In this feedback control strategy L^R_Sð Þt ≥0, L^F_Sð Þt ≥0, L S R (t)≥0,L~^R_Sð Þt ≥0 and L~^F_Sð Þt ≥0, are the control variables andωð Þ ¼t ðω₁ð Þ,t ω₂ð Þt Þ∈ð0, 1Þ. K tð Þ>0 is the state variable. In feedback control process, it is assumed that players at every point in time have access to the current system and can make decisions accordingly to that state. Consequently, the players can respond to any disturbance in an optimal way. Hence, feedback strategies are robust for deviations and players can react to disturbances during the evolution of the game and adapt their actions accordingly, [34].

ξ∈ð0, 1�, is the attenuation coefficient of hybrid-enabling technology. Let ϑ₁ð Þ ¼t �ϑ^R₁ð Þ þt ϑ^F₁ð Þt�

andϑ₂ð Þ ¼t �ϑ^R₂ð Þ þt ϑ^F₂ð Þt�

denote the influence of the effort level of hybrid-enabling technology sharing on collaboration innovation between Player I and Player II, at time t. W tð Þis a standard Brownian motion andφ ffiffiffiffi

pK ð Þt

� �

random interference factor on hybrid-enabling technology.

LetΠð Þt denotes the total profit under the hybrid-enabling technology system at time t. Letðα₁ð Þt,α₂ð ÞtÞandðβ₁ð Þt,β₂ð ÞtÞdenote the influence of the effort level hybrid-enabling technology on the total profit of Player I and player II, respectively, at time t, namely, the marginal return coefficient of hybrid-enabling technology.

Total profit function can be expressed as:

Πð Þ ¼t �α₁ð ÞLt ^Rð Þ þt α₂ð ÞLt ^Fð Þt�

þ�β₁ð Þt ~L^Rð Þ þt β₂ð Þt L~^Fð Þt �

þðΓþδÞK tð Þ, (20) where

α₁ð Þ ¼t ΠR,Rð Þt

ΠF,Fð Þ �t ΠR,Fð Þ �t ΠF,Rð Þ þt ΠR,Rð Þt , (21) α₂ð Þ ¼t �ΠF,Rð Þt

ΠF,Fð Þ �t ΠR,Fð Þ �t ΠF,Rð Þ þt ΠR,Rð Þt , Γ ¼Γð ÞI þΓð ÞII,δ¼δ_{ð ÞI} þδ_{ð ÞII}, and

(22)

β₁ð Þ ¼t ΠR,Rð Þt

ΠF,Fð Þ �t ΠR,Fð Þ �t ΠF,Rð Þ þt ΠR,Rð Þt , (23) β₂ð Þ ¼t �ΠR,Fð Þt

ΠF,Fð Þ �t ΠR,Fð Þ �t ΠF,Rð Þ þt ΠR,Rð Þt : (24) Γis the influence of the hybrid-enabling technology innovation on total revenue δ∈ð0, 1�;δis the total government subsidy coefficient of hybrid-enabling technol- ogy based on increments of advances in hybrid-enabling technology.

Proposition 4. At least one of the Power Plants has a second mover advantage.

Proof. Demand function D_ij�p_ij, p_ji�

>0, given by Eq. (4), is twice continuously differentiable and

∂Dij�p_ij, p_ji�

∂p_ij ¼ �β_ij<0, and

∂Dij�p_ij, p_ji�

∂p_ji ¼γ_ij>0∀�p_ij, p_ji�

∈P_I�P_II: (25)

The first inequality says that each demand is downward sloping in own price, and the second that goods are substitutes (each demand increases with the price of the other good). [32] shows that in case of symmetric firms, there is a second-mover (first-mover) advantage for both players when each profit function is strictly con- cave in own action and strictly increasing (decreasing) in rival’s action, and reaction curves are upward (downward) sloping.

Then a sufficient condition on the super-modularity of the profit function is obtained via the profit functionΠij, given by Eq. (4):

∂Dij�p_ij, p_ji�

∂p_ji þ p_ij�Ci�vi

� �∂²Dij�p_ij, p_ji�

∂p_jip_ij 2

4

3

5E K tð ð ÞÞ>0, (26)

where E is the expectations. The main implication of this is that it leads to reaction correspondences that are non-decreasing (in the sense that each selection is non-decreasing) but need not be single-valued or continuous. This has a very appealing and precise interpretation: The price elasticity of Power Plant i’s demand increases in the rival’s price, [33]. This is a very intuitive and general condition, though clearly not a universal one. It is satisfied in particular if^∂2Dijð^{pij, pji}Þ

∂p_jip_ij >0, if a higher price by a Power Plant’s rival does not lower the responsiveness of the Power plant’s demand to a change in own price.

We further assume that the total revenue is allocated between two players and θð Þt is the payoff distribution coefficient of player I at time t andθð Þt ∈½0, 1�.

Although Player II has heterogeneous resources of hybrid-enabling technology, Player I can produce electricity more efficiently with lower GHG-emission, ensure immediate reliability and affordability in energy production. Then Player II, can acquire practical outcomes of this hybrid-enabling technological advances. To pro- mote the hybrid-enabling technology, Player II (leader) determine an optimal shar- ing effort level and an optimal subsidy. Then Player I (follower) choose their optimal sharing effort level according to the optimal sharing effort level and sub- sidy. This leads to a Stackelberg equilibrium. Letωð Þ ¼t ðω₁ð Þ,t ω₂ð ÞtÞ, denote the subsidy for hybrid-enabling technology, with Player II willing to pay to Payer I under collaboration. The objective functions of power plant I and power plant II satisfy the following partial differential equations

J_{ð ÞI}ðK0Þ

¼ max

L^R_S, L^F_S

f g^≥0E ð_∞

0e^�ρ1thθð Þt �α₁ð ÞtL^Rð Þ þt α₂ð Þt L^Fð Þ þt β₁ð ÞtL~^Rð Þ þt β₂ð Þt ~L^Fð Þt

�

þðΓþδÞK tð ÞÞÞ �1

2β^Rð Þtð1�ω₁Þ�L^Rð Þt�2

�1

2β^Fð Þt ð1�ω₂Þ�L^Fð Þt �2� dt

� ,

(27) and

J_{ð ÞII}ðK₀Þ ¼ max

L~^RS,~L^FS,ωð Þt

� �

≥0

E ð_∞

0e^�ρ2thð1�θð ÞtÞ�α₁ð Þt L^Rð Þ þt α₂ð ÞtL^Fð Þ þt β₁ð ÞtL~^Rð Þt

�

þβ₂ð ÞtL~^Fð Þ þt ðΓþδÞK tð Þ�

�1

2~β^Rð Þt �L^Rð Þt �2

�1

2~β^Fð Þt�L~^Fð Þt�₂

�1

2ω₁β^Rð Þt�L^Rð Þt�2

�1

2ω₂β^Fð Þt�L^Fð Þt�2� dt

� ,

(28) whereρ₁andρ₂are the discount rates of Player I and Player II, respectively. In this feedback control strategy L^R_Sð Þt ≥0, L^F_Sð Þt ≥0, L S R (t)≥0,L~^R_Sð Þt ≥0 and

~L^F_Sð Þt ≥0, are the control variables andωð Þ ¼t ðω₁ð Þ,t ω₂ð ÞtÞ∈ð0, 1Þ. K tð Þ>0 is the state variable. In feedback control process, it is assumed that players at every point in time have access to the current system and can make decisions accordingly to that state. Consequently, the players can respond to any disturbance in an optimal way. Hence, feedback strategies are robust for deviations and players can react to disturbances during the evolution of the game and adapt their actions accordingly, [34].