Impact of policy incentives on electric vehicles development: a system dynamics-based evolutionary game theoretical analysis

Abstract

A system dynamics-based evolutionary game theoretical analysis is proposed to examine the impact of policy incentives, i.e., price subsidy and taxation preference on electric vehicles (EVs) industry development. Two case scenarios were used to distinguish policy performance by dividing it into a static and dynamic incentive. The result reflected that the game in implementation of the static incentive policy did not achieve stable equilibrium, indicating that such a policy is not effective for driving the development of the EVs industry. However, the game had stable equilibrium when dynamic incentive policy was implemented. The taxation preference had better performance in incentivizing EVs production than the direct subsidy. The study is expected to provide insight into policy making in the industrial transition toward low-carbon consumption. Limitations are given to indicate opportunities for further research.

Graphical abstract

Appendix

Evolutionary equilibrium stability analysis

• (1) Scenario 1

The stability analysis is to verify the SD simulation. By substituting the original values (Table 3) in Eq. (13), the replicated dynamic equations under the static policies are obtained as follows:

$$\left\{ {\begin{array}{*{20}l} {F\left( x \right) = \frac{{{\text{d}}x}}{{{\text{d}}t}} = x\left( {1 - x} \right)\left( {0.47y - 0.21} \right)} \hfill \\ {F\left( y \right) = \frac{{{\text{d}}y}}{{{\text{d}}t}} = y\left( {1 - y} \right)\left( {0.2 - 0.42x} \right)} \hfill \\ \end{array} } \right.$$

(15)

Let X = [F(x) F(y)] = 0; the equilibrium points of the game are:

$$X_{1} = \, \left( {0, \, 0} \right), \, X_{2} = \, \left( {0, \, 1} \right), \, X_{3} = \, \left( {1, \, 0} \right), \, X_{4} = \, \left( {1, \, 1} \right), \, X_{5} = \left( {\frac{10}{21},\;\frac{21}{47}} \right)$$

The stability of equilibrium strategy is derived from the Jacobian matrix. Any equilibrium point that satisfies detJ > 0 and trJ < 0 is considered as asymptotically stable, which is deemed as an evolutionary stable strategy (Weinstein 1986). The Jacobian matrix J is given as follows:

$$J = \left[ {\begin{array}{*{20}c} {\frac{\partial F\left( x \right)}{\partial x}} & {\frac{\partial F\left( x \right)}{\partial y}} \\ {\frac{\partial F\left( y \right)}{\partial x}} & {\frac{\partial F\left( y \right)}{\partial y}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\left( {1 - 2x} \right)\left( {0.47y - 0.21} \right)} & {0.47x\left( {1 - x} \right)} \\ { - \,0.42y\left( {1 - y} \right)} & {\left( {1 - 2y} \right)\left( {0.2 - 0.42x} \right)} \\ \end{array} } \right]$$

(16)

The stability of the five strategic pairs derived from the Scenario 1 is given in Table 4. There are four unstable equilibrium points and one center point, indicating that no evolutionary stable strategy(ESS) exists.

Table 4 Stability analysis for the equilibrium points under the situation of static policies

Figure 8 shows the evolutionary game process under the implementation of the static policy incentives. Such process shows a periodic circle, indicating that enterprises and consumers may be easily impacted by the policies to adjust their strategies. This phenomenon has verified the SD simulation results of the Scenario 1.

Fig. 8

Evolutionary game process under the static policy incentives

• (2) Scenario 2

The simulation results of the Scenario 2 indicate that the dynamic incentive policies have better performance than that of the static ones. The replicated dynamic equations and the corresponding Jacobian matrix under the dynamic incentive policies are obtained as follows:

• i. The dynamic subsidy to enterprises

The replicated dynamic equation set is obtained by substituting $W_{\text{e}}^{\prime }$ for $W_{\text{e}}$ in Eq. (13).

$$\left\{ {\begin{array}{*{20}l} {F\left( x \right) = \frac{{{\text{d}}x}}{{{\text{d}}t}} = x\left( {1 - x} \right)\left[ {y\left( {C_{g} + C_{n} + \varPi_{\text{e}}^{g} + \varPi_{\text{e}}^{c} + W_{\text{e}}^{\prime } + T_{\text{e}} } \right) - \left( {\varPi_{\text{e}}^{c} + C_{g} } \right)} \right]} \hfill \\ {F\left( { y} \right) = \frac{{{\text{d}}y}}{{{\text{d}}t}} = y\left( {1 - y} \right)\left[ {x\left( {U_{\text{c}}^{g} + W_{\text{c}} + U_{\text{c}}^{n} - R_{n} - R_{g} } \right) + \left( {R_{n} - U_{\text{c}}^{n} } \right)} \right] } \hfill \\ \end{array} } \right.$$

(17)

Consequently, the equilibrium are obtained as follows:

$$X_{1}^{\prime } = \left( {\begin{array}{*{20}c} { 0 } \\ 0 \\ \end{array} } \right),\;X_{2}^{\prime } = \left( {\begin{array}{*{20}c} { 0 } \\ 1 \\ \end{array} } \right),\;X_{3}^{\prime } = \left( {\begin{array}{*{20}c} { 1 } \\ 0 \\ \end{array} } \right),\;X_{4}^{\prime } = \left( {\begin{array}{*{20}c} { 1 } \\ 1 \\ \end{array} } \right),\;X_{5}^{\prime } = \left( {\begin{array}{*{20}c} { x } \\ y \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} { \frac{{U_{\text{c}}^{n} - R_{n} }}{{U_{\text{c}}^{g} + W_{\text{c}} + U_{\text{c}}^{n} - R_{n} - R_{g} }} } \\ {\frac{{\varPi_{\text{e}}^{c} + C_{g} }}{{C_{g} + C_{n} + \varPi_{\text{e}}^{g} + \varPi_{\text{e}}^{c} + W_{\text{e}}^{\prime } + T_{\text{e}} }}} \\ \end{array} } \right)$$

Similarly, the corresponding Jacobian matrix J is:

$$J_{1}^{\prime } = \left[ {\begin{array}{*{20}c} {\frac{\partial F\left( x \right)}{\partial x}} & {\frac{\partial F\left( x \right)}{\partial y}} \\ {\frac{\partial F\left( y \right)}{\partial x}} & {\frac{\partial F\left( y \right)}{\partial y}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\left( {1 - 2x} \right)\left( {yA - B} \right) - xy\left( {1 - x} \right)C} & {x\left( {1 - x} \right)A} \\ {y\left( {1 - y} \right)D} & {\left( {1 - 2y} \right)\left( {xD + E} \right)} \\ \end{array} } \right]$$

(18)

where $A = C_{g} + C_{n} + \varPi_{\text{e}}^{g} + \varPi_{\text{e}}^{c} + W_{\text{e}}^{\prime } + T_{\text{e}}$; B = $\varPi_{\text{e}}^{c} + C_{g}$; C = $W_{\text{e}}$; $D = U_{\text{c}}^{g} + W_{\text{c}} + U_{\text{c}}^{n} - R_{n} - R_{g}$; E = $R_{n} - U_{\text{c}}^{n}$.

• ii. The dynamic subsidy to consumers

The replicated dynamic equation set is obtained by substituting $W_{\text{c}}^{\prime }$ for $W_{\text{c}}$ in Eq. (13).

$$\left\{ {\begin{array}{*{20}l} {F\left( x \right) = \frac{{{\text{d}}x}}{{{\text{d}}t}} = x\left( {1 - x} \right)\left[ {y\left( {C_{g} + C_{n} + \varPi_{\text{e}}^{g} + \varPi_{\text{e}}^{c} + W_{\text{e}} + T_{\text{e}} } \right) - \left( {\varPi_{\text{e}}^{c} + C_{g} } \right)} \right] } \hfill \\ {F\left( { y} \right) = \frac{{{\text{d}}y}}{{{\text{d}}t}} = y\left( {1 - y} \right)\left[ {x\left( {U_{\text{c}}^{g} + W_{\text{c}}^{\prime } + U_{\text{c}}^{n} - R_{n} - R_{g} } \right) + \left( {R_{n} - U_{\text{c}}^{n} } \right)} \right] } \hfill \\ \end{array} } \right.$$

(19)

Consequently, the equilibrium are obtained as follows:

$$X_{1}^{c} = \left( {\begin{array}{*{20}c} { 0 } \\ 0 \\ \end{array} } \right),\;X_{2}^{c} = \left( {\begin{array}{*{20}c} { 0 } \\ 1 \\ \end{array} } \right),\;X_{3}^{c} = \left( {\begin{array}{*{20}c} { 1 } \\ 0 \\ \end{array} } \right),\;X_{4}^{c} = \left( {\begin{array}{*{20}c} { 1 } \\ 1 \\ \end{array} } \right),\;X_{5}^{c} = \left( {\begin{array}{*{20}c} { x } \\ y \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} { \frac{{U_{\text{c}}^{n} - R_{n} }}{{U_{\text{c}}^{g} + W_{\text{c}}^{\prime } + U_{\text{c}}^{n} - R_{n} - R_{g} }} } \\ {\frac{{\varPi_{\text{e}}^{c} + C_{g} }}{{C_{g} + C_{n} + \varPi_{\text{e}}^{g} + \varPi_{\text{e}}^{c} + W_{\text{e}} + T_{\text{e}} }}} \\ \end{array} } \right)$$

Similarly, the corresponding Jacobian matrix J is:

$$J_{2}^{\prime } = \left[ {\begin{array}{*{20}c} {\frac{\partial F\left( x \right)}{\partial x}} & {\frac{\partial F\left( x \right)}{\partial y}} \\ {\frac{\partial F\left( y \right)}{\partial x}} & {\frac{\partial F\left( y \right)}{\partial y}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\left( {1 - 2x} \right)\left( {yA^{c} - B^{c} } \right)} & {x\left( {1 - x} \right)A^{c} } \\ {y\left( {1 - y} \right)D^{c} } & {\left( {1 - 2y} \right)\left( {xD^{c} + E^{c} } \right) - xy\left( {1 - y} \right)C^{c} } \\ \end{array} } \right]$$

(20)

where $A^{c} = C_{g} + C_{n} + \varPi_{\text{e}}^{g} + \varPi_{\text{e}}^{c} + W_{\text{e}} + T_{\text{e}}$; $B^{c} = \varPi_{\text{e}}^{c} + C_{g}$; $C^{c} = W_{\text{c}}$; $D^{c} = U_{\text{c}}^{g} + W_{\text{c}}^{\prime } + U_{\text{c}}^{n} - R_{n} - R_{g}$; $E^{c} = R_{n} - U_{\text{c}}^{n}$.

• iii. The dynamic preferential tax on enterprises

The replicated dynamic equation set is obtained by substituting $T_{\text{e}}^{\prime }$ for $T_{\text{e}}$ in Eq. (13).

$$\left\{ {\begin{array}{*{20}l} {F\left( x \right) = \frac{{{\text{d}}x}}{{{\text{d}}t}} = x\left( {1 - x} \right)\left[ {y\left( {C_{g} + C_{n} + \varPi_{\text{e}}^{g} + \varPi_{\text{e}}^{c} + W_{\text{e}} + T_{\text{e}}^{\prime } } \right) - \left( {\varPi_{\text{e}}^{c} + C_{g} } \right)} \right] } \hfill \\ {F\left( { y} \right) = \frac{{{\text{d}}y}}{{{\text{d}}t}} = y\left( {1 - y} \right)\left[ {x\left( {U_{\text{c}}^{g} + W_{\text{c}} + U_{\text{c}}^{n} - R_{n} - R_{g} } \right) + \left( {R_{n} - U_{\text{c}}^{n} } \right)} \right] } \hfill \\ \end{array} } \right.$$

(21)

Consequently, the equilibrium are obtained as follows:

$$X_{1}^{\text{T}} = \left( {\begin{array}{*{20}c} { 0 } \\ 0 \\ \end{array} } \right),\;X_{2}^{\text{T}} = \left( {\begin{array}{*{20}c} { 0 } \\ 1 \\ \end{array} } \right),\;X_{3}^{\text{T}} = \left( {\begin{array}{*{20}c} { 1 } \\ 0 \\ \end{array} } \right),\;X_{4}^{\text{T}} = \left( {\begin{array}{*{20}c} { 1 } \\ 1 \\ \end{array} } \right),\;X_{5}^{\text{T}} = \left( {\begin{array}{*{20}c} { x } \\ y \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} { \frac{{U_{\text{c}}^{n} - R_{n} }}{{U_{\text{c}}^{g} + W_{\text{c}} + U_{\text{c}}^{n} - R_{n} - R_{g} }} } \\ {\frac{{\varPi_{\text{e}}^{c} + C_{g} }}{{C_{g} + C_{n} + \varPi_{\text{e}}^{g} + \varPi_{\text{e}}^{c} + W_{\text{e}} + T_{\text{e}}^{\prime } }}} \\ \end{array} } \right)$$

Similarly, the corresponding Jacobian matrix J is:

$$J_{3}^{\prime } = \left[ {\begin{array}{*{20}c} {\frac{\partial F\left( x \right)}{\partial x}} & {\frac{\partial F\left( x \right)}{\partial y}} \\ {\frac{\partial F\left( y \right)}{\partial x}} & {\frac{\partial F\left( y \right)}{\partial y}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\left( {1 - 2x} \right)\left( {yA^{T} - B^{T} } \right) - xy\left( {1 - x} \right)C^{T} } & {x\left( {1 - x} \right)A^{T} } \\ {y\left( {1 - y} \right)D^{T} } & {\left( {1 - 2y} \right)\left( {xD^{T} + E^{T} } \right)} \\ \end{array} } \right]$$

(22)

where $A^{\text{T}} = C_{g} + C_{n} + \varPi_{\text{e}}^{g} + \varPi_{\text{e}}^{c} + W_{\text{e}} + T_{\text{e}}^{\prime }$; $B^{\text{T}} = \varPi_{\text{e}}^{c} + C_{g}$; $C^{\text{T}} = T_{\text{e}}$; $D^{\text{T}} = U_{\text{c}}^{g} + W_{\text{c}} + U_{\text{c}}^{n} - R_{n} - R_{g}$; $E^{\text{T}} = R_{n} - U_{\text{c}}^{n}$.

• iv. The combination of dynamic policy incentives

The replicated dynamic equation set is obtained by substituting $W_{\text{e}}^{\prime }$, $W_{\text{c}}^{\prime }$ and $T_{\text{e}}^{\prime }$ for $W_{\text{e}}$, $W_{\text{c}}$ and $T_{\text{e}}$ in Eq. (13), respectively.

$$\left\{ {\begin{array}{*{20}l} {F\left( x \right) = \frac{{{\text{d}}x}}{{{\text{d}}t}} = x\left( {1 - x} \right)\left[ {y\left( {C_{g} + C_{n} + \varPi_{\text{e}}^{g} + \varPi_{\text{e}}^{c} + W_{\text{e}}^{\prime } + T_{\text{e}}^{\prime } } \right) - \left( {\varPi_{\text{e}}^{c} + C_{g} } \right)} \right] } \hfill \\ {F\left( { y} \right) = \frac{{{\text{d}}y}}{{{\text{d}}t}} = y\left( {1 - y} \right)\left[ {x\left( {U_{\text{c}}^{g} + W_{\text{c}}^{\prime } + U_{\text{c}}^{n} - R_{n} - R_{g} } \right) + \left( {R_{n} - U_{\text{c}}^{n} } \right)} \right] } \hfill \\ \end{array} } \right.$$

(23)

Consequently, the equilibrium are obtained as follows:

$$X_{1}^{\theta } = \left( {\begin{array}{*{20}c} { 0 } \\ 0 \\ \end{array} } \right),\;X_{2}^{\theta } = \left( {\begin{array}{*{20}c} { 0 } \\ 1 \\ \end{array} } \right),\;X_{3}^{\theta } = \left( {\begin{array}{*{20}c} { 1 } \\ 0 \\ \end{array} } \right),\;X_{4}^{\theta } = \left( {\begin{array}{*{20}c} { 1 } \\ 1 \\ \end{array} } \right),\;X_{5}^{\theta } = \left( {\begin{array}{*{20}c} { x } \\ y \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} { \frac{{U_{\text{c}}^{n} - R_{n} }}{{U_{\text{c}}^{g} + W_{\text{c}}^{\prime } + U_{\text{c}}^{n} - R_{n} - R_{g} }} } \\ {\frac{{\varPi_{\text{e}}^{c} + C_{g} }}{{C_{g} + C_{n} + \varPi_{\text{e}}^{g} + \varPi_{\text{e}}^{c} + W_{\text{e}}^{\prime } + T_{\text{e}}^{\prime } }}} \\ \end{array} } \right)$$

Similarly, the corresponding Jacobian matrix J is:

$$J_{4}^{\prime } = \left[ {\begin{array}{*{20}c} {\frac{\partial F\left( x \right)}{\partial x}} & {\frac{\partial F\left( x \right)}{\partial y}} \\ {\frac{\partial F\left( y \right)}{\partial x}} & {\frac{\partial F\left( y \right)}{\partial y}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\left( {1 - 2x} \right)\left( {yA^{\theta } - B^{\theta } } \right) - xy\left( {1 - x} \right)C^{\theta } } & {x\left( {1 - x} \right)A^{\theta } } \\ {y\left( {1 - y} \right)D^{\theta } } & {\left( {1 - 2y} \right)\left( {xD^{\theta } + E^{\theta } } \right) - xy\left( {1 - y} \right)F^{\theta } } \\ \end{array} } \right]$$

(24)

where $A^{\theta } = C_{g} + C_{n} + \varPi_{\text{e}}^{g} + \varPi_{\text{e}}^{c} + W_{\text{e}}^{\prime } + T_{\text{e}}^{\prime }$; $B^{\theta } = \varPi_{\text{e}}^{c} + C_{g}$; $C^{\theta } = W_{\text{e}} + T_{\text{e}}$; $D^{\theta } = U_{\text{c}}^{g} + W_{\text{c}}^{\prime } + U_{\text{c}}^{n} - R_{n} - R_{g}$; $E^{\theta } = R_{n} - U_{\text{c}}^{n}$; $F^{\theta } = W_{\text{c}}$.

The stability of the five strategic pairs under the different dynamic incentive policies is given in Table 5. There is an ESS existed (x, y), which verifies the simulation results of the Scenario 2. Figure 9 shows evolutionary process of the game under the different dynamic incentive policies. As the rounds of the game increase, the trend of the curves gradually reaches an equilibrium point, which indicates that the game has asymptotic stability under the dynamic incentive policies.

Table 5 Stability analysis for the equilibrium points under the dynamic incentive policies

Fig. 9

Players’ behaviors in evolutionary game with the different dynamic incentive policies