An evolutionary analysis of low-carbon technology investment strategies based on the manufacturer-supplier matching game under government regulations

Abstract

Developing a low-carbon economy is the only way for countries to achieve sustainable development. Carbon emission reduction policies and low-carbon technology (LCT) innovation play key roles in developing low-carbon economy. Under government reward and punishment regulations, based on the bilateral matching and evolutionary theories, this paper constructs an evolution model consisting of a manufacturer investing LCT and a supplier offering LCT to analyze multi-phase LCT investment strategies. Firstly, the profit optimization model of a green supply chain is constructed from the perspectives of centralized-matching (CM), decentralized-matching (DM), and mismatching (MM), and the spatial information internal evolution law of multi-phase LCT investment is described by the Markov chain. Then, a bilateral matching algorithm is proposed to solve the equilibrium solutions, and the evolution process of the three modes is analyzed by numerical simulation. Finally, based on the product green degree, we analyze the impact of subsidies and taxes on investment-production decisions. Analytical results show that the matching mechanism proposed in this paper can help supply chain firms to obtain stable matching and has a significant effect on the realization of “triple wins” of society, economy, and environment. The investment utility of CM is higher than that of DM and MM. Manufacturers are inclined to adopt LCT, and the investment level tends to be stable over time. Government reward and punishment regulations are helpful to motivate supply chain firms to invest in LCT, and the synergistic effect of subsidies and taxes is better than that of a single policy.

Appendices

Appendix 1

Proof of Lemma 2

When there is a matching between the manufacturer and supplier but no subsidy or tax, i.e., ${\mathrm{x}}_{s_i}^{\theta_k}=1$, G^t = 0, the green degree can be derived, and it is equal to the equilibrium output in this condition. The manufacturer's objective profit function is formulated by,

$$\begin{aligned}\max_{q_{s_i}^t,s_i^t}{\Pi}_{m,\left(s_i,\theta_k\right)}^t &=\left(p_{s_i}^t-c_{s_i}\right)q_{s_i}^t+\left(e^0-e_{s_i}^t\right)w-C_{s_i,K}^t\\&=\left(1-q_{s_i}^t+\alpha s_i^t-c_{s_i}\right)q_{s_i}^t+\left[e^0-\left(\gamma-\beta s_i^t\right)q_{s_i}^t\right]w-\frac{1}{2}\eta\left(s_i^t\right)^2\end{aligned}$$

(18)

The Hessian matrix is obtained of Eq. (18) regarding ${q}_i^t$ and ${\mathrm{s}}_i^t$, i.e., ${H}_{m,\left({s}_i,{\theta}_k\right)}=\left(\begin{array}{cc}-2& \alpha +\beta w\\ {}\alpha +\beta w& -\eta \end{array}\right)$.

• ①When 2η > (α + βw)², the matrix is negative definite, and there must exist optimal solutions.

$${q}_{s_i}^{\ast t}=\frac{\eta \left(1-{c}_{s_i}-\gamma w\right)}{2\eta -{\left(\alpha +\beta w\right)}^2}$$

(19)

$${s}_i^{\ast t}=\frac{\left(1-{c}_{s_i}-\gamma w\right)\left(\alpha +\beta w\right)}{2\eta -{\left(\alpha +\beta w\right)}^2}$$

(20)

Consequently, the subsidy threshold is derived,

$${\Gamma}_g={q}_{s_i}^{\ast t}=\frac{\eta \left(1-{c}_{s_i}-\gamma w\right)}{2\eta -{\left(\alpha +\beta w\right)}^2}$$

(21)

• ②When 2η ≤ (α + βw)², the matrix is positive definite, and the optimal output can be derived when investment level equals to the maximum budget $\bar{s}$.

$${q}_{s_i}^{\ast t}=\frac{1}{2}\left(1-{c}_{s_i}-\gamma w+\alpha \bar{s}+\beta \bar{s}w\right)$$

(22)

$${\mathrm{s}}_i^{\ast t}=\bar{s}$$

(23)

Therefore, the green degree holds

$${\Gamma}_g={q}_{s_i}^{\ast t}=\frac{1}{2}\left(1-{c}_{s_i}-\gamma w+\alpha \bar{s}+\beta \bar{s}w\right)$$

(24)

Appendix 2

Proof of Proposition 1

It is derived that the Hessian matrix of Eq.(7) in terms of ${q}_c^t$,${\mathrm{s}}_c^t$ and ${u}_k^t$ is a piecewise function. Formally,

$$H_{sc}=\begin{pmatrix}\begin{array}{ccc}\frac{\partial^2\pi_{sc}^{\ast t}}{\partial\left(q_c^t\right)^2}&\frac{\partial^2\pi_{sc}^{\ast t}}{\partial q_c^t\partial s_c^t}&\frac{\partial^2\pi_{sc}^{\ast t}}{\partial q_c^t\partial u_k^t}\end{array}\\\begin{array}{ccc}\frac{\partial^2\pi_{sc}^\ast}{\partial s_c^t\partial q_c^t}&\frac{\partial^2\pi_{sc}^\ast}{\partial\left(s_c^t\right)^2}&\frac{\partial^2\pi_{sc}^\ast}{\partial s_c^t\partial u_k^t}\end{array}\\\begin{array}{ccc}\frac{\partial^2\pi_{sc}^\ast}{\partial u_k^t\partial q_c^t}&\frac{\partial^2\pi_{sc}^\ast}{\partial u_k^t\partial s_c^t}&\frac{\partial^2\pi_{sc}^\ast}{\partial\left(u_k^t\right)^2}\end{array}\end{pmatrix}=\left\{\begin{array}{l}\begin{array}{cc}\left(\begin{array}{l}\begin{array}{ccc}-2\left(1-d/\Gamma_g\right)&y^tz&0\\y^tz&-y^t\eta\left(1-\xi\right)&0\\0&0&-\rho\end{array}\end{array}\right),&\quad\quad if 0 < q_c^t<\Gamma_g\end{array}\\\begin{array}{cc}\begin{pmatrix}-2&y^tz&0\\y^tz&-y^t\eta\left(1-\xi\right)&0\\0&0&-\rho\end{pmatrix},& \quad\quad if q_c^t=\Gamma_g\end{array}\\\begin{array}{cc}\begin{pmatrix}-2y^t\left(1-b/\Gamma_g\right)-2\left(1-y^t\right)\left(1-d/\Gamma_g\right)&y^tz&0\\y^tz&-y^t\eta\left(1-\xi\right)&0\\0&0&-\rho\end{pmatrix},& \quad if q_c^t>\Gamma_g\end{array}\end{array}\right.$$

(25)

where z = α + βw.

Calculating the determinant gives $H_{sc}=\left\{\begin{array}{l}\begin{array}{cc}y^tz^2-2\eta\left(1-\xi\right)\left(1-d/\Gamma_g\right), & if 0 < q_c^t<\Gamma_g\end{array}\\\begin{array}{cc}y^tz^2-2\eta\left(1-\xi\right), & if q_c^t=\Gamma_g\end{array}\\\begin{array}{cc}y^tz^2-2\eta\left(1-\xi\right)\left[y^t\left(1-b/\Gamma_g\right)+\left(1-y^t\right)\left(1-d/\Gamma_g\right)\right],&\;\;\;\;ifq_c^t>\Gamma_g\end{array}\end{array}\right.$. Considering the complexity of the expression, with reference to the studies of Liu et al. 2020a, 2021a), the parameters are set as follows: y⁰ = 0.5, e⁰ = 15, γ = 0.1, β = 0.06, η = 0.1, α = 0.05, c = 0.2, h = 0.2, ρ = 0.4, ξ = 0.8, λ = 0.8, w = 0.2, b = 0.2, d = 0.1.

• ① when Γ_g = 0.3507, ${H}_{sc}=\left\{\begin{array}{c}-0.024, if0<{q}_c^t<{\Gamma}_g\\ {}-0.038, if{q}_c^t={\Gamma}_g\\ {}-0.021, if{q}_c^t>{\Gamma}_g\end{array}\right.$

• ② when Γ_g = 0.385, ${H}_{sc}=\left\{\begin{array}{c}-0.027, if0<{q}_c^t<{\Gamma}_g\\ {}-0.038, if{q}_c^t={\Gamma}_g\\ {}-0.022, if{q}_c^t>{\Gamma}_g\end{array}\right.$

Hessian matrixes in the two conditions are all negative definite. Therefore, there must exist the optimal output, investment level, and LCT R&D effort level. Solve the derivative of Eq.(7) to ${q}_c^t$,${\mathrm{s}}_c^t$ and ${u}_k^t$, and equal them to zero to can derive the optimal solutions of CM mode (Table 5).

Similarly, the optimal solutions of DM and MM modes are derived, as shown in Tables 6 and 7.

Appendix 3

Proof of Proposition 2

The equilibrium solutions of output, carbon emissions and profits of participants under MM mode can be obtained below. Profit functions of the manufacturer and supplier are

$$\underset{q_{\varnothing}^t}{\max }{\varPi}_{m,\varnothing}^t=\left({p}_{\varnothing}^t-c\right){q}_{\varnothing}^t+\left({e}^0-{e}_{\varnothing}^t\right)w+\lambda {G}_{\varnothing}^t=\left(1-{q}_{\varnothing}^t-\mathrm{c}\right){q}_{\varnothing}^t+\left({e}_{\varnothing}^0-\gamma {q}_{\varnothing}^t\right)w+\lambda d\frac{q_{\varnothing}^t-{\Gamma}_g}{\Gamma_g}{q}_{\varnothing}^t$$

(26)

$$\underset{u_k^t}{\max }{\varPi}_{s,\varnothing}^t={R}_{\theta_k}^t-{C}_{\theta_k}^t+\left(1-\lambda \right){G}_{\varnothing}^t=h{u}_k^t-\frac{1}{2}\rho {\left({u}_k^t\right)}^2+\left(1-\lambda \right)d\frac{q_{\varnothing}^t-{\Gamma}_g}{\Gamma_g}{q}_{\varnothing}^t$$

(27)

Performing the first-order derivation as shown in Proposition 1 derives the optimal output and LCT R&D effort level.

$${q}_{\varnothing}^{\ast t}=\frac{\eta \left(1-c-\gamma w\right)\left(1-c-\gamma w-\lambda d\right)}{2\lambda d{\left(\alpha +\beta w\right)}^2+2\eta \left(1-c-\gamma w-2\lambda d\right)}$$

(28)

$${\mathrm{u}}_k^{\ast t}=\frac{h}{\rho }$$

(29)

The corresponding carbon emissions are

$${e}_{\varnothing}^{\ast t}=\frac{\gamma \eta \left(1-c-\gamma w\right)\left(1-c-\gamma w-\lambda d\right)}{2\lambda d{\left(\alpha +\beta w\right)}^2+2\eta \left(1-c-\gamma w-2\lambda d\right)}$$

(30)

Combining the simulation results of Figs.2 and 3 (y⁰ = 0.5,e⁰ = 15,γ = 0.1,β = 0.06,η = 0.1,α = 0.05,c = 0.2,h = 0.2,ρ = 0.4,ξ = 0.8,λ = 0.8,w = 0.2,b = 0.2,d = 0.1), the equilibria under CM, DM and MM modes can be obtained: $\left({q}_c^{\ast t},{e}_c^{\ast t},{\varPi}_{sc}^{\ast t}\right)=\left(0.4674,0.0187,30.175\right)$, $\left({q}_d^{\ast t},{e}_d^{\ast t},{\varPi}_m^{\ast t},{\varPi}_s^{\ast t}\right)=$ (0.4101, 0.0238, 30.0984,0.0619), $\left({q}_{\varnothing}^{\ast t},{e}_{\varnothing}^{\ast t},{\Pi}_{m,\varnothing}^{\ast t},{\Pi}_{s,\varnothing}^{\ast t},{\Pi}_{sc,\varnothing}^{\ast t}\right)=$ (0.3368, 0.0337, 30.0876, 0.0497, 30.1373). Intuitively, ${q}_{\mathrm{c}}^{\ast t}>{q}_d^{\ast t}>{q}_{\varnothing}^{\ast t}$, ${e}_c^{\ast t}<{e}_d^{\ast t}<{e}_{\varnothing}^{\ast t}$, ${\varPi}_{sc}^{\ast t}>{\varPi}_{m,{\theta}_k}^{\ast t}+{\varPi}_{s,{\theta}_k}^{\ast t}>{\varPi}_{m,\varnothing}^{\ast t}+{\varPi}_{s,\varnothing}^{\ast t}={\varPi}_{sc,\varnothing}^{\ast t}$, ${\Pi}_m^{\ast t}>{\Pi}_{m,\varnothing}^{\ast t}$, ${\Pi}_s^{\ast t}>{\Pi}_{s,\varnothing}^{\ast t}$.

Appendix 4

Proof of Fig. 4. As mentioned above, the manufacturer is risk-averse, and its utility function is drawn by referring to the utility theory (Gao 2011, Fig. 6).

When increasing or decreasing x units of LCT investment profit, the utility gained or lost by the manufacturer is represented by area A and B respectively, which obviously shows that, for the manufacturer, the “satisfaction” gained by increasing x unit of profit is greater than the “sense of loss” by decreasing x unit of profit (A>B). Therefore, in the context of sustainable development, manufacturers may be more willing to invest in LCT to gain long-term market competitiveness and seize opportunities for sustainable development. The trend shown in Fig. 4 is consistent with the above analysis results of utility theory.

Fig. 6

Utility theory