# Manufacturers’ digital transformation under carbon cap-and-trade policy: investment strategy and environmental impact

From the above, we can see that the equilibrium solutions in different scenarios are related to the level of digital technology. When \(p_e \,< \,\frac\varphi \,-\,\alpha \,+\,2\lambda \beta =p_e0\), *m*_{1} > *m*_{2}. Otherwise, *m*_{2} > *m*_{1}. To ensure that the equilibrium solution is non-negative, and the Hessian matrix is negative definite, \(m \,> \,\frac(\varphi \,+\,\lambda )(\varphi \,+\,\alpha \,+\,\beta p_e)2=m_0^h\) when *p*_{e} < *p*_{e0} and \(m \,> \,\max \left\\frac(\alpha \,+\,\beta p_e)(\varphi \,+\,\alpha \,+\,\beta p_e)2,\frac(\varphi \,+\,2\alpha \,+\,2\beta p_e-\lambda )(\varphi \,+\,\alpha \,+\,\beta p_e)6\right\=m_0^l\) when *p*_{e} > *p*_{e0}. Next, we will conduct a comparative analysis of the equilibrium solutions. All the propositions have been proven and are provided in the Appendix.

### Comparison of equilibrium solutions

*Proposition 1:**Given M2 (M1)‘s strategy (N or D), the relationship between product prices of M1 (M2) in different scenarios is shown in* Table 3. *Where* \(p_e1=\frac2\varphi \,-\,2\alpha \,+\,\lambda 2\beta \).

Proposition 1 indicates that when the CTP is low (*p*_{e} < *p* _{e1}), given M2 (M1)‘s strategy (N or D), the product pricing of M1 (M2) will be higher after DX, regardless of the digital technology level. The primary reason is that the investment cost of DX leads to increased costs, resulting in higher product prices. When the CTP is high (*p*_{e} > *p*_{e1}), M1 (M2) will reduce their product prices. This is because when the CTP is sufficiently high, DX helps manufacturers reduce carbon emissions, bringing more cost advantages. Manufacturers can then lower their product prices to attract more consumers to buy products, thereby promoting their business development.

*Proposition 2:**Given M2 (M1)‘s strategy (N or D), the relationship between the outputs of M1 (M2) in different scenarios is shown in* Table 4.

Proposition 2 indicates that, given M2 (M1)‘s strategy (N or D), the output of M1 (M2) will increase after DX, regardless of the CTP or digital technology level. The reasons are: (1) DX has a direct market expansion effect. (2) DX helps reduce costs (production costs and carbon emission costs). When *p*_{e} > *p*_{e1}, M1 (M2) will lower the product price, which will ultimately have a positive impact on market demand. (3) When *p*_{e} < *p*_{e1}, product pricing may increase, suppressing market demand, but the output of M1 (M2) will still rise, as the market expansion effect of the DX dominates. Therefore, when manufacturers implement the DX, their product output will always increase.

*Proposition 3:**Given M2 (M1)‘s strategy (N or D), the profit relationship of M1 (M2) in different scenarios is shown in* Table 5.

Proposition 3 indicates that, given M2 (M1)‘s strategy (N or D), the profit of M1 (M2) will increase after DX, regardless of the CTP or digital technology level. The reasons are: (1) DX has a direct market expansion effect and the utility of cost reduction (production costs and carbon emission costs). The profit generated is greater than the investment cost of the DX, resulting in profit growth. (2) While *p*_{e} < *p*_{e1} may lead to an increase in product pricing, the market demand for M1 (M2) will still increase, leading to profit growth. Therefore, when manufacturers implement the DX, their profits will always increase.

### Equilibrium strategy

*Proposition 4:**Strategy DD is the only Nash equilibrium*.

Proposition 4 indicates that the scenario in which both manufacturers implement DX is the only equilibrium situation. This is because when M1 (M2) chooses traditional technology or DX, M2 (M1) will realize profit improvement. In order to maximize the benefits, M2 (M1) will implement DX on its own initiative or out of necessity, so as to achieve competitive balance in the market. This conclusion also corresponds to the actual situation. For example, both Gree and Midea have implemented DX. Manufacturers will implement DX regardless of digital technology level. However, the digital technology level determines the degree of DX and the investment cost of DX. In a market with low digital technology level, both manufacturers will partially transformation, degree of DX is \(\theta _iL^DD\ast \) and investment costs of DX is \(C_iL^DD\ast \). When the digital technology level exceeds a certain threshold, both manufacturers will implement complete transformation, investment costs of DX is \(C_iH^DD\ast \).

*Proposition 5:**When m* > *m*_{3} *and p*_{e} < *p*_{e2}*, the two manufacturers fall into a prisoner’s dilemma, i.e.,* \(\pi _i^DD\ast \,< \,\pi _i^NN\ast \).

*Where* \(m_3=(\varphi +\lambda )(2Q+\varphi +\lambda )\), \(p_e2=\frac3\varphi\, -\,\alpha \,+\,4\lambda \beta \).

Proposition 5 illustrates that a prisoner’s dilemma might occur amongst two manufacturers in a competitive environment. Specifically, when *m* > *m*_{3} and *p*_{e} > *p*_{e2}, the equilibrium strategy of both manufacturers is to implement DX, but their respective profits are smaller than that of scenario NN. Even if traditional technology can potentially lead to greater profits, both manufacturers will still adopt DX. This is because both manufacturers pursue their own profit maximization and will choose to implement DX. However, when the cost coefficient of DX is large, it will lead to a smaller degree of manufacturers’ DX, and the increase of CTP will also lead to an increase in carbon emission costs. If both manufacturers choose traditional technologies, they will also pay higher carbon emissions costs, but will still be able to maintain high profits because they do not invest in expensive digital technologies. Consequently, in scenario DD, each manufacturer’s profits are lower than in scenario NN, meaning both manufacturers succumb to the prisoner’s dilemma. This proposition indicates that the existence of a prisoner’s dilemma is not advantageous for manufacturers seeking to maximize their profits. Hence, under the aforementioned conditions, both manufacturers should seek strategies to mitigate the effects of the prisoner’s dilemma.

Figure 2 showcases the outcomes of Propositions 4 and 5. The shaded region signifies the infeasible domain. The top segment (indicated by a blue arrow) suggests partial transformation by manufacturers, while the central area (indicated by a red arrow) indicates complete transformation by manufacturers. Furthermore, the upper right corner signifies that both manufacturers are caught in a prisoner’s dilemma.

### Impact of CTP on equilibrium results

Since scenario DD is the only Nash equilibrium, we only need to discuss the impact of the CTP on the equilibrium solution of scenario DD.

*Proposition 6:**The impact of CTP on equilibrium strategy.*

(1) \(\frac\partial m_1\partial p_e\, > \,0\)*; (2)* \(\frac\partial \theta _iL^DD\ast \partial p_e \,> \,0\)*; (3)* \(\frac\partial C_iL^DD\ast \partial p_e\, > \,0\).

Proposition 6(1) implies that an increase in the CTP will serve as a catalyst for manufacturers to undergo a complete transformation, a phenomenon also depicted in Fig. 2. Propositions 6(2) and (3) suggest that at a lower digital technology level, an upsurge in the CTP can stimulate manufacturers to augment their DX investment, thereby enhancing their degree of DX. This is predicated on the idea that a higher CTP encourages manufacturers to actively seek ways to diminish carbon emissions for the dual purpose of reducing costs and boosting profits. By enhancing the degree of DX, manufacturers can use resources more efficiently and considerably decrease carbon emissions. Therefore, they will increase their investment in DX to achieve a higher level of DX and consequently, realize an increase in profits.

*Proposition 7:**The effect of CTP on manufacturers’ profits under scenario DD.*

*(1) When p*_{e} < *p*_{e3}, \(\frac\partial \pi _iL^DD\ast \partial p_e\, > \,0\)*, otherwise*, \(\frac\partial \pi _iL^DD\ast \partial p_e\, < \,0\)*;* *(2)* \(\frac\partial \pi _iH^DD\ast \partial p_e \,> \,0\).

*Where* \(p_e3=\frac6m\beta (\varphi\, -\,\lambda )+\frac(2\times 2^1/3\times 3^2/3\beta m^2Q^2)\beta \Theta -\frac2^2/3\times 3^1/3\Theta \beta (\varphi \,-\,\lambda )^3R_i-\frac\varphi \,+\,\alpha \beta \),

$$\beginarrayl\Theta =\left[\right.9\beta (\varphi\, +\,\lambda )^5(m-(\varphi \,+\,\lambda )^2)m^2Q^2R_i^2\\\left.\qquad+\,\sqrt3\sqrt\beta ^2(\varphi \,+\,\lambda )^9m^4Q^4R_i^3 [27(\varphi\, +\,\lambda )(m-(\varphi\, +\,\lambda )^2)^2R_i+4\beta m^2Q^2]\right]^1/3.\endarray$$

Proposition 7 reveals that in a market with a low digital technology level, as the CTP increases, manufacturers’ profits initially increase and then decrease post-DX. This is attributed to the fact that DX can curtail carbon emissions, and a rise in the CTP will decrease the manufacturer’s carbon emission expenses, thereby leading to an increment in profits. As CTP continue to rise, manufacturers will increase investment to achieve a higher degree of DX. However, the returns from DX fail to exceed the investment costs, hence leading to a decrease in manufacturers’ profits. Therefore, when the CTP is *p*_{e3}, the manufacturers’ profit is maximized post-DX. In a market with a high digital technology level, CTP always positively impact manufacturers’ profits. This is because, in a high digital technology market, manufacturers always achieve a complete degree of DX, and an increase in CTP will reduce the manufacturers’ carbon emission costs, resulting in increased profits.

### Analysis of total carbon emission

In the context of global carbon neutrality, environmental issues are increasingly garnering attention from governments worldwide. Manufacturers are compelled to focus on carbon emission issues too. Although DX can mitigate resource consumption, its impact on total carbon emissions remains uncertain. Therefore, we will discuss the effect of manufacturers’ DX on total carbon emissions in this section. As strategy DD is the only Nash equilibrium, we will only compare the total carbon emissions of scenario DD with scenario NN. The formula for calculating total carbon emissions is as follows:

$$CE=q(e-\beta \theta )$$

(13)

Based on Eq. (13), we can calculate the total carbon emissions under both scenarios. In the DD scenario, total carbon emissions from manufacturers operating in markets with low and high levels of digital technology are given as follows:

$$CE_iL^DD\ast =\frac2mQ[2me-\beta Q(\varphi +\alpha +\beta p_e)-e(\varphi +\lambda )(\varphi +\alpha +\beta p_e)][2m-(\varphi +\lambda )(\varphi +\alpha +\beta p_e)]^2$$

(14)

$$CE_iH^DD\ast =(e-\beta )(Q+\varphi +\lambda )$$

(15)

In scenario NN, the total carbon emissions of manufacturers are:

$$CE_i^NN\ast =Qe$$

(16)

*Proposition 8:**The relationship between total carbon emissions before and after the DX of manufacturers.*

*(1) When m* > *m*_{1}, \(CE_iL^DD\ast \, > \,CE_i^NN\ast \) *if e* > *e*_{1}*; otherwise*, \(CE_iL^DD\ast \, < \,CE_i^NN\ast \)*;*

*(2) When m* ≤ *m*_{1}, \(CE_iH^DD\ast \, > \,CE_i^NN\ast \) *if e* > *e*_{2}*; otherwise*, \(CE_iH^DD\ast \,< \,CE_i^NN\ast \).

*Where* \(e_1=\frac2m\beta Q(\varphi \,+\,\lambda )[2m-(\varphi \,+\,\lambda )(\varphi \,+\,\alpha \,+\,\beta p_e)]\), \(e_2=\frac\beta (Q\,+\,\varphi \,+\,\lambda )\varphi \,-\,\lambda \).

Proposition 8 indicates that while DX can reduce carbon emissions of the unit product, the change in manufacturers’ total carbon emissions is uncertain. In the low digital technology level market, the relationship between total carbon emissions before and after manufacturers’ DX is related to unit carbon emissions. When unit carbon emissions are small (*e* < *e*_{1}), DX can reduce manufacturers’ total carbon emissions, benefiting environmental protection. When unit carbon emissions are large (*e* > *e*_{1}), the manufacturers’ total carbon emissions will increase after DX, which is not conducive to environmental protection. This is due to the market expansion effect caused by manufacturers’ DX. Raising CTP can increase this threshold (*e*_{1}), so a higher CTP can reduce manufacturers’ total carbon emissions. Similar conclusions can be drawn for the high digital technology level market. When unit carbon emissions are small (*e* < *e*_{2}), DX can reduce manufacturers’ total carbon emissions. When unit carbon emissions are large (*e* > *e*_{2}), the manufacturers’ total carbon emissions will increase after DX. Interestingly, due to *e*_{1} < *e*_{2}, it’s more likely that manufacturers will reduce total carbon emissions after DX at the high digital technology level market. The government must strengthen digital infrastructure, encourage digital technology research and development, and enhance the market’s digital technology level. In conclusion, the DX of traditional manufacturers is inevitable. In the context of global carbon neutrality, it is also inevitable for manufacturers to reduce total carbon emissions. Therefore, manufacturers must continuously introduce and develop green technologies, reduce unit carbon emissions in production, effectively reduce total carbon emissions, and achieve the goals of transformation, upgrading, and sustainable development.

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