## 6 Conclusions  
This study explored the practicality of a sustainable production-inventory model considering carbon tax policy and collaborative investment in carbon emission reduction technology. Raw material inventory stage and price-dependent demand were also considered suitable for the model in a real-life situation. Moreover, the model was based on a manufacturer Stackelberg game where the manufacturer is the leader and the retailer is the follower. The manufacturer's optimal production, shipping, and carbon emission technology investment strategies, along with the retailer's optimal pricing and replenishment strategies, under the Stackelberg game, were mathematically analyzed using a proposed algorithm. Some main results of managerial and instructive cases were also obtained from the sensitivity analyses:  
1. From a macro perspective, it can be concluded that increased carbon tax does help the manufacturer and retailer reduce carbon emissions. However, this may inspire the government's carbon reduction policy. In addition, a change in the carbon tax has a relatively significant impact on manufacturer's optimal equilibrium technology investment for reducing carbon emissions. 2. For companies, the most intuitive conclusion is that the optimal equilibrium technology investment for carbon emission reduction will increase with higher values of most carbon emission parameters; however, such increases will negatively affect both the manufacturer's and retailer's total profits. However, changes in carbon emission parameters are relatively insensitive to the optimal equilibrium solutions or profits. Nevertheless, for the sustainable development of the enterprise, it is necessary to take carbon emission parameters into consideration of the proposed model. 3. From the retailer's perspective, the retailer may raise the optimal equilibrium selling price to respond to increases in the values of the carbon emission parameters or carbon tax. The retailer's carbon reduction technology investment may seem to be subsidized from selling prices. Moreover, the retailer's optimal equilibrium order quantity decreases with higher values of most carbon emission parameters, whereas the retailer's order cost or fixed shipping cost positively impacts the manufacturer's total profit but lessens the retailer's total profit. Greater retailer's holding cost or deterioration rate of finished goods causes the optimal equilibrium order quantity to decrease, resulting in less total profits of both players. Obviously, the retailer tends to decrease inventory when holding cost or deterioration rate is high. 4. From the manufacturer's perspective, an increase in the unit sales price will raise the optimal equilibrium technology investment for reducing carbon emissions, total profit, and retailer's optimal equilibrium selling price. Contrarily, the retailer's optimal equilibrium order quantity decreases, resulting in less total profit. Furthermore, it is found that changes in manufacturer's parameters except for its unit sales price are relatively insensitive to the optimal equilibrium solutions, especially for the retailer. 5. As to the impact of changes in market demand, autonomous consumption positively affects both the manufacturer's and retailer's optimal equilibrium policies, whereas induced consumption does the opposite. Furthermore, an increase in autonomous consumption or a decrease in induced consumption results in higher manufacturer's and retailer's total profits and carbon emissions. There are some limitations in this study. For example, the proposed model is assumed that the manufacturer is the leader and the retailer is the follower. In practice, the retailer of supply chain systems may be a leader, such as Wal-Mart. This situation may be considered in the future research. Furthermore, this study only considers the production-inventory model under the carbon tax policy and other carbon emission reduction policies, i.e., carbon cap-and-trade, carbon offset, or total carbon control can be extended based on the proposed model.