Abstract: Carbon emission reduction could improve the performance of the supply chain, but it also provided opportunities for retailers to enjoy the benefit without cost, which would destroy the manufacturer's emission reduction motivation. Unilateral shareholding and cross-shareholding between supply chain members are common in practice. Studies have shown that the supply chain members could form an interest community through cross-shareholding. Some papers investigate the impact of unilateral shareholding on the supply chain's emission reduction and pricing decisions and found that unilateral shareholding could not coordinate the supply chain. Thus, the cross-shareholding strategy is adopted to study the supply chain's emission reduction and pricing game under the cap-and-trade regulation. The games between the manufacturer and the retailer with Stackelberg models are modeled. The supply chain's optimal decision in the centralized decision scenario, and the equilibriums of decentralized decision scenarios with and without cross-shareholding are explored. Also, the effects of cross-shareholding and carbon-trade price on the enterprises' decision and performance are demonstrated by the numerical examples. It is found that the cross-shareholding between the manufacturer and the retailer could transform the effects of carbon-trade price on the manufacturer's emission reduction, increase the emission reduction per unit product and the market demand, and decrease the wholesale price and retail price under certain conditions. The optimal carbon emission reduction per unit product increased with the increase of shares holding by manufacturer and retailer in each other. If the retailer's shares hold by the manufacturer increases, the manufacturer's profit increases certainly, while the retailer's profit increases only if that shareholding ratio is lower relatively. The impact of the strategy that the retailer hold the manufacturer's shares on the supply chain members' profits is uncertain. But fortunately, the cross-shareholding strategy could simultaneously increase the profit of the manufacturer and the retailer and coordinate the supply chain when the cross-shareholding ratio meet certain conditions.

Key words: cap-and-trade, cross-shareholding, carbon emission, pricing, supply chain coordination