In this study, we consider a carbon emission cap-and-trade system in which the policymaker decides the cap for carbon emissions for each company and also has the power to regulate the carbon price in the carbon trading market for the purpose of minimizing total carbon emissions. We assume that there are n companies regulated in terms of carbon emissions by the policymaker, each of which emits carbon when producing its own product. After learning the carbon cap and carbon price regulated by the

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... olicymaker, each company makes simultaneous pricing and production decisions using the quick response strategy, and can trade some of its carbon emissions in the carbon market at the carbon price set by the policymaker, if the carbon emissions are below the cap. We model this non-cooperative game between the policymaker and companies as a Stackelberg game in which the policymaker is the leader and the companies are the followers. We show that there exists an equilibrium for the policymaker's carbon pricing decisions and each company's production and pricing decisions. From this equilibrium, we derive a carbon cap for the company at which the amount of traded carbon emissions is zero. This implies that some company's production and pricing decisions, even under carbon emission restrictions, will be equal to those without the carbon emission restrictions. Also, we find that companies participating in the carbon cap-and-trade system would reduce their carbon emissions through reduced production, but can have a chance to improve profit through control of the product's selling price.