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Investment in carbon emissions reduction is an important decision for many firms due to climate change and growing pressures from various stakeholders. In this paper, we examine the problem of jointly determining optimal order quantity and investment in carbon emissions reduction in the EOQ model with a cap-and-price regulation policy, in which the firm is penalised (rewarded) if its carbon footprint is larger (smaller) than a threshold. Cap-and-trade, carbon tax, cap-and-offset, and carbon cap can be viewed as special cases of the cap-and-price policy. The investment reduces the carbons emitted per replenishment and per unit produced. We characterise the optimal solutions and compare different regulations in terms of the firm's annual total cost and carbon footprint.