This paper examines optimal monetary policy in a two-country New Keynesian model with international trade in intermediate inputs. We derive the loss function of a cooperative monetary policymaker and find that the optimal monetary policy must target intermediate-goods price inflation rates, final-goods price inflation rates, final-goods output gaps, and relative-price gaps. We use the welfare loss under the optimal monetary policy as a benchmark to evaluate the welfare implications of three Taylor-type monetary policy rules. A main finding is that the degree of price stickiness at the stage of intermediate-goods production is a key factor to determine which policy rule should be followed. Specifically, when the degree of price stickiness at the stage of intermediate-goods production is high, the policymaker should follow intermediate-goods PPI-based Taylor rule, whereas CPI-based Taylor rule should be followed when the degree of price stickiness at the stage of intermediate-goods production is intermediate or low.