What should be the characteristics of the optimal monetary policy under commitment in the situation of a global liquidity trap when Central Banks do not coordinate their actions? Using a two-country open economy model, we perform a numerical exercise in order to address this question and study the differences between this setting and a cooperative situation, when monetary authorities are not only worried with the national household utility, but instead maximize a measure of world welfare. Our findings points towards differences of history and international dependence features of optimal monetary prescriptions in each of the observed cases. We also execute a welfare analysis of our experiment that suggests that a local Central Bank prefers, not only to stay restrained by zero lower bound (ZLB) on nominal interest rates for a longer period, but also that the foreign country exists this situation as early as possible. Lastly, we make a robustness analysis of our results varying the size of each nation and the degree of substitution of the composite goods produced in each locality.