Purpose - The purpose of this paper is to study strategic trade policies under international process research and development (R&D) competition with or without market leaders for free trade and a subsidy regime and compare the effects of R&D subsidies and export subsidies on the equilibrium levels of firm profit and social welfare. Design/methodology - For the analysis, we use previous work by Haaland and Kind (2008) and construct a differentiated goods duopoly model, wherein two firms compete via quantity in a third-country market for free trade and the subsidy regime. We consider simultaneous-move quantity competition when the two firms choose their quantities simultaneously and sequential-move quantity competition when they choose their quantities sequentially. The results are compared to those of Balboa, Daughety and Reinganum (2004), who studied export subsidies. Findings - The following are the findings. First, the results of firm preference orderings regarding firm position from Dowrick (1986) and Balboa, Daughety and Reinganum (2004) may not hold in our model when the firms' strategies are strategic substitutes under free trade. Second, the preference rankings under Cournot competition for free trade and a subsidy regime are the same as those in the strategic trade policy of export subsidy. Third, except for the cases of too close substitutes and complements, the results of firm and government preferences regarding firm position are different from those of Balboa, Daughety and Reinganum (2004) in that Stackelberg leadership in a subsidy regime is advantageous when the goods are substitutes but is disadvantageous when the goods are complements. Moreover, the equilibrium level of firm profit is the highest in the Cournot-Nash play when the goods are substitutes in a subsidy regime. Fourth, except for the cases of too close substitutes and complements, the results of firms' and their respective governments' trade regime preferences are similar to those of Balboa, Daughety and Reinganum (2004) in that a Stackelberg leader firm and government prefer free trade if the goods are substitutes and prefer a subsidy regime if the goods are complements. Furthermore, a Stackelberg follower firm and government strongly prefer a subsidy regime to free trade. Originality/value - By analyzing the effects of R&D subsidies and export subsidies in international markets, we can find similarities and differences between them in international markets.