We construct an asymmetri c duopolistic R&D and production behavior model subject to knowledge spillovers. This model is an extension to the symmetric model of d'Aspremont and Jacquemin (A&J (1988)) and aims to determine the cooperative and non-cooperative R&D strategies for two agents of different size. The paper concludes that the introduction of asymmetry into the A&J (1988) model leads to different R&D expenditures and production decisions made by the firms. Simulations show that the bigger agent has larger R&D expenditures and higher output. If firms choose the monopoly collusion or the welfare-maximizing strategy, the optimal solution implies that R&D is conducted asymmetrically by both agents, but that production is conducted only by the largest agent.