In this paper, we analyse the optimal foreign direct investment (FDI) of a firm operating in a duopolistic market. FDI induces technological spillovers to a competitor in the foreign country; the intensity of which depends on the absorptive capacity of the foreign firm and the size of the technological gap. We characterize a technology spillover threshold and show that for an intensity of spillovers below this threshold, there is a unique locally asymptotic stable steady state with a positive capital stock in the developing country. Furthermore, we characterize how optimal foreign investment patterns and the investor's value function depend on the level of technology transferred and characterize the optimal level to be used for the FDI.