This paper analyzes how firms decide where to patent in a heterogeneous firm model of trade with endogenous rival entry. In the model, innovating firms compete with rival firms on price, where rivals force the innovating firm to reduce markups and lower the innovating firm's probability of obtaining monopolistic profits. Patenting allows the innovating firm to reduce the number of rival rms by increasing their fixed overhead costs, thereby providing higher expected profits and increased markups from reduced competition. Countries with higher states of technology, more competition and better patent protection have a greater proportion of entrants who patent. Industries tend to follow a U-shaped pattern of patenting where industries with high heterogeneity in production and low substitution, along with industries with low heterogeneity in production and high substitution patent more frequently. Using a generalized framework of the model, I estimate market-based measures of country-level patent protection, which when compared with other IP indices, suggests that not enough international patenting is taking place. Finally, I test the predictions of the model using a newly available technology-to-industry concordance on bilateral patent flows and show that firms are increasingly sensitive to foreign IP protection. Countries that choose to maximize their IP protection can increase the number of foreign patents by almost 10%.