(1) Problem Definition: Pharmaceutical companies are reluctant to invest in production capacity for vaccines and therapeutic drugs to combat diseases prevalent in developing countries because demand is uncertain and the affected countries have limited ability-to-pay to cover product development and capacity infrastructure costs. (2) Academic/Practical Relevance: We derive equilibrium decisions of pharmaceutical manufacturers and global health agencies within the framework of an advance market commitment (AMC) contract via formal models. We also compare several variants of the AMC contract that global health agencies have considered implementing. (3) Methodology: We formulate and solve Stackelberg games between a global health agency and either one or two manufacturers. Our two-manufacturer model also involves a simultaneous capacity game between the manufacturers, embedded in a leader-follower setting involving the global health agency and the manufacturers. (4) Results: We derive contract terms that induce a single manufacturer with a late-stage product to invest in an amount of capacity that maximizes the number of doses supplied. We also show that among variants of the base AMC contract, minimum capacity requirement contract outperforms the minimum purchase guarantee contract. When there are two manufacturers, we demonstrate that demand allocation rules matter. Also, if both manufacturers' products meet the necessary functional requirements, it may not be optimal for the global health agency to strictly prioritize one manufacturer over the other for demand allocation. (5) Managerial Implications: Global health agencies have considered proposals to offer purchase guarantees as part of AMC contracts. We show that a better alternative might be to require minimum capacity investments from manufacturers. Similarly, with two manufacturers, it might be beneficial for the health agency to consider allocating demand proportional to each manufacturer's capacity investment.