Despite recent improvements in the ecological status of wild-capture fisheries, a significant share (33%) of global marine stocks remains overexploited. While top-down management has been shown to work in many settings, there is growing interest in demand-side interventions, which work through consumer-originated price signals to incentivize reduced fishing pressure. The effectiveness of demand-side interventions would rely, in part, on a large enough supply elasticity of fisheries, though this crucial statistic is notoriously difficult to estimate and remains elusive in the literature. Using plausibly exogenous variation in fish prices and extensive data on the world's fisheries, we derive an empirical approach to estimate this elasticity using an instrumental variables estimator. We find it is very low -- similar to that observed in comparable sectors. This suggests that even if prices did respond to demand-side interventions, the supply response would be small. To determine whether the ensuing supply response would have meaningful ecological consequences, we combine a bioeconomic model of fisheries with a global model of supply and demand for  seafood calibrated with our estimates. We find that even interventions that lead to dramatic demand shifts are unlikely to achieve more than marginal improvements to fisheries status. In contrast, we find that supply-side policies (such as well-enforced fishing quotas), even imperfectly designed or implemented, can result in substantial recovery.