Mortgage borrowers are "locked in": forgoing moves to keep low rates. We show missing downsizers stay in larger homes, raising net demand. We design a spatial housing ladder model with long-term mortgages, generating a distribution of locked-in rates and causal mobility effects consistent with the data. A temporary rate hike causes lock-in, increasing house prices by 4.4% and rents by 1.4% relative to a counterfactual without lock-in, offsetting a third of the house-price decline caused by higher rates. A starter-home seller subsidy modestly increases mobility at a high cost per marginal move, suggesting demand-based policies are poorly targeted responses.