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We estimate local housing supply elasticities for 325 local authorities and 6,788 census tract areas in England. We examine how housing supply responds to price changes across small areas and how this varies according to a rich set of geographic and policy constraints. Our central estimate for the average elasticity of relative local supply with respect to price across local authorities is 0.14 over a period of 25 years between 1996 and 2021. This is low compared to estimates from other countries. Elasticities are lower in areas with less land available for development, greater differences in elevation, higher historical population density and in areas where local planning authorities had a greater historic tendency to reject new developments. We also find that urban density and constraints on the amount of available land have stronger negative effects on the supply of larger properties than properties with fewer bedrooms.