Although the volatility of house prices is often ascribed to demand-side factors, constraints on housing supply have important and little-studied implications for housing dynamics. I illustrate the strong relationship between the volatility of house prices and the regulation of new housing supply. I then employ a dynamic structural model of housing investment to investigate the mechanisms underlying this relationship. I find that supply constraints increase volatility through two channels: First, regulation lowers the elasticity of new housing supply by increasing lags in the permit process and adding to the cost of supplying new houses on the margin. Second, geographic limitations on the area available for building houses, such as steep slopes and water bodies, lead to less investment on average relative to the size of the existing housing stock, leaving less scope for the supply response to attenuate the effects of a demand shock. My estimates and simulations confirm that regulation and geographic constraints play critical and complementary roles in decreasing the responsiveness of investment to demand shocks, which in turn amplifies house price volatility.