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We analyse the economics and epidemiology of different scenarios for a phased restart of the UK economy. Our economic model is designed to address the unique features of the COVID-19 pandemic. Social distancing measures affect both supply and demand, and input-output constraints play a key role in restricting economic output. Standard models for production functions are not adequate to model the short-term effects of lockdown. A survey of industry analysts conducted by IHS Markit allows us to evaluate which inputs for each industry are absolutely necessary for production over a two month period. Our model also includes inventory dynamics and feedback between unemployment and consumption. We demonstrate that economic outcomes are very sensitive to the choice of production function, show how supply constraints cause strong network effects, and find some counter-intuitive effects, such as that reopening only a few industries can actually lower aggregate output. Occupation-specific data and contact surveys allow us to estimate how different industries affect the transmission rate of the disease. We investigate six different re-opening scenarios, presenting our best estimates for the increase in R0 and the increase in GDP. Our results suggest that there is a reasonable compromise that yields a relatively small increase in R0 and delivers a substantial boost in economic output. This corresponds to a situation in which all non-consumer facing industries reopen, schools are open only for workers who need childcare, and everyone who can work from home continues to work from home.