We apply an incomplete demand system to supermarket scanner data to estimate the effects of a fat tax on dairy products for different demographic groups. We find own-price elasticities of demand are relatively inelastic and vary little across groups. A fat tax may be an effective means to raise revenue, but will not result in a significant reduction in fat consumption. The welfare effects associated with a fat tax are large and vary greatly across demographic groups. These fat taxes are regressive in nature, as the elderly and poor suffer greater welfare losses.