Consumption of antibiotics in food animals is increasing worldwide and is approaching, if not already surpassing, the volume consumed by humans. It is often suggested that reducing the volume of antibiotics consumed by food animals could have a public health benefits. Although this notion is widely regarded as intuitively obvious there is a lack of robust, quantitative evidence to either support or contradict the suggestion. As a first step towards addressing this knowledge gap, we develop a simple mathematical model for exploring the relationship between antibiotic consumption by food animals and levels of resistant bacterial infections in humans. We use the model to investigate the impact of restricting antibiotic consumption by animals and to identify which model parameters most strongly determine that impact. Our results suggest that for a wide range of scenarios curtailing the volume of antibiotics consumed by food animals has, as a stand-alone measure, little impact on the level of resistance in humans. We also find that reducing the rate of transmission of resistance from animals to humans may often be more effective than an equivalent reduction in the consumption of antibiotics in food animals. Moreover, the response to any intervention is strongly determined by the rate of transmission from humans to animals, an aspect which is rarely considered.