A fat and a healthy good provide immediate gratification, and cause health costs or benefits in the long run, which are misperceived. Additionally, the fat good (healthy good) increases (decreases) health care costs by increasing (decreasing) the probability of suffering from a chronic disease in the future. Individuals differ in income and in their degree of misperceptions concerning the health effects of the consumption of fat and of healthy goods. The level of the fat tax is determined through majority voting. Individuals vote according to their misperceived utility function. Consequently, excessive fat consumption is not due to a self-control problem but due to information deficiencies or cognitive inability to process information. A fraction of the fat tax proceeds is "earmarked" to reduce health insurance premiums while the remaining fraction finances a subsidy on the healthy good. This earmarking rule is determined at a constitutional stage to maximize utilitarian or Rawlsian welfare, anticipating the induced political equilibrium. We show that the fat tax in the political equilibrium is always lower than the utilitarian fat tax. This is no longer necessarily true with a Rawlsian objective. The determination of the optimal earmarking rule is quite complex. Even in the utilitarian case, it is not just used to boost political support for the fat tax. Instead, it may involve a tradeoff between the fat tax and the healthy good subsidy.