The present paper discusses the concept of fuel poverty taking into account the arbitrages made by households when they are facing economic constraints. Fuel poverty is still lacking a common definition throughout Europe: while the UK and France have (different) official definitions, there is still no definition in a country like Germany, or at the European level. Where definitions exist, they often consider that fuel poor households have high energy needs. The possibility of being fuel poor even without having high energy needs and the various arbitrage possibilities of households - i.e. to under-spend and use too little energy - are not systematically discussed. Our paper tries to fill that gap by putting fuel poverty into the larger context of constraints faced by households. Based on a graphical analysis, it shows that different situations of fuel poverty might occur. It results in the identification of two distinct fuel poverty problems: an 'energy inequality' problem, reflected by the fact that some households pay disproportionately high energy bills, and an 'energy affordability' problem that can affect a larger share of the population. It finally explores the two types of fuel poverty for European countries and discusses policy implications.