The impacts of poverty and material scarcity on human decision making appear paradoxical. One set of findings associates poverty with risk aversion, whilst another set associates it with risk taking. We present an idealised rational-choice model, the Desperation Threshold Model (DTM), that explains how both these accounts can be correct. The DTM assumes that there are basic needs whose satisfaction is not fully divisible. This generates an S-shaped utility function for material resources. The value of gaining a dollar is at first small (because even with the extra dollar, basic needs still cannot be met); then large (because the extra dollar enables basic needs to be met); and then small again. Just above the basic needs threshold, people’s main concern is not falling below, and they are predicted to avoid risk especially strongly. Below the threshold, their most important concern is jumping above, and they are predicted to take risks that would otherwise be avoided. Versions of the DTM have been proposed under various names across biology, anthropology, economics and psychology. We review a broad range of relevant empirical evidence from a variety of societal contexts. Though the model primarily concerns individual decision making, it connects to a range of population-scale and societal issues such as: the consequences of economic inequality; the deterrence of crime; and the optimal design and behavioural consequences of the welfare state. We discuss interpretative issues, and suggest areas for future DTM research that bridges disciplines.