This article discusses the hypothesis of underconsumption, which is a particular form of the concept of demand constraint. Rising income inequality and falling shares of labour incomes in GDP have been observed in developed countries in recent decades. In a traditional approach related to the hypothesis, wages constitute a major source of financing consumption spending, while a falling share of wages brings about a diminishing share of consumption in GDP and leads to the development of unsustainable forms of financing consumption with credit. As a consequence, high capital incomes fail to generate new investment and economies fall into stagnation. Since these developments are typical of contemporary advanced economies, the hypothesis gains credibility and is worthy of careful analysis. The article presents different interpretations of the hypothesis and related growth and equilibrium theories through the history of economic thought from early classical economists to contemporary research. The review shows that the hypothesis has been a challenge for economists and is still far from fully comprehended. The last part of the paper explains why statistical analysis is of little help in either conditionally accepting or disproving the hypothesis.