In the 1950s, more well-defined models were built on discounted utility theory and approached the question of inter-temporal consumption as a lifetime income optimization problem.
Solving this problem mathematically, assuming that individuals are rational and have access to complete markets, Modigliani & Brumberg (1954), Albert Ando, and Milton Friedman (1957) developed what became known as the life-cycle model.
The main discrepancies between predicted and actual behaviour is that people drastically 'underconsume' early and late in their lifetime by failing to borrow against future earnings and not saving enough to adequately finance retirement incomes respectively.
People also seem to 'overconsume' during their highest earning years, the elderly do not consume from their assets as would be expected (particularly from their household equity) and also treat windfall gains in a manner inconsistent with the life-cycle model.
Specific alterations to the theory have been proposed to help it accommodate the data; a bequest motive, capital market imperfections such as liquidity constraints, a changing individual utility function over time or a particular form of expectation as to future income.