equation Given a finite time horizon of length , we set with the understanding the consumer spends all his wealth by the end of the last period. Solving the consumer's budget constraint forward to the last period, we determine the consumption function is given by Over an infinite time horizon, we instead impose a no Ponzi game condition, which prevents the consumer from continuously borrowing and rolling over their debt to future periods, by requiring The resulting consumption function is then Both expressions () and () capture the essence of the permanent income hypothesis: current consumption is determined by a combination of current non human wealth and human capital wealth . The fraction of total wealth consumed today further depends on the interest rate and the length of the time horizon over which the consumer is optimizing. Liquidity constraints Some have attempted to improve Friedman's original hypothesis by including liquidity constraints, most notably Christopher D. Carroll. Empirical evidence Observations, recorded from 1888 to 1941, of stagnant average propensity to consume in the face of rising real incomes provide strong evidence for the existence of the permanent income hypothesis. An early test of the permanent income hypothesis was reported by Robert Hall in 1978, and, assuming rational expectations, finds consumption follows a martingale sequence. Hall & Mishkin (1982) analyze data from 2,000 households and find consumption responds much more strongly to permanent than to transitory movements of income, and reinforce the compatibility of the PIH with 80% of households in the sample. Bernanke (1984) finds 'no evidence against the permanent income hypothesis' when looking at data on automobile consumption. In contrast, Flavin (1981) finds consumption is very sensitive to transitory income shocks ('excess sensitivity'), while Mankiw & Shapiro (1985) dispute these findings, arguing that Flavin's test specification (which assumes income is stationary) is biased towards finding excess sensitivity. Souleles (1999) uses income tax refunds to test the PIH. Since a refund depends on income in the previous year, it is predictable income and should thus not alter consumption in the year of its receipt. The evidence finds that consumption is sensitive to the income refund, with a marginal propensity to consume between 35 and 60%. Stephens (2003) finds the consumption patterns of social security recipients in the United States is not well explained by the permanent income hypothesis. Stafford (1974) argues that Friedman's explanation cannot account for market failures such as