# Intertemporal Choices

In economics, the intertemporal choice is the study of the relative values people assign to two or more payoffs at different points in time. It is usually simplified to two periods: today and some future date. The prime objective behind studying the intertemporal choice is to realize that how the various choices are made by the consumers. It also suggests how the choices should be made. A conventional study on intertemporal choice says that the delaying effect of the future outcome value that is subjective in nature is generally addressed by the discount function. This plays a vital role in the decision making in case of intertemporal choice.

John Rae introduced intertemporal choice in the year of 1834 in his theory named "Sociological Theory of Capital". The theory was further elaborated by Eugen von Bohm-Bawerk in the year of 1889 and Irving Fisher in 1930. In the 1950s, more well-defined models 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.

# Permanent Income Hypothesis Model

Consider a household whose life has only two periods. His utility function is u(c1; c2) = U(c1) + βU(c2), where where c1 is consumption in the first period of his life,c2 is consumption in the second period of his life and β is between zero and one and measures household's degree of impatience.

The life-long budget constraint is given by:

${\displaystyle c_{1}+{\frac {c_{2}}{1+r}}=y_{1}+{\frac {y_{2}}{1+r}}+A}$

where y1 and y2 are incomes in periods 1 and 2 and A is the endowment he received at birth. The price of the consumption good in the first period is normalised to 1. Gross interest rate 1 + r is the relative price of consumption goods today to consumption goods tomorrow. This household's lifetime utility can be maximised by using Lagrangian multiplier method. The solution to this optimisation problem is often known as the Euler's Equation:

${\displaystyle {\frac {u'_{c_{1}}}{u'_{c_{2}}}}=\beta (1+r)}$

The main prediction from this hypothesis model is that that the choices made by consumers regarding their consumption patterns are determined not by current income but by their longer-term income expectations. The key conclusion of this theory is that transitory, short-term changes in income have little effect on consumer spending behavior. Measured income and measured consumption contain a permanent element and a transitory element. Friedman concluded that the individual will consume a constant proportion of his/her permanent income; and that low income earners have a higher propensity to consume; and high income earners have a higher transitory element to their income and a lower than average propensity to consume. The key determinant of consumption is an individual's real wealth, not his current real disposable income.