INTRODUCTION INTER TEMPORAL CHOICE

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1 INTRODUCTION The theories that were developed to explain the observed phenomena (already noted in the first lecture) all have basic foundations in the microeconomic theory of consumer choice. In particular, both approaches/theories developed by Friedman and Modigliani begin with the explicit assumption that observed consumer behaviour is the result of an attempt by rational consumers to maximise utility by allocating a lifetime stream of earnings to an optimum lifetime pattern of consumption. INTER TEMPORAL CHOICE Thus, along with Friedman and Modigliani, and before them Irvin Fisher (who pioneered this approach of inter-temporal consumer choice) we can begin with the case of single consumer with a utility function Where lifetime utility U is a function of his real consumption in all time periods up to T, the instant before the consumer dies. The objective of the consumer to maximise his/her utility, that is, to obtain the highest level of utility, subject to the budget constraint that the present value of his total consumption in life cannot exceed the present value of his total income in life. Symbolically, this constraint can be written as Where T is the individual s expected life time. This budget constraint says that the consumer can allocate his income stream to a consumption stream by borrowing and lending, but the present value of consumption is limited by the present value of income. 1

2 An important point worth making! For this restriction to hold as a strict equality we make a further assumption that if a person receive receives an inheritance, he/she passes on a bequest of an equal amount. To simplify our lives, we restrict the analysis to a two-period case. Here we suppose that the consumer has an income stream, and wants to maximise subject to the borrowing-lending constraint. 1 Illustration of this is provided below in Figure 1. Income in Period 1 y 1 A c 1 B Slope = -(1+r) y 0 c 0 Income in Period 0 1 On the use of time notation, period 0 refers to the present, in t = 0, 1. This is analogous to period 1, in t =1, 2 2

3 In the figure above, the income stream locates point A. This point shows the amount of income the individual will earn in period 0,, and the amount he/she will earn in period 1,. We also assume that the individual can either borrow or lend at the interest r. Thus, if his/her income in period 0 is greater than the value of goods and services he/she wants to consume in that period he can lend, that is, save his unspent income: income lent in period 0. (Also note that if the consumer borrows in period 0, then ). By lending this amount in period 0 at the interest rate r, he/she will receive in period 1 an amount equal to, so that his/her consumption in period 1 can exceed his/her income in by that amount, which is period 1 dissaving, :. We introduce the minus sign because of the dissaving in period 1, which, can easily be understood as the opposite sign to the saving in period 0, and. Diving the expression for by that for yields the trade-off between present and future consumption,. From the right hand of the equation above, by cancelling the through by, we obtain s and multiplying This says that by reducing consumption in period 0 below income by the amount, the consumer can enjoy in period 1 consumption in excess of income,, by the amount. In other words the consumer can trade from a income point to a consumption point along a budget line that has a slope of (1+r). Another way to construct the this budget line is to suppose that the individual wants to consume 100 percent of his/her income stream in period 0, by borrowing against his/her period 1 income. In that case, the maximum amount s/he can. 3

4 consume in period 0 will then be, which is the intercept of the budget line on the period 0 axis. Conversely, if the individual decides not to consume at all in period 0, putting off all consumption until period 1, the maximum s/he can consume in period 1 will be, which is the intercept of the budget line on the period 1 axis. Thus, the budget line above bounds the consumption possibilities open to the individual with an income stream, facing an interest rate, r. We also know that from the individual s utility function we can obtain a set of indifference curves that show the points at which s/he is indifferent between additional consumption in period 1 or period 0 at each level of utility. Higher levels of indifference curves indicate higher levels of utility for the individual consumer. Figure 2 illustrates utility maximization in the two-period consumption case. Income in Period 1 Y 1 A C 1 B U 2 U 1 U 0 Y 0 C 0 Income in period 0 4

5 Important Things We Need To Note (some familiar, some new): All points on or below the budget line are attainable Utility maximization ensures that individual consumes on the budget line, at that point, the indifference curve must be tangent to the budget line. This occurs at point B, at which point the individual s consumption pattern is Since the individual s consumption is skewed toward period 1 (y 1 is much greater than y 0 ), the individual borrows c 0 y 0 in period at the interest rate, r. In period 1, the individual pays back The individual s pattern of consumption, including present consumption c 0, is determined by the position of the budget line and the shape of the indifference curves. The position of the budget line is determined by two variables the income in each period and the interest rate. Hence, we know that if the individual s income should increase in any period, the present value of his/her income stream will increase. In other words, any increase in income will shift the budget line up parallel to the old budget line. The consumer can thus reach a new, higher level of utility at a new point. The second variable is the interest rate. We know that the slope of the budget line is. If consumption in any period is not an inferior good, then whenever any period s income rises, all periods consumptions rise. So for example, the level of present consumption will rise with a rise in future (expected) income. 5

6 One implication of the forgoing analysis is that current period consumption will vary less than does income. In the two-period case we utilise, an increase in present income would be spread across an increase in both present and future consumption. Thus, if we extend the analysis to many periods, say 25 years, spreading an increase in present income over 25 years of consumption increments would give a present consumption increase that is very small relative to the increase in present income The relationship between the present value of the income stream and current consumption gives us the first general formulation of the consumption function,, where PV 0 is the present value of current and expected future income at time 0. Thus, PV 0 is formulated as. This consumption function, simply says that an individual s consumption in time 0 is an increasing function of the present value of his/her income in time 0. Symbolically, is the slope of the consumption function. 6