The inter-temporal choice is an economic term describing how current decisions affect what options become available in the future. A preference for focusing on current consumption leads many individuals to make intertemporal choices that accommodate near-term needs and wants. In other words, decisions that have consequences in multiple time periods are intertemporal choices. Decisions about savings, work effort, education, nutrition, exercise, and health care are all intertemporal choices, that is, choices involving different periods of time.
Irving Fisher developed the inter-temporal choice model with two periods, the present and the future. When people consume and save a part of their income, they are thinking about their present and future consumption. Higher consumption in the present time implies lower consumption in the future and vice-versa which is a trade-off. Irving Fisher, thus, developed this model to show how a rational consumer recognises this trade-off and distributes his/her consumption over time, that is, making intertemporal choices. Consumers always face a budget constraint due to which they consume less than they desire. When they decide how much to consume at present and how much to save for the future, they face an intertemporal budget constraint (measures the total resources available for consumption today and in the future). Suppose that a consumer lives for only two periods wherein period 1 represents the consumer’s younger stage and period 2 represents the consumer’s old age. The consumer’s real income and real consumption in period 1 are represented as Y1 and C1 and in period 2 as Y2 and C2, respectively. Since the consumer can borrow and save, consumption in any one period can be either greater or less than the income in that period.
In period 1, savings is given as, S1 = Y1 – C1
In period 2, consumption equals the accumulated savings in period 1, including the interest rate earned on that savings plus income in period 2, that is, C2 = (1 + r) S1 + Y2, where r is the real interest rate. Since only two periods are taken, the consumer consumes S1 in period 2 and does not save in period 2.
If C1 is less than Y1 in period 1 (C1 < Y1), the consumer is saving and S1 will be greater than zero. If C1 is greater than Y1 in period 1 (C1 > Y1), the consumer is borrowing and S1 will be less than zero.
Substituting equation S1 = Y1 – C1 in C2 = (1 + r) S1 + Y2, we can derive consumer’s budget constraint as,
C2 = (1 + r) (Y1 – C1) + Y2
C2 = (1 + r) Y1 – (1 + r) C1 + Y2
By rearranging the similar terms on one side, we get,
(1 + r) C1 + C2 = (1 + r) Y1 + Y2
Dividing both sides by (1 + r), we get,
The above equation shows the consumption to income in the two periods which is the standard way of expressing the consumer’s intertemporal budget constraint. If r = 0, total consumption equals total income in the two periods. If r > 0, future consumption and future income are discounted by a factor 1 + r. This discounting arises from the interest rate earned on savings. The consumer earns interest on savings (S1) from current income in period 1. Since the consumer is not saving in period 2, no interest rate is earned in period 2 and so, his future income is worth less than his current income. Similarly, future consumption is paid from the savings (S1) that have already earned interest in period 1, and therefore, future consumption costs less than current consumption.
The factor 
is the price of C2 measured in terms of C1, that is, it is the amount of C1 that the consumer must sacrifice to obtain one unit of C2. This is because, in period 1, the consumer with the given income Y1 consumes a part of the income C1 and also saves the rest of the income S1. This savings S1 plus the interest rate earned, (1 + r) S1 is used for consumption in period 2, that is, for future consumption. In period 2, since the consumer does not save, the consumer does not earn any interest rate. Therefore, in order to consume more in period 2, that is, C2, consumption C1 must be reduced and savings S1 must be increased in period 1. In other words, current consumption must be reduced and current savings must be increased in order to consume more in the future (with the future income, current-period savings plus its interest).
Fisher’s intertemporal two-period choice model is explained with the help of the intertemporal budget constraint and the indifference curves where the x-axis and y-axis measure C1 and C2, respectively.
The consumer prefers the best possible combination of consumption in the two periods which is obtained at the point where the highest possible indifference curve is tangent to the budget constraint. The slope of the indifference curve is the marginal rate of substitution and the slope of the budget line is 1 + r. Therefore, at the optimum point, MRS = 1 + r, which means that the consumer chooses consumption in the two periods in such a way that the marginal rate of substitution equals 1 + r.
Change in consumption due to change in income: An increase in income, either Y1 or Y2 shifts the budget constraint upwards. With a higher budget constraint, the consumer can reach a higher indifference curve. If the consumer increases his consumption with an increase in income, the goods consumed are called normal goods. Fisher’s model states that consumption is based on the income that the consumer expects over his entire lifetime. Hence, irrespective of whether the increase in income occurs in period 1 or 2, the consumer spreads his income over the consumption in both periods. This is called consumption smoothing. Therefore, it can be said that consumption depends on the present value of current and future income and can be presented as,
Change in consumption due to change in real interest rate: An increase in the real interest rate rotates the consumer’s budget line around the point (Y1, Y2) and, thereby, changes the amount of consumption in both periods. The consumer moves from point A to point B and at point B, it can be observed that C1 falls and C2 increases. The impact of an increase in real interest rates on consumption can be divided into two effects: Income and Substitution Effects.
The income effect is the change in consumption that results from the movement to a higher indifference curve. Since the consumer is a saver, as C1 < Y1, the increase in the interest rate makes him better-off as he moved to a higher indifference curve. If normal goods are consumed in both C1 and C2, the consumer will want to spread this improvement over both periods (consumption smoothing). Thus, the income effect tends to make the consumer want more consumption in both periods.
The substitution effect is the change in consumption that results from the change in the relative price of consumption in the two periods. C2 becomes less expensive relative to C1 when the interest rate rises because the real interest rate earned on saving is higher in period 1 which is used for consumption in period 2. Thus, with the savings plus interest rate from period 1 and the income from period 2 (Y2), the consumer can now increase consumption in period 2. So, the consumer need not lower his consumption in period 1 (C1) in order to obtain an extra unit of C2. Thus, the substitution effect tends to make the consumer increase his consumption in period 2 and lessen his consumption in period 1.
The consumer’s choice depends on both the income effect and the substitution effect. Because both effects act to increase the amount of C2, it can be concluded that an increase in the real interest rate raises C2. But the two effects have opposite impacts on C1, so the increase in the interest rate could either lower (substitution effect) or raise it (income effect). Hence, depending on the relative size of income and substitution effects, an increase in the interest rate could either increase or decrease savings.
Fisher’s model assumes that the consumer can borrow as well as save. The ability to borrow allows current consumption to exceed current income (C1 > Y1). When the consumer borrows, he consumes some of his future income today. But if this assumption was removed, then, the inability to borrow prevents current consumption from exceeding current income, which can be expressed as, C1 ≤ Y1. This inequality states that C1 must be less than or equal to Y1 and is known as borrowing constraint or liquidity constraint.
The consumer’s choice must satisfy both the intertemporal budget constraint and the borrowing constraint. The shaded area represents the combinations of first-period consumption and second-period consumption that satisfy both constraints. This is how the borrowing constraint restricts the consumer’s set of choices.
When the consumer faces a borrowing constraint, there are two possible scenarios. In Fig. (a), the consumer chooses C1 < Y1, so the borrowing constraint is not binding (it means that the consumer has some savings, so borrowing is not necessary) and does not affect consumption in either period. In Fig. (b), the borrowing constraint is binding. The consumer would like to consume at point D where his consumption will be greater than his income. But since there is a borrowing constraint, the best available choice is at point E, where his C1 = Y1.
Therefore, for the consumer for whom the borrowing constraint is not binding, consumption in both periods depends on the present value of lifetime income,
and for the consumers for whom the borrowing constraint is binding, current consumption equals current income C1 = Y1 and C2 = Y2. Thus, for those consumers who would like to borrow but cannot, consumption depends only on current income. Thus, consumers’ current consumption choice depends not only on their current income but also on their future consumption, future income and borrowing constraint.
References
Levacic, R., & Rebmann, A. (2015), ‘Macroeconomics: An Introduction to Keynesian-Neoclassical Controversies’, Macmillan International Higher Education.
Mankiw, G. (2010), ‘Macroeconomics’, Worth Publishers, 7th ed.
FISHER’S INTER-TEMPORAL CHOICE MODEL (in Malayalam)
World of Economics 