Introduction to Consumer Choices
5. Intertemporal Choices in Financial Capital Markets
5.1. Using Marginal Utility to Make Intertemporal Choices
Savings behavior varies considerably across households. One factor is that households with higher incomes tend to save a larger percentage of their income. This pattern makes intuitive sense; a well-to-do family has the flexibility in its budget to save 20–25% of income, while a poor family struggling to keep food on the table will find it harder to put money aside.
Another factor that causes personal saving to vary is personal preferences. Some people may prefer to consume more now, and let the future look after itself. Others may wish to enjoy a lavish retirement, complete with expensive vacations, or to pile up money that they can pass along to their grandchildren. There are savers and spendthrifts among the young, middle-aged, and old, and among those with high, middle, and low income levels.
Consider this example: Yelberton is a young man starting off at his first job. He thinks of the "present" as his working life and the "future" as after retirement. Yelberton's plan is to save money from ages 30 to 60, retire at age 60, and then live off his retirement money from ages 60 to 85. On average, therefore, he will be saving for 30 years. If the rate of return that he can receive is 6% per year, then $1 saved in the present would build up to $5.74 after 30 years (using the formula for compound interest, $1(1 + 0.06)30 = $5.74). Say that Yelberton will earn $1,000,000 over the 30 years from age 30 to age 60 (this amount is approximately an annual salary of $33,333 multiplied by 30 years). The question for Yelberton is how much of those lifetime earnings to consume during his working life, and how much to put aside until after retirement. This example is obviously built on simplifying assumptions, but it does convey the basic life-cycle choice of saving during working life for future consumption after retirement.
Figure 6.9 and Table 6.9 show Yelberton's intertemporal budget constraint. Yelberton's choice involves comparing the utility of present consumption during his working life and future consumption after retirement. The rate of return that determines the slope of the intertemporal budget line between present consumption and future consumption in this example is the annual interest rate that he would earn on his savings, compounded over the 30 years of his working life. (For simplicity, we are assuming that any savings from current income will compound for 30 years). Thus, in the lower budget constraint line on the figure, future consumption grows by increments of $574,000, because each time $100,000 is saved in the present, it compounds to $574,000 after 30 years at a 6% interest rate. If some of the numbers on the future consumption axis look bizarrely large, remember that this occurs because of the power of compound interest over substantial periods of time, and because the figure is grouping together all of Yelberton's saving for retirement over his lifetime.
Figure 6.9 Yelberton's Choice: The Intertemporal Budget Set Yelberton will make a choice between present and future consumption. With an annual rate of return of 6%, he decides that his utility will be highest at point B, which represents a choice of $800,000 in present consumption and $1,148,000 in future consumption. When the annual rate of return rises to 9%, the intertemporal budget constraint pivots up. Yelberton could choose to take the gains from this higher rate of return in several forms: more present saving and much higher future consumption (J), the same present saving and higher future consumption (K), more present consumption and more future consumption (L), or more present consumption and the same future consumption (M).
| Present Consumption | Present Savings | Future Consumption (6% annual return) | Future Consumption (9% annual return) |
|---|---|---|---|
| $1,000,000 | 0 | 0 | 0 |
| $900,000 | $100,000 | $574,000 | $1,327,000 |
| $800,000 | $200,000 | $1,148,000 | $2,654,000 |
| $700,000 | $300,000 | $1,722,000 | $3,981,000 |
| $600,000 | $400,000 | $2,296,000 | $5,308,000 |
| $400,000 | $600,000 | $3,444,000 | $7,962,000 |
| $200,000 | $800,000 | $4,592,000 | $10,616,000 |
| 0 | $1,000,000 | $5,740,000 | $13,270,000 |
Table 6.9 Yelburton's Intertemporal Budget Constraint
Yelberton will compare the different choices along the budget constraint and choose the one that provides him with the highest utility. For example, he will compare the utility he would receive from a choice like point A, with consumption of $1 million in the present, zero savings, and zero future consumption; point B, with present consumption of $800,000, savings of $200,000, and future consumption of $1,148,000; point C, with present consumption of $600,000, savings of $400,000, and future consumption of $2,296,000; or even choice D, with present consumption of zero, savings of $1,000,000, and future consumption of $5,740,000. Yelberton will also ask himself questions like these: "Would I prefer to consume a little less in the present, save more, and have more future consumption?" or "Would I prefer to consume a little more in the present, save less, and have less future consumption?" By considering marginal changes toward more or less consumption, he can seek out the choice that will provide him with the highest level of utility.
Let us say that Yelberton's preferred choice is B. Imagine that Yelberton's annual rate of return raises from 6% to 9%. In this case, each time he saves $100,000 in the present, it will be worth $1,327,000 in 30 years from now (using the formula for compound interest that $100,000 (1 + 0.09)30 = $1,327,000). A change in rate of return alters the slope of the intertemporal budget constraint: a higher rate of return or interest rate will cause the budget line to pivot upward, while a lower rate of return will cause it to pivot downward. If Yelberton were to consume nothing in the present and save all $1,000,000, with a 9% rate of return, his future consumption would be $13,270,000, as shown on Figure 6.9.
As the rate of return rises, Yelberton considers a range of choices on the new intertemporal budget constraint. The dashed vertical and horizontal lines running through the original choice B help to illustrate his range of options. One choice is to reduce present consumption (that is, to save more) and to have considerably higher future consumption at a point like J above and to the left of his original choice B. A second choice would be to keep the level of present consumption and savings the same, and to receive the benefits of the higher rate of return entirely in the form of higher future consumption, which would be choice K.
As a third choice Yelberton could have both more present consumption - that is, less savings - but still have higher future consumption because of the higher interest rate, which would be choice like L, above and to the right of his original choice B. Thus, the higher rate of return might cause Yelberton to save more, or less, or the same amount, depending on his own preferences. A fourth choice would be that Yelberton could react to the higher rate of return by increasing his current consumption and leaving his future consumption unchanged, as at point M directly to the right of his original choice B. The actual choice of what quantity to save and how saving will respond to changes in the rate of return will vary from person to person, according to the choice that will maximize each person's utility.