Unit 9 Lenders and borrowers and differences in wealth

9.4 Reasons to borrow: The value of spending now

Why people want to smooth their consumption

consumption smoothing

Actions taken by an individual, family, or other group in order to sustain their customary level of consumption. Actions include borrowing or reducing savings to offset negative shocks, such as unemployment or illness; and increasing saving or reducing debt in response to positive shocks, such as promotion or inheritance.

diminishing marginal utility

If the value to the individual of an additional unit of some good declines the more that is consumed, holding constant the amount of other goods, we say that the good has diminishing marginal utility.

utility

A numerical indicator of the value that one places on an outcome. Outcomes with higher utility will be chosen in preference to lower valued ones when both are feasible.

marginal rate of substitution (MRS)

The trade-off that a person is willing to make between two goods. At any point, the MRS is the absolute value of the slope of the indifference curve. See also: marginal rate of transformation.

The desire to smooth one’s consumption (consumption smoothing) means preferring both some now and some later rather than everything now and nothing later or the reverse—having nothing now and everything later. For example, a powerful motivation for wanting to bring forward buying power is to be able to purchase the necessities of food, rent, phone contract, and electricity.

Think about food—the first few bites of a dish are likely to be much more pleasurable than bites from your third serving. This is a fundamental psychological reality, sometimes termed the law of satiation of wants: the first bite satisfies more intense wants than the 25th bite.

The desire to smooth consumption is explained by diminishing marginal utility of consumption: at a given point in time, the value to the individual of an additional unit of consumption declines as more is consumed. Diminishing marginal utility implies that we prefer to smooth our consumption, having the same in each period of time rather than having a lot sometimes and a little at other times.

Remember that indifference curves join points of equal utility. The preference for smoothing means that the desire to bring forward consumption from the future to the present falls as we move along an indifference curve from left to right. The slope of the indifference curve is the marginal rate of substitution (MRS) between consumption now and in the future. Where future consumption is high and present consumption is low (towards the left), an additional unit of present consumption brings more utility than an additional unit of future consumption. So the MRS is high and the indifference curve is steep. The MRS falls as we move towards the right, so the curve becomes flatter. As a result, the indifference curves are bowed to the origin.

impatience

A preference for consuming something sooner rather than later. Impatience may by situational (because the person has little now and will have more later); or intrinsic, in which case they would prefer to consume more now rather than the same amounts now and later.

While diminishing marginal utility (the preference for smoothing) and the associated bowed shape of the indifference curves is universal, when we compare people with the same endowment, their preference for shifting consumption (that is, the slope of the indifference curve at that point) can differ. This brings us to the difference between situational and intrinsic impatience.

To make the analysis more concrete and memorable, let’s return to Julia. She, like anyone else, could be impatient for two reasons:

• Situational impatience. Her situation is that she is poor now and will be better off later; she prefers to smooth her consumption between now and later, instead of consuming everything later and nothing now.
• Intrinsic impatience. If she is able to consume the same amount in both periods, and is given the choice to consume more now and less later, she is intrinsically impatient if she chooses to consume more now.

Situational impatience: Being present-oriented because you have less now and more later

Situational impatience means that if, like Julia, our situation is that we have nothing now and $100 in the next period then the marginal utility of a dollar now is greater than the marginal utility of a dollar next period. As a result, we would be willing to give up more than a dollar in the future to get a dollar now. This would allow us to smooth our consumption between the two periods.

This is also true if we have lots later and little now, as Figure 9.4a shows.

Intrinsic impatience: How impatient you are as a person

There is a second reason for preferring the good now, called intrinsic impatience. To determine whether someone is intrinsically impatient, we ask whether, if they initially had the same amount of the good in both periods, they would value more highly having more of the good now than more of it later. Two reasons for intrinsic impatience are:

• Myopia (short-sightedness): People experience the present satisfaction of hunger or some other desire more strongly than they imagine the same satisfaction at a future date.
• Prudence: People know that the opportunity to consume the good in the future may not be available and so choosing present consumption may be a good idea. (For example, you are offered a ‘dream’ holiday, which you could take now or next summer—prudence may lead you to take it now.)

The terms ‘myopia’ and ‘prudence’ are descriptive—they are not judgemental. Whether because of myopia or prudence or some combination, most people are intrinsically impatient.

To understand what intrinsic impatience means, we compare two points on the same indifference curve in Figure 9.4b. Julia’s indifference curve is in black. At point A, she has $50 now and $50 later, so her consumption is completely smooth at her initial endowment. We ask how much extra consumption she would need to have later in order to compensate her for losing $10 now. Point B on the same indifference curve gives us the answer. If she had only $40 now, she would need $62 later in order to stay on the same indifference curve and be equally happy. So she needs $12 later to compensate for losing $10 now. Julia has intrinsic impatience because rather than preferring to perfectly smooth her consumption, she places more value on an additional unit of consumption today than in the future.

We can use the same reasoning to compare the intrinsic impatience of different people. The blue indifference curve that also passes through point A shows the preferences of another person. Point C shows that if they gave up $10 today, they would need to have $20 tomorrow to be equally happy. In contrast, Julia only needs $12 tomorrow (her indifference curve through point A is shallower), so she is more patient than the other person.

There is another way to find out how intrinsically impatient a person is. Returning to Julia, who will receive $100 in the future, we know she wants to borrow. The situation that she is in gives her a strong desire to smooth by borrowing. Think about what the shape of Julia’s indifference curve, passing through her endowment point, might be. As shown in Figure 9.5, at point A she has a strong preference for increasing consumption now.

reservation indifference curve

A curve that indicates combinations of goods that are as highly valued as one’s reservation option.

This is called Julia’s reservation indifference curve, because it is made of all the points at which Julia would be just as well off as at her reservation position, which is her endowment with no borrowing or lending. At point A, with no expenditure at all on consumption now, we assume Julia has some way of maintaining herself.

Figure 9.5 shows three of Julia’s indifference curves:

• Her reservation indifference curve: The curve closest to the origin.
• Her reservation indifference curve if she had the $100 now: The next highest curve assumes that she has already earned $100.
• A higher indifference curve.

For any particular point in the figure, the individual’s impatience is indicated by the steepness of the indifference curve. At her endowment point—$100 later, nothing now—shown by point A, her indifference curve is very steep. Because she has nothing now, she is very impatient. It means she would be willing to give up a substantial amount of consumption later to gain a little bit of consumption now. This could be illustrated by a move from point A to A′. The fact that the steepness of the indifference curves differ at the various points in the figure illustrates situational impatience (each point in the figure is a different situation).

discount rate

A measure of someone’s impatience: how much the person values an additional unit of consumption now relative to an additional unit of consumption later. It is equal to the slope of the indifference curve for consumption now and consumption later, minus one. Also known as: subjective discount rate.

To measure someone’s impatience, namely how much they value an extra unit of consumption now, relative to an extra unit of consumption later, we calculate their discount rate. A person’s discount rate, \(\rho\) (economists use the Greek letter \(\rho\), or rho, pronounced ‘row’), is calculated as the slope (remember, we take the absolute—positive—value) of the indifference curve minus one.

Notice in Figure 9.5 that if Julia hypothetically had the $100 now (point B), she would be much less impatient; at B, her indifference curve is very flat. Given her very different hypothetical situation at B, her desire to smooth her consumption would mean that she would like to have more consumption in the future and less now. So she would be willing to give up a dollar now, even if she got less than a dollar in return later. This means that smoothing can happen in either direction, moving consumption from the future to the present or from the present to the future.

The figure shows that not only would Julia be less impatient if she were hypothetically at point B (with $100 now) than at point A (in her real situation, having $100 later), but she would also be better off. In Figure 9.5, the indifference curve through B is above the indifference curve through A. To understand this, notice that when Julia’s utility is at the same level as when she has the $100 in the future, she must be on the same indifference curve, that is, the one going through A. You can see that on that indifference curve at B′, her consumption is much lower than $100. For the indifference curves shown in Figure 9.5, she values $100 later the same as she values half that amount now (B′ is one-half of B). This is because, as a person, she has a degree of intrinsic impatience. This helps illustrate why we say that most people are intrinsically impatient.

Borrowing allows smoothing by bringing consumption to the present

How much will Julia borrow? If we combine Figure 9.2 and Figure 9.3, we will have the answer. Just like other constrained choice models, where there is a feasible set and indifference curves, in the model of intertemporal choice, Julia wishes to get to the highest possible indifference curve but is limited by her feasible frontier. The highest feasible indifference curve will be the one that is tangent to the feasible frontier, shown as point E in Figure 9.6. Recall that the slope of the feasible frontier depends on the interest rate. This means that Julia borrows just enough so that:

\[\begin{align} \text{slope of the indifference curve (MRS) } = \text{ slope of the feasible frontier (MRT)} \end{align}\]

We know that:

\[\begin{align*} \text{MRS} &= 1 + \rho \\ \text{MRT} &= 1 + r \\ \end{align*}\]

Therefore:

\[\begin{align*} \text{MRS} &= \text{MRT} \\ 1 + \rho &= 1 + r \\ \end{align*}\]

If we subtract 1 from both sides of this equation, we have:

\[\begin{align*} \rho &= r \\ \text{discount rate} &= \text{rate of interest} \\ \end{align*}\]

Here, Julia chooses to borrow and consume $56 and repay $62 later, leaving her $38 to consume later.

Now consider how much she would borrow if she had to pay not the 10% interest rate, but the 78% that was the average among the Chambar farmers. Figure 9.6 shows that, as before, finding the point of tangency between the new feasible frontier given by the 78% interest rate and one of Julia’s indifference curves, she will choose point G, meaning that she will borrow much less—$35—to consume now, paying $62 with interest, and having $38 to consume later.

The higher ‘price’ of moving consumption forward in time means two things:

• She is less well off than with the lower interest rate: Compare her two indifference curves (one through E, the other through G).
• She will borrow less and consume less now: $35 rather than $56.

income effect

The effect that an increase in income has on an individual’s demand for a good (the amount that the person chooses to buy) because it expands the feasible set of purchases. When the price of a good changes, this has an income effect because it expands or shrinks the feasible set, and it also has a substitution effect. See also: substitution effect.

substitution effect

When the price of a good changes, the substitution effect is the change in the consumption of the good that occurs because of the change in the good’s relative price. The price change also has an income effect, because it expands or shrinks the feasible set. See also: income effect.

In the example we considered above, Julia’s consumption later does not change. But, that need not be the case: depending on the shape of her indifference curves, it could be lower or higher. What is certain is that with the higher interest rate, she is less well off, so this would tend to push down her consumption later (this is an income effect). But the higher interest rate makes it costlier to bring consumption forward, which would tend to push up her consumption later (this is a substitution effect). In the example shown, the two effects are of equal size—which is why there is no change in consumption later.

Use the analysis in Figure 9.6 to understand how Julia will choose consumption when the interest rate is 10% and when it is 78%.