Unit 9 Lenders and borrowers and differences in wealth

9.7 Investing: Another way to move consumption to the future

If Marco has an investment project, he could do even better. To make the example transparent, assume Marco owns some land, and instead of $100 cash, he has $100 worth of grain. He could consume the grain. Alternatively, he could invest the grain (planting it as seed and feeding it to his draught animals to help him work the fields until harvest). If he takes this opportunity to invest (rather than consuming the grain) he will further expand his feasible set. Suppose that if he were to invest all of his grain, he could harvest $150 worth of grain later, as shown in Figure 9.9. He has invested $100, harvested $150, and so earned a profit of $150 − $100 = $50, or a profit rate (profits divided by the investment required) of $50/$100 = 50%. The slope of Marco’s budget constraint (the red line) is −1.5, where the absolute value (1.5) is the MRT of investment into returns, or 1 plus the rate of return on the investment.

In this diagram, the horizontal axis shows consumption now in dollars, and ranges from 0 to 120. The vertical axis shows consumption later in dollars, and ranges from 0 to 160. Coordinates are (consumption now, consumption later). Point (100, 0) is Marco’s endowment. A downward-sloping straight line connects points (0, 150) and (100, 0) and is labelled FF (invest grain, 50% return). The slope of the line shows that one dollar of consumption now can be exchanged for 1.5 dollars of consumption later. A downward-sloping, convex curve labelled Marco’s IC is tangential to FF (invest grain, 50% return) at point K (60, 60) where MRS = MRT. The horizontal distance between the vertical axis and point K is consumption now. The horizontal distance between point K and point (100, 0) is investment. The vertical distance between point K and the horizontal axis is consumption later.

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Figure 9.9 Investing in a high-return project.

The return on investment: In this diagram, the horizontal axis shows consumption now in dollars, and ranges from 0 to 120. The vertical axis shows consumption later in dollars, and ranges from 0 to 160. Coordinates are (consumption now, consumption later). Point (100, 0) is Marco’s endowment. A downward-sloping straight line connects points (0, 150) and (100, 0) and is labelled FF (invest grain, 50% return).

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The return on investment

If Marco were to invest all of his $100 worth of grain now (and so consume nothing now), he could harvest $150 worth of grain later.

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The return on investment

The slope of Marco’s budget constraint (the red line) is 1 plus the rate of return on the investment.

Marco’s choice: In this diagram, the horizontal axis shows consumption now in dollars, and ranges from 0 to 120. The vertical axis shows consumption later in dollars, and ranges from 0 to 160. Coordinates are (consumption now, consumption later). Point (100, 0) is Marco’s endowment. A downward-sloping straight line connects points (0, 150) and (100, 0) and is labelled FF (invest grain, 50% return). The slope of the line shows that one dollar of consumption now can be exchanged for 1.5 dollars of consumption later. A downward-sloping, convex curve labelled Marco’s IC is tangential to FF (invest grain, 50% return) at point K (60, 60) where MRS = MRT. The horizontal distance between the vertical axis and point K is consumption now. The horizontal distance between point K and point (100, 0) is investment. The vertical distance between point K and the horizontal axis is consumption later.

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Marco’s choice

Marco chooses to invest $40 and so to consume $60 now and $60 later, as shown by point K. Note the labels on the horizontal axis for amounts consumed and invested. At this point, the feasible frontier is tangent to an indifference curve.

If Marco could get a loan at 10%, he would quickly realise that he would be able to reach a higher indifference curve with an entirely new plan: invest everything he has, with a harvest next year of $150, but also borrow now in order to be able to consume more both now and in the future. This ‘invest-it-all’ plan is shown in Figure 9.10. The plan shifts Marco’s feasible frontier out even further, so he can now consume a maximum of $136 today, as shown by the dotted red line. Marco ends up consuming at a new point, L, with more both now and in the future.

In this diagram, the horizontal axis shows consumption now in dollars, and ranges from 0 to 150. The vertical axis shows consumption later in dollars, and ranges from 0 to 150 160. Coordinates are (consumption now, consumption later). Point (100, 0) is Marco’s endowment. A downward-sloping straight line connects points (0, 150) and (100, 0) and is labelled FF (invest grain, 50% return). The slope of the line shows that one dollar of consumption now can be exchanged for 1.5 dollars of consumption later. A downward-sloping, convex curve labelled Marco’s IC is tangential to FF (invest grain, 50% return) at point K (60, 60). Another downward-sloping, straight line connects points (0, 150) and (136, 0) and is labelled FF (invest grain, 50% return; borrow at 10%). A downward-sloping convex curve labelled Marco’s IC (very high utility) is tangential to this FF line at point L (80, 62). The horizontal distance between the vertical axis and point K is consumption now. The horizontal distance between the vertical axis and point L is investment.

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Figure 9.10 Borrowing to invest in a high-return project.

Marco’s choice when he can invest: In this diagram, the horizontal axis shows consumption now in dollars, and ranges from 0 to 150. The vertical axis shows consumption later in dollars, and ranges from 0 to 150. Coordinates are (consumption now, consumption later). Point (100, 0) is Marco’s endowment. A downward-sloping straight line connects points (0, 150) and (100, 0) and is labelled FF (invest grain, 50% return). The slope of the line shows that one dollar of consumption now can be exchanged for 1.5 dollars of consumption later. A downward-sloping, convex curve labelled Marco’s IC (high utility) is tangential to FF (invest grain, 50% return) at point K (60, 60).

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Marco’s choice when he can invest

His choice when he can invest is at point K.

Marco gets a loan: In this diagram, the horizontal axis shows consumption now in dollars, and ranges from 0 to 150. The vertical axis shows consumption later in dollars, and ranges from 0 to 150. Coordinates are (consumption now, consumption later). Point (100, 0) is Marco’s endowment. A downward-sloping straight line connects points (0, 150) and (100, 0) and is labelled FF (invest grain, 50% return). The slope of the line shows that one dollar of consumption now can be exchanged for 1.5 dollars of consumption later. A downward-sloping, convex curve labelled Marco’s IC (high utility) is tangential to FF (invest grain, 50% return) at point K (60, 60). Another downward-sloping, straight line connects points (0, 150) and (136, 0) and is labelled FF (invest grain, 50% return; borrow at 10%).

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Marco gets a loan

With the loan at the 10% interest rate, he could consume up to $\frac{150}{1\ +\ 0.1}=136$ now, so he would be better off by investing everything he has. The ability to borrow expands his feasible set so that he can consume up to $136 today, as shown by the dotted red line.

Marco’s best choice after getting a loan: In this diagram, the horizontal axis shows consumption now in dollars, and ranges from 0 to 150. The vertical axis shows consumption later in dollars, and ranges from 0 to 150 160. Coordinates are (consumption now, consumption later). Point (100, 0) is Marco’s endowment. A downward-sloping straight line connects points (0, 150) and (100, 0) and is labelled FF (invest grain, 50% return). The slope of the line shows that one dollar of consumption now can be exchanged for 1.5 dollars of consumption later. A downward-sloping, convex curve labelled Marco’s IC is tangential to FF (invest grain, 50% return) at point K (60, 60). Another downward-sloping, straight line connects points (0, 150) and (136, 0) and is labelled FF (invest grain, 50% return; borrow at 10%). A downward-sloping convex curve labelled Marco’s IC (very high utility) is tangential to this FF line at point L (80, 62). The horizontal distance between the vertical axis and point K is consumption now. The horizontal distance between the vertical axis and point L is investment.

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Marco’s best choice after getting a loan

Marco ends up consuming at point L, with $80 now and $62 in the future.

As before, storage is also an option to move consumption to the future, but in the case of grain it may be costly. The grain could rot or mice could eat some of it. What the mice eat is a form of depreciation: a reduction in Marco’s wealth due to the passage of time. So, taking account of the mice, if he consumed nothing at all during this period, he would have just $80 worth of grain a year later. This means that the cost of moving grain from the present to the future is 20% per year.

Figure 9.11 summarizes how the ‘invest-it-all and borrow’ plan works compared to the other options.

Plan (points in Figures 9.7 and 9.10)	Rate of return or interest	Consumption now, consumption later	Investment	Ranking by utility (or combined consumption)
Storage (M)	−20% (loss)	$68, $26	n/a	Worst ($94)
Lending only (D)	10%	$65, $39	n/a	Third best ($104)
Investment only (K)	50%	$60, $60	$40	Second best ($120)
Investment and borrowing (L)	50% (investment), −10% (lending)	$80, $62	$100	Best ($142)

Figure 9.11 Storage, lending, investment, and borrowing provide Marco with many feasible sets.

The feasible sets for all of Marco’s options are shown in Figure 9.12.

In this diagram, the horizontal axis shows consumption now, in dollars, and ranges between 0 and 150. The vertical axis shows consumption later, in dollars, and ranges from 0 to 150. Coordinates are (consumption now, consumption later). Five points are labelled: H with coordinates (68, 26), J with coordinates (65, 39), K with coordinates (60, 60), L with coordinates (80, 62), and Marco’s endowment of (100, 0). There are four downward-sloping straight lines. The first is the feasible frontier for storing grain with a 20% loss. It passes through the points (0, 80), point H, and Marco’s endowment. The second is the feasible frontier for lending at 10%. It passes through the points (0, 110), point J, and Marco’s endowment. The third is the feasible frontier for investing grain at a 50% return. It passes through the points (0, 150), point K, and Marco’s endowment. The fourth is the feasible frontier for investing grain at a 50% return and borrowing at 10%. It passes through the points (0, 150), point L, and (136, 0). There are four parallel downward-sloping convex curves representing Marco’s indifference curves. The first is tangent to the feasible frontier for storing grain with a 20% loss at point H and is labelled Storing. The second is tangent to the feasible frontier for lending at 10% at point J and is labelled Lending. The third is tangent to the feasible frontier for investing grain at a 50% return at point K and is labelled Investing. The fourth is tangent to the feasible frontier for investing grain at a 50% return and borrowing at 10% at point L and is labelled Investing it all and borrowing.

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Figure 9.12 Options for the individual (Marco) who starts with assets.

Borrowing, the interest rate, and the feasible set.

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Figure 9.3 Borrowing, the interest rate, and the feasible set.

Let’s return to how Marco differs from Julia. Compare the feasible sets of Julia shown in Figure 9.3 and of Marco, whose options are shown in Figure 9.12.

Three differences between Marco and Julia, who share the same intrinsic impatience, explain the disparity in their outcomes:

Marco starts with an asset while Julia starts with nothing: Julia has the prospect of a similar asset later, but this puts the two on opposite sides of the credit market.
Marco has a productive investment opportunity, while Julia does not.
Marco and Julia may face different interest rates: The less-obvious difference is that if Marco (after investing his entire asset at a 50% return) wants to move his buying power forward in time, he borrows against his future income at a rate of 10%. Julia, lacking assets like the poor farmers in Chambar, may have no alternative but to borrow at the higher rate of 78%. The paradox is that Marco can borrow at a low interest rate because he does not need to borrow.

From the cases considered so far, we can generalize from the illustration of Marco and Julia to conclude that borrowing, lending, storing, and investing are ways of moving goods consumption from the future to the present from the present to the future.

People engage in these activities because:

They can increase their utility by smoothing consumption: Or, if they have intrinsic impatience, by moving consumption to the present.
They can increase their consumption across both periods: By lending, or investing.

People differ in which of these activities they engage (some borrowing, some lending) because:

They have differences in their situation: For example, having an income now or later will affect their discount rates and their opportunities. Also, some have investment opportunities (like Marco), while others do not.
They differ in their level of intrinsic impatience (although this is not the case in comparing Marco and Julia).

Building block

Section 3.7 provides an explanation of income and substitution effects.

Exercise 9.7 Income and substitution effects

Use a diagram like Figure 9.10 to show the income and the substitution effects of an increase in the interest rate for Marco who receives his endowment today.
Explain whether or not both of these effects work in the same direction.
Compare these effects for Marco with those for Julia in Exercise 9.3 and explain your results.

Question 9.9 Choose the correct answer(s)

Figure 9.10 depicts Marco’s choice of consumption in periods 1 (now) and 2 (later). He has $100 worth of grain in period 1 and no income in period 2. Marco has two choices. In scheme 1, he can invest the grain he does not consume in period 1, which gives a 50% return. In scheme 2, he can invest the grain that he does not consume and borrow money today at a 10% interest rate. Based on this information, read the following statements and choose the correct option(s).

With scheme 1, if Marco consumes $68 worth of grain in period 1, he can consume $48 worth of grain in period 2.
With scheme 2, if Marco consumes $68 worth of grain in period 1, he can consume $80 worth of grain in period 2.
The marginal rate of transformation is higher under scheme 1 than under scheme 2.
Marco will always be on a higher indifference curve under scheme 2 than under scheme 1.

If Marco consumes $68 worth of grain, he invests $32 with a 50% interest rate, giving him 32 × 1.5 = $48 to consume in period 2.
If Marco borrows to consume $68 in period 1, he will have to repay 68 × 1.1 = $75 in period 2, leaving him with $75 to consume in period 2.
The MRT is the slope of the feasible frontier, so it is higher under scheme 1 than under scheme 2.
The budget constraint in scheme 2 is higher than in scheme 1 at any positive level of saving.

Question 9.10 Choose the correct answer(s)

Figure 9.12 depicts four possible feasible frontiers for Marco, who has $100 worth of grain in period 1 (now) and no income in period 2 (later). In scheme 1, he can store the grain that he does not consume in period 1. This results in 20% loss of the grain due to pests and rotting. In scheme 2, he can sell the grain that he does not consume and lend the money at 10%. In scheme 3, he can invest the remaining grain for a return of 50%. Finally, in scheme 4, he can invest the entire amount of grain and borrow against his future income at 10%. Based on this information, read the following statements and choose the correct option(s).

20% depreciation from storage means that Marco is worse off at H than at his initial endowment of consuming all $100 worth of grain in period 1.
The consumption choice J can only be attained under scheme 2.
If the rate of lending increases, the feasible frontier for scheme 2 tilts inwards from the point $100 on the horizontal axis (becomes flatter).
If the rate of borrowing increases, the feasible frontier for scheme 4 tilts inwards from the point $150 on the vertical axis (becomes steeper).

Marco is on a higher indifference curve at H than at his endowment point. Therefore he is better off.
While J is the best option attainable under scheme 2, it is also within the feasible set of schemes 3 and 4. Therefore it can be attained under those two schemes as well.
With a higher rate of lending, the feasible frontier becomes steeper, pivoting from the point $100 on the horizontal axis. In particular, the intercept with the vertical axis will be higher than $110.
$150 in period 2 is feasible whatever the interest rate so that point is fixed, but a higher interest rate implies that he can borrow less in period 1. So the budget constraint intersects the horizontal axis at less than 136, becoming steeper.

Can Julia borrow to invest?

Like Julia in our model, payday borrowers in New York City often buy groceries or clothes for their children with their borrowed funds; farmers in Chambar also often borrow for purposes of consumption, for example, to pay for a wedding. But both the Chambar farmers and New York payday borrowers sometimes use the borrowed funds for an investment. For the Pakistani farmers, this could be the purchase of equipment that would improve the crop yield.

Now suppose that Julia is considering becoming a driver for a ridesharing company like Uber or Lyft, and to qualify for getting the job, she needs to make some cosmetic repairs on her brother’s car that she will drive. She goes to a payday lender, who, as in the example in Figure 9.3, will charge her an interest rate of 78%. Figure 9.13 shows Julia’s feasible set and frontier based on borrowing at this rate.

Julia has a new option—she can borrow and then split how much she has borrowed between consuming some now and investing the rest. This is how she does her planning:

Take some of the $56 that she can borrow and invest this in fixing up the car.
The more she spends fixing up the car, the more money she will make as a driver, so she now has a new feasible frontier.

Suppose it is the case that every dollar Julia spends on the car will result in $3 more in income next year assuming that she drives the same number of hours (that is, a rate of return of 200%). With this investment, she can move along the new dashed feasible frontier.

To determine the full range of her new options, we deduce that if she invested the entire $56 (so that she would have no consumption now), she would have $168 (3 × 56) next year. All the points on the dashed feasible frontier with investment are now open to her. The slope of the feasible frontier with investment is 3, which is the ratio of income later to the amount invested. The steeper the slope is, the better for Julia. This slope is the marginal rate of transformation of current investment into future income.

In this diagram, the horizontal axis shows consumption now, in dollars, and ranges between 0 and 100. The vertical axis shows consumption later, in dollars, and ranges from 0 to 170. Coordinates are (consumption now, consumption later). A downward-sloping straight line connects points A (0, 100) and (56, 0) and is labelled FF (borrow; interest rate at 78%). Point A is Julia’s endowment. Another downward-sloping straight line connects points (0, 168) and (56, 0) and is labelled FF (borrow at 78%, invest with 200% return). There are three parallel, downward-sloping, convex curves. One is tangential to the FF line for borrowing and investing at point I (35, 63) where MRS = MRT. The second is tangential to the FF line for borrowing at point G (35, 38). The third passes through point A and is labelled Julia’s reservation IC. The vertical distance between point (35, 0) and G is consumption later (borrowing only). The vertical distance between points G and I is extra consumption later (with investment).

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Figure 9.13 Options for the individual (Julia) who starts without assets but can borrow and invest.

Julia’s options: In this diagram, the horizontal axis shows consumption now, in dollars, and ranges between 0 and 100. The vertical axis shows consumption later, in dollars, and ranges from 0 to 170. Coordinates are (consumption now, consumption later). A downward-sloping straight line connects points A (0, 100) and (56, 0) and is labelled FF (borrow; interest rate at 78%). Point A is Julia’s endowment. Another downward-sloping straight line connects points (0, 168) and (56, 0) and is labelled FF (borrow at 78%, invest with 200% return).

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Julia’s options

She can borrow at an interest rate of 78% and can also choose to invest some of her income with a return of 200%. The dotted line shows her feasible frontier when she chooses to borrow and invest.

How much will Julia invest?: In this diagram, the horizontal axis shows consumption now, in dollars, and ranges between 0 and 100. The vertical axis shows consumption later, in dollars, and ranges from 0 to 170. Coordinates are (consumption now, consumption later). A downward-sloping straight line connects points A (0, 100) and (56, 0) and is labelled FF (borrow; interest rate at 78%). Point A is Julia’s endowment. Another downward-sloping straight line connects points (0, 168) and (56, 0) and is labelled FF (borrow at 78%, invest with 200% return). There are three parallel, downward-sloping, convex curves. One is tangential to the FF line for borrowing and investing at point (35, 63). The second is tangential to the FF line for borrowing at point (35, 38). The third passes through point A and is labelled Julia’s reservation IC. The vertical distance between point (35, 0) and (35, 38) is consumption later (borrowing only). The vertical distance between points (35, 38) and (38, 63) is extra consumption later (with investment).

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How much will Julia invest?

She will use the same rule that she used in deciding how much to borrow when there was no option to invest. She will find the highest feasible indifference curve by finding the tangency of an indifference curve and the feasible frontier, or what is the same thing, equating the MRT (slope of the feasible frontier) with the MRS (slope of the indifference curve).

Julia’s choice: In this diagram, the horizontal axis shows consumption now, in dollars, and ranges between 0 and 100. The vertical axis shows consumption later, in dollars, and ranges from 0 to 170. Coordinates are (consumption now, consumption later). A downward-sloping straight line connects points A (0, 100) and (56, 0) and is labelled FF (borrow; interest rate at 78%). Point A is Julia’s endowment. Another downward-sloping straight line connects points (0, 168) and (56, 0) and is labelled FF (borrow at 78%, invest with 200% return). There are three parallel, downward-sloping, convex curves. One is tangential to the FF line for borrowing and investing at point I (35, 63) where MRS = MRT. The second is tangential to the FF line for borrowing at point G (35, 38). The third passes through point A and is labelled Julia’s reservation IC. The vertical distance between point (35, 0) and G is consumption later (borrowing only). The vertical distance between points G and I is extra consumption later (with investment).

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Julia’s choice

Doing this, Julia chooses point I in the figure. How does this work out for her? Having borrowed the maximum—$56—she invests $21 and consumes $35 now. The investment of $21 will yield her an income of $63 later.

The investment opportunity has clearly made Julia better off. She consumes the same amount now as before—$35—but she can now consume $63 later, rather than only $38. Note that she will only invest if she has an investment project with a rate of return higher than 78%—the rate of return has to be higher than the borrowing rate in order to expand her feasible set.

In our example, Julia consumes the same amount now under the borrowing only and borrowing and investment options; her current consumption could be either greater than or less than in the case without the investment opportunity. What is certain is that she is better off with the investment opportunity because her feasible set is expanded.

We can therefore contrast three situations in which Julia might have found herself:

Borrowing and investing: She has the opportunity to bring buying power forward in time to the present (by borrowing) and then shifting some of it back in time while tripling its value (by investing), enabling her to achieve point I.
Borrowing: When Julia can borrow but not invest, her feasible set includes point A, and also all the other points in the solid line in Figure 9.13, including point G—her choice when she can borrow but not invest.
Exclusion from the credit market: She is unable to get a loan of any kind. Her feasible set is just a single point (A) in Figure 9.13. Section 9.10 explains why many would-be borrowers are simply excluded from borrowing at any rate of interest. Her reservation indifference curve indicates her wellbeing in that situation.

Question 9.11 Choose the correct answer(s)

Figure 9.13 depicts two possible feasible frontiers for Julia, who has no income in period 1 (now) and $100 in period 2 (later). The solid line (option 1) shows her feasible frontier if she borrows at an interest rate of 78%. The dotted line shows her feasible frontier if she borrows at an interest rate of 78% and can invest for a return of 200% (option 2). Based on this information, read the following statements and choose the correct option(s).

When borrowing only, Julia is worse off than at her initial endowment (point A) because of the high interest rate.
The consumption choice G can only be attained under option 1.
If the interest rate for borrowing increases to 100%, ceteris paribus the feasible frontiers become steeper and the vertical axis intercept under option 2 is now $150.
If the return on investment increases to 250%, ceteris paribus the vertical axis intercept under option 2 is now $200.

Julia is on a higher indifference curve at G than at her endowment point. Therefore, she is better off when borrowing.
G is in the feasible set of options 1 and 2, so it is a possible choice under both options.
With an interest rate of 100%, Julia can borrow a maximum of $50 in period 1, so her maximum consumption in period 2 is 50 + 2(50) = $150.
With a return of 250%, Julia’s maximum consumption in period 2 is 56 + 2.5(56) = $196.