Unit 9 Lenders and borrowers and differences in wealth

9.3 Borrowing: Bringing consumption forward in time to the present

Borrowing and lending are about shifting consumption and production over time. A person’s wealth affects their opportunities to shift consumption and production. The moneylender in Chambar offers funds to the farmer to purchase fertilizer now, to pay back after the crop matures. The payday borrower will be paid at the end of the month, but needs to buy food now. The borrower brings some future buying power to the present.

feasible set
All of the combinations of goods or outcomes that a decision-maker could choose, given the economic, physical, or other constraints that they face. See also: feasible frontier.
indifference curve
A curve that joins together all the combinations of goods that provide a given level of utility to the individual.
preferences
A description of the relative values a person places on each possible outcome of a choice or decision they have to make.
intertemporal choice model
A model representing decision making concerning borrowing, lending, and investing as ways of moving purchasing power forward (to the present) or backward (to the future) in time.
opportunity cost
What you lose when you choose one action rather than the next best alternative. Example: ‘I decided to go on vacation rather than take a summer job. The job was boring and badly paid, so the opportunity cost of going on vacation was low.’

To understand borrowing and lending, we will use feasible sets and indifference curves. In Unit 3, Karim makes choices between his conflicting objectives of free time and consumption. He makes choices from the feasible set, based on preferences described by indifference curves that represent how much he values one objective relative to the other.

Here, the same feasible set and indifference curve analysis applies to choosing between having something now and having something later. The model of constrained choice is applied to the choice of two goods: consumption now and consumption in the future. This is called the intertemporal choice model. In earlier units, giving up free time is a way of getting more goods, or more grain. In this unit, we show how giving up some goods to be enjoyed now will sometimes allow us to have more goods later. The opportunity cost of having more goods now is having fewer goods later.

To illustrate how the model works, we introduce one of the actors, who needs to borrow some money; we’ll call her Julia. She can count on her family (now and in the future) to provide the bare necessities. But she would like to consume more now. She could represent a payday borrower in New York City or a farmer in Chambar at planting time, or perhaps she has just graduated and needs to finance a period before her first job begins. To focus attention, we use the example of Julia as a payday borrower.

The term endowment is used to refer to anything that a person has—such as a savings account, ownership of a company or a university degree—that affects the level of their consumption.

Julia knows that, in the next period (‘later’), she will have $100 after she is paid at the end of the year. Julia’s situation is shown in Figure 9.3. Each point in the figure shows a given combination of Julia’s consumption beyond the bare necessities provided by her family—both now (measured on the horizontal axis) and later (measured on the vertical axis). What she has to begin with is termed her endowment.

We will use a figure like this throughout the unit, and often refer to the ‘slope’ of lines and curves that we draw. You may remember from studying geometry that, when a line slopes downwards from left to right, the slope is negative. But when economists talk about the ‘slope’ of the trade-off between now and later, they usually simplify things in the description by using the positive value of this number. This is called taking the absolute value of the slope. When we refer to the ‘slope’ of a line or curve in this unit, we will refer to the absolute value, and so the slope is always a positive number. You will find this is easier when you are describing the trade-off that borrowers face.

We begin by showing the choices that are available to Julia in a diagram.

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 between 0 and 110. Coordinates are (consumption now, consumption later). Julia’s endowment is the point (0, 100). There are two downward-sloping straight lines. One line represents the feasible frontier for a 10% interest rate and passes through Julia’s endowment and the points (30, 67), (70, 23), and (91, 0). The other line represents the feasible frontier for a 78% interest rate and passes through Julia’s endowment and (56, 0). The feasible set, defined as the region enclosed by the feasible frontier and both axes, is smaller for a 78% interest rate compared to a 10% interest rate.
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Figure 9.3 Borrowing, the interest rate, and the feasible set.

Julia has nothing: 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 between 0 and 110. Coordinates are (consumption now, consumption later). Julia’s endowment is the point (0, 100).
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Julia has nothing

Julia has no money now but she knows that, in the next period, she will have $100 because she is paid at the end of the year. Given this state of affairs, her consumption now is $0 and $100 later. This point is labelled as her endowment. It is what she has now or expects to get before any other interaction, such as borrowing.

Bringing future income to the present: 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 between 0 and 110. Coordinates are (consumption now, consumption later). Julia’s endowment is the point (0, 100). The point (91, 0) indicates the maximum amount Julia can consume today.
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Bringing future income to the present

Assuming an interest rate of 10%, Julia could, for example, borrow a maximum of \(\frac{100}{(1 + 0.10)}\) = $91 now (rounding to the nearest dollar) and promise to pay the lender the $100 that she will have later, assuming an interest rate of 10%.

Borrowing less: 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 between 0 and 110. Coordinates are (consumption now, consumption later). Julia’s endowment is the point (0, 100). The point (91, 0) indicates the maximum amount Julia can consume today. Julia can also choose the point (70, 23).
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Borrowing less

At the same interest rate (10%), she could also borrow $70 to spend now, and repay $77 at the end of the year. In that case, she would have $23 to spend next year.

Borrowing even less: 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 between 0 and 110. Coordinates are (consumption now, consumption later). Julia’s endowment is the point (0, 100). The point (91, 0) indicates the maximum amount Julia can consume today. Julia can also choose the points (30, 67) or (70, 23).
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Borrowing even less

At the same interest rate (10%), she could also borrow $30 to spend now, and repay $33 at the end of the year. In that case, she would have $67 to spend next year.

Julia’s feasible set: 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 between 0 and 110. Coordinates are (consumption now, consumption later). Julia’s endowment is the point (0, 100). A downward-sloping straight line passing through Julia’s endowment and the points (30, 67), (70, 23), and (91, 0) is the feasible frontier for a 10% interest rate. The feasible set is the region enclosed by the feasible frontier and both axes.
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Julia’s feasible set

By repeating these hypothetical borrowing and repayment combinations, the boundary of Julia’s feasible set—called her feasible frontier—is formed. This is shown for the assumed interest rate of 10%.

Julia’s feasible frontier: 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 between 0 and 110. Coordinates are (consumption now, consumption later). Julia’s endowment is the point (0, 100). A downward-sloping straight line passing through Julia’s endowment and the points (30, 67), (70, 23), and (91, 0) is the feasible frontier for a 10% interest rate. The feasible set is the region enclosed by the feasible frontier and both axes.
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Julia’s feasible frontier

If Julia can borrow at 10%, she can move from her endowment by borrowing now and choose any combination on her feasible frontier.

A higher interest rate: 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 between 0 and 110. Coordinates are (consumption now, consumption later). Julia’s endowment is the point (0, 100). A downward-sloping straight line passing through Julia’s endowment and the points (30, 67), (70, 23), and (91, 0) is the feasible frontier for a 10% interest rate. The feasible set is the region enclosed by the feasible frontier and both axes.
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A higher interest rate

If, instead of 10%, the interest rate is 78%, Julia can only borrow a maximum of $56 now.

The feasible set: 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 between 0 and 110. Coordinates are (consumption now, consumption later). Julia’s endowment is the point (0, 100). There are two downward-sloping straight lines. One line represents the feasible frontier for a 10% interest rate and passes through Julia’s endowment and the points (30, 67), (70, 23), and (91, 0). The other line represents the feasible frontier for a 78% interest rate and passes through Julia’s endowment and (56, 0). The feasible set, defined as the region enclosed by the feasible frontier and both axes, is smaller for a 78% interest rate compared to a 10% interest rate.
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The feasible set

The feasible set with the interest rate of 78% is the dark-shaded area, while the feasible set with an interest rate of 10% is the dark-shaded area plus the light-shaded area. A higher interest rate will reduce the borrower’s ability to shift buying power to the present.

A closer analysis of borrowing

In Figure 9.3, Julia is at the point labelled ‘Julia’s endowment’. To consume at least something now, Julia considers taking out a loan, as shown.

interest rate
The interest rate is the price paid by the borrower to the lender (or saver) for a loan. It is expressed as the payment per period, as a percentage of the loan amount. For a borrower it is the cost of bringing buying power forward from the future; for a saver it is the benefit of deferring buying power to the future. See also: nominal interest rate, real interest rate.

If the interest rate were 10%, Julia could, for example, borrow $91 now and promise to pay the lender the whole $100 that she will have later. Her total repayment of $100 would include the principal (how much she borrowed, namely $91) plus the interest charge ($9) at the rate r, or:

real interest rate
The interest rate corrected for inflation (that is, the nominal interest rate minus the rate of inflation). It represents how many goods in the future one gets for the goods not consumed now. See also: nominal interest rate, interest rate.
nominal interest rate
The interest rate uncorrected for inflation. It is the interest rate quoted by high-street banks. See also: real interest rate, interest rate.

We use r here to indicate the real interest rate (corrected for inflation); i is used to refer to the nominal interest rate (what you get on a savings account, for example). In the intertemporal model, we assume there is no inflation so that the real and nominal interest rates are identical.

\[\begin{align*} \text{repayment} &= \text{principal + interest} \\ &= 91 + 91r \\ &= 91(1 + r) \\ &= \$100 \end{align*}\]

And if ‘later’ means in one year from now, then the annual interest rate, r, is:

\[\begin{align*} \text{interest rate} &= \frac{\text{repayment}}{\text{principal}} – 1 \\ &= \frac{100}{91} -1 \\ &= 0.1 = 10\% \end{align*}\]

You can think of the interest rate as the price of bringing some spending power forward in time.

At the same interest rate (10%), Julia could instead borrow $70 to spend now, and repay $77 at the end of the year, that is:

\[\begin{align*} \text{repayment} &= 70 + 70r \\ &= 70(1 + r) \\ &= \$77 \end{align*}\]

In this case, she would have $23 to spend next year. Another possible combination is to borrow and spend just $30 now, which would leave Julia with $67 to spend next year, after repaying her loan.

feasible frontier
The curve or line made of points that defines the maximum feasible quantity of one good for a given quantity of the other. See also: feasible set.

All of her possible combinations of consumption now and consumption later, for example ($91, $0), ($70, $23), ($30, $67), are the points that make up the feasible frontier shown in Figure 9.3. This is the boundary of the feasible set when the interest rate is 10%.

The fact that Julia can borrow means that she does not have to consume only in the later period. She can borrow now and choose any combination on her feasible frontier. But the more she consumes now, the less she can consume later. With an interest rate of r = 10%, the opportunity cost of spending one dollar now is that Julia will have to spend 1.10 = 1 + r dollars less later.

marginal rate of transformation (MRT)
The quantity of a good that must be sacrificed to acquire one additional unit of another good. At any point, it is the absolute value of the slope of the feasible frontier. See also: marginal rate of substitution.

One plus the interest rate (1 + r) is the marginal rate of transformation (MRT) of goods from the future to the present, because to have one unit of the good now, you have to give up 1 + r goods in the future. This is the same concept as the MRT of goods or grain into free time that we use in Unit 3 and Unit 5.

A higher interest rate raises the price of bringing buying power forward

Suppose that, instead of 10%, the interest rate is 78%, the average rate paid by the farmers in Chambar. At this interest rate, Julia can now only borrow a maximum of $56, because the interest on a loan of $56 is $44, using up all $100 of her future income. Her feasible frontier therefore pivots inward from her endowment and the feasible set becomes smaller. Because the price of bringing buying power forward in time has increased, the capacity to consume in the present has fallen. The difference between the two feasible frontiers is shown in Figure 9.3.

Exercise 9.1 Julia’s feasible frontier

Construct Julia’s feasible frontier by finding all the combinations of consumption now and next period, given her endowment and the interest rate.

  1. Complete the table below, using the information given. Round your answers to the nearest dollar.
  2. Using your completed table, draw a diagram similar to Figure 9.3, showing the feasible frontier, consumption amounts, and the amount of repayment.
Point on the feasible frontier Consumption now Consumption later Repayment
Calculation = amount borrowed = income later − repayment = income later − consumption later
Endowment point
(0, 100)
Interest rate = 15%
(87, 0)
(72, 17)
(20, 77)
Interest rate = 110%
(48, 0)
(36, 24)
(20, 58)

Question 9.4 Choose the correct answer(s)

Anna will get her next pay cheque in two weeks but she needs at least $375 to tide her over until then. She is aware of two payday loan agencies in her neighbourhood and is considering taking a loan from one of them. My Payday will charge her $50 to lend her $375 for two weeks, while Quick Cash will charge her $80 to lend her $400 for three weeks. The interest rate on her credit card is currently 22%. She wants to compare the APR (annual percentage rate) of the payday loans to this rate. The payday lenders are not clearly advertising their APR, but Anna finds some instructions for how to calculate it on a government website. She starts by writing down the amount of credit she will receive (amount financed), the dollar amount the credit will cost (finance charge) and the duration (‘term’) of the loan (in days). She then follows this process:

Step 1: Divide the total finance charge by the amount financed. Step 2: Multiply the answer by the number of days in the year (365). Step 3: Divide the answer by the term of the loan (in days). Step 4: Multiply by 100 and add a % sign.

Based on this information, read the following statements and choose the correct option(s).

  • The APR that Anna would be charged by My Payday is equal to 521%.
  • Both My Payday and Quick Cash would be charging the same APR of 348%.
  • Assuming that Anna does not currently have other suitable options for borrowing, has not maxed out her credit card, and does not need to pay cash for her purchases, comparing the APR of 22% on her credit card with the APR charged by the payday loan companies would be a sensible way for her to decide how to finance her necessary purchases for the next two weeks.
  • Anna does need to pay cash for her purchases in the next two weeks (for example, her landlord only accepts cash). She discovers that if she uses her credit card to withdraw cash, she will start paying interest of 28% APR on that amount immediately. She also learns that there is a cash advance fee of 5% of the total amount borrowed as well as bank ATM fees of $2.50. It would therefore be better for her to withdraw $375 using her credit card, compared to taking one of the payday loans.
  • My Payday would charge an APR of 348%, as ($50/$375) × (365/14) × 100 = 348%.
  • Both My Payday and Quick Cash would charge the same APR. For My Payday, ($50/$375) × (365/14) × 100 = 348% and for Quick Cash, ($80/$400) × (365/21) × 100 = 348%.
  • Yes, comparing the APR of either option would help Anna make a good decision since it would indicate clearly which option would be more expensive for her (the option with the higher APR would be more expensive, so she should choose the other option).
  • Paying interest at 28% for two weeks would amount to a finance charge of $4.03 (28% × $375 × 14/365). She would also have to pay $18.75 for the cash advance fee (5% × $375) and $2.50 for the ATM fee for a total of $25.28 if she uses her credit card. This compares to the finance charge of $50 from My Payday, so using her credit card would be the better option.