# Borrowing and lending rates

## Different borrowing and lending rates

Note that when the borrowing and lending rates are equal, our budget constraint is a straight downward sloping line. However, this is not the case if the borrowing and lending rates are different. For an initial example, let’s imagine a scenario where you make $50 today and $50 one year from today. In this scenario, your endowment point lies at \((50, 50)\). Try adjusting the slider to different borrowing and lending rates and observe how the budget constraint changes. The lending rate typically doesn’t exceed the borrowing rate.

### Budget constraint with different borrowing and lending rates

## Practicing with an example

Let’s consider an example where the lending rate is 5% and the borrowing rate is 50%. We have two sources of income: the $50 of present income and the $50 of future income. Enter this lending and borrowing rate into the figure above to visualize the budget constraint. We will calculate the intercepts manually to demonstrate how total consumption changes with the choice to borrow or lend.

### Maximizing consumption now (\(x\)-intercept)

If we want to maximize consumption now, we would spend all $50 of our current income. We would also take out a loan that we would later repay with all $50 of our future income. Although we make $50 in the future, we only consume a fraction of that income because part of it is used to pay back the interest on our loan. We can calculate the maximum amount we can borrow (\(B\)) by plugging the borrowing rate (\(r\)) and total amount ($50 future income) into the formula below:

\[\begin{align} \text{Total amount} &= B (1 + r) \\ \therefore B &= \frac{\text{Total amount}}{(1 + r)}\\ B &= \frac{50}{1.5} \\ &= 33.33 \end{align}\]If we have $50 of income now and the maximum amount we can borrow using our future income is $33.33:

\[\begin{align} \text{Maximum consumption now} &= 50 + 33.33 \\ &= \$83.33 \end{align}\]### Maximizing consumption later (\(y\)-intercept)

We maximize consumption later by consuming $0 of our current income and saving all our money. We can lend these savings and make money when others pay back the $50 with interest. In this case, the interest rate (\(r\)) is the lending rate of 5%. The amount being borrowed from us is $50. Therefore, the total amount of money repaid is:

\[\begin{align} \text{Total amount} &= B(1 + r)\\ \text{Total amount} &= 50(1.05) \\ &= \$52.50 \end{align}\]The 5% lending interest generates an extra $2.50 in income. Therefore, the maximum amount we could consume one year from now is:

\[\begin{align} \text{Maximum consumption later } =& \text{ Current income} \\ &+ \text{Interest generated from lending current income} \\ &+ \text{Future income} \\ \text{Maximum consumption later } =& \ 50 + 52.5 \\ =& \ \$102.50 \end{align}\]### Connecting the intercepts

The \(y\)-intercept of $102.50 is greater than the \(x\)-intercept of $83.33. Interest increases total consumption for the lender but decreases total consumption for the borrower. As a result, saving and lending all your current income allows you to consume $19.17 more than if you maximize consumption now by taking out a loan.

Observe that if we simply connect the two intercepts in the figure above, the line we draw doesn’t match the true shape of the budget constraint. The difference in the interest rate for lenders and borrowers prevents us from drawing a straight line to connect the \(x\)- and \(y\)-intercepts. The differing rates result in two line segments with distinct slopes. There is a greater cost to borrowing than there is a benefit to lending; the tradeoff between consumption now and later (the slope) changes based whether an individual is borrowing or lending.

### Exercise

Suppose you have $30 of earnings now and expect to earn $460 next year. If your bank’s borrowing rate is 15% and its lending rate is 10%, what is the maximum quantity of money you are able to consume in the current time period?

- The maximum that you can borrow is \(\$460 \div 1.15 = \$400\). Your current income is $30 so your maximum consumption in the current period is $430.
- The maximum that you can borrow is \(\$460 \div 1.15 = \$400\). Your current income is $30 so your maximum consumption in the current period is $430.
- The maximum that you can borrow is \(\$460 \div 1.15 = \$400\). Your current income is $30 so your maximum consumption in the current period is $430.

### Exercise

Suppose you have $30 of earnings now and expect to earn $460 next year. If your bank’s borrowing rate is 15% and its lending rate is 10%, what is the maximum quantity of money you are able to consume next year?

- You can lend your current income, which is $30. The lending rate is 10% so you generate $3 of additional income from interest on the $30 loan. Your future income is $460 so your maximum consumption next year is \(\$460 + \$30 + \$3 = \$493\).
- You can lend your current income, which is $30. The lending rate is 10% so you generate $3 of additional income from interest on the $30 loan. Your future income is $460 so your maximum consumption next year is \(\$460 + \$30 + \$3 = \$493\).
- You can lend your current income, which is $30. The lending rate is 10% so you generate $3 of additional income from interest on the $30 loan. Your future income is $460 so your maximum consumption next year is \(\$460 + \$30 + \$3 = \$493\).