Saving and borrowing are choices between consumption now and consumption later. The budget comes from Present Value, and the best choice comes from the tangency in The Best Affordable Bundle.
Key idea
With income $m_1$ now and $m_2$ later, and interest rate $r$ for both saving and borrowing: $$c_1 + \frac{c_2}{1 + r} = m_1 + \frac{m_2}{1 + r}.$$ The right side is the present value of income. The line passes through the endowment $(m_1, m_2)$ with slope $-(1 + r)$, and at an interior optimum $MRS = 1 + r$.
Why it works
Saving $s = m_1 - c_1$ (negative for borrowing) earns interest, so $c_2 = m_2 + (1 + r)s$. Substitute $s$ and divide by $1 + r$ to get the budget. Giving up one dollar of $c_1$ buys $1 + r$ dollars of $c_2$, so $1 + r$ is the price of consumption now in units of consumption later. A rise in $r$ rotates the line around the endowment, which is always affordable. For a borrower, a higher rate is bad news on both counts: substitution and income effects both cut $c_1$. For a saver, the substitution effect says "save more," but the higher return makes her richer, and the income effect says "consume more now." Saving can go either way.
Worked example: A balanced choice
$U = \ln c_1 + \ln c_2$, $m_1 = 100$, $m_2 = 110$, $r = 10\%$.
- Present value of income: $100 + \dfrac{110}{1.1} = 200$.
- With $U = \ln c_1 + \ln c_2$ (Cobb-Douglas in logs), half of that goes to each period in present value: $c_1 = 100$.
- Saving $= 100 - 100 = 0$; $c_2 = 110$. Check: $MRS = c_2/c_1 = 1.1 = 1 + r$.
Answer: $c_1 = 100$, no saving.
Worked example: A saver and a rate rise
$U = \ln c_1 + \ln c_2$, $m_1 = 150$, $m_2 = 0$. Compare $r = 25\%$ and $r = 50\%$.
- Present value of income is 150 either way (no future income to discount).
- $c_1 = 75$ at both rates, so saving stays at 75.
- $c_2 = 1.25(75) = 93.75$, then $1.5(75) = 112.5$. For log utility with income only now, the income and substitution effects of a rate change cancel.
Answer: Saving stays at 75; only $c_2$ rises.
Common mistake
Thinking a higher interest rate must make every saver save more. The higher return also makes a saver richer, and that income effect raises consumption now. In the example the two effects canceled exactly; with other preferences either can win. Check: ask whether the person is a saver or a borrower first, then sort the two effects.
How much household saving responds to interest rates is an empirical question, and estimates differ. The model's sure lesson is about borrowers and lenders: a higher rate always hurts a borrower and helps a saver.