Note: These explanations are in the process of being adapted from my textbook.
I'm trying to make them each a "standalone" treatment of a concept, but there may still
be references to the narrative flow of the book that I have yet to remove.
This work is under development and has not yet been professionally edited.
If you catch a typo or error, or just have a suggestion, please submit a note here. Thanks!

# The Intertemporal Budget Constraint

To model the tradeoff between present and future consumption, let’s think of “good 1” as “present consumption,” denoted $c_1$; and “good 2” as “future consumption,” denoted $c_2$. For the purposes of interest rates, we might think of this as “consumption this year” and “consumption next year,” or “consumption in period 1” vs “consumption in period 2,” or just more generally “consumption today” and “consumption tomorrow,” in the metaphorical sense of “tomorrow.”

We’ll assume that an agent — let’s call her Rita — has an income stream of a certain amount of money now, and a certain amount she expects to receive in the future. We can think of this as, for example, a weekly paycheck; or that she has a certain amount of money in the bank now, and expects a future payment. Either way, we can represent her income stream as her endowment point of $m_1$ dollars today and $m_2$ dollars in the future, as shown as the point $M$ in the graph below.

If Rita has no access to banking or financial markets, she has a simple choice: she can either choose to spend all $m_1$ of her current income today, or save some of it as cash to increase her consumption tomorrow. That is, if she saves $s$ dollars today, she can consume $c_1 = m_1 - s$ dollars today and $c_2 = m_2 + s$ dollars tomorrow; that is, $$c_2 = m_2 + (m_1 - c_1)$$ or more simply $$c_1 + c_2 = m_1 + m_2$$

## Saving with interest

Now suppose that Rita has a bank account that will pay her an interest rate of $r$ on her money: that is, if she saves $s$ at interest rate $r$, in the future she will receive $(1 + r)s$. Now her future consumption will be $$c_2 = m_2 + (1 + r)s = m_2 + (1 + r)(m_1 - c_1)$$ Collecting the $c$ terms on the left, this gives us $$(1+r)c_1 + c_2 = (1+r)m_1 + m_2$$ or $$c_1 + {c_2 \over 1 + r} = m_1 + {m_2 \over 1 + r}$$ Note that if $c_1 = 0$, we have $c_2 = (1+r)m_1 + m_2$. This is the vertical intercept of the budget line, and is called future value of the income stream.

## Borrowing with interest

Finally, let’s suppose that instead of saving, Rita borrows some amount $b$ in the present. Like most loans, it comes with an interest rate $r$: that is, she needs to repay $(1 + r)b$ in the future. With this, her present consumption can be $$c_1 = m_1 + b$$ and in the future she’ll have to pay back the loan, so $$c_2 = m_2 - (1+r)b$$ Substituting $b = c_1 - m_1$ into the second equation gives us $$c_2 = m_2 - (1+r)(c_1 - m_1)$$ or $$(1 + r)c_1 + c_2 = (1 + r)m_1 + m_2$$ which is exactly what we had before. Essentially, as long as Rita can borrow or save at the same interest rate $r$, her budget constraint will be a straight line passing through her endowment point with a slope of $1 + r$:

Notice that if we divide the entire budget constraint through by $1 + r$, we get $$c_1 + {c_2 \over 1 + r} = m_1 + {m_2 \over 1 + r}$$ The right-hand side of this equation is called the present value of the income stream, and is the horizontal intercept of the budget line. It represents the value of the income stream if it were all spent today: that is, the largest amount Rita could afford to borrow would be $b = m_2/(1+r)$, since that would grow to $(1 + r)b = m_2$ in the future, which is the maximum amount she could pay back with her future income.

Note that this is just like a normal budget line from consumer theory $$p_1x_1 + p_2x_2 = m$$ if \begin{aligned} x_1 &= c_1 & \text{ (present consumption)}\\ x_2 &= c_2 & \text{ (future consumption)}\\ p_1 &= 1 & \text{ (value of a dollar today)}\\ p_2 &= \frac{1}{1 + r} & \text{ (value of a dollar in the future)}\\ m &= m_1 + \frac{m_2}{1 + r} & \text{ (present value of the income stream)} \end{aligned}

Remember that the slope of any budget line is price ratio $p_1/p_2$, which is always measured in terms of “units of good 2 per unit of good 1.” In this case, since “good 2” is future consumption and “good 1” is present consumption, the price ratio tells how many future dollars are equivalent to one present dollar: $${p_1 \over p_2} = {1 \over 1/(1 + r)} = 1 + r$$ In other words, the financial system is allowing Rita to trade 1 dollar today for $1 + r$ dollars in the future.

## Effect of a change in the interest rate

How does a change in the interest rate affect the budget line? If Rita wants to save money, it helps her: for any given amount of savings, $s$, she gets more in the future. On the other hand, if she wants to borrow money, it hurts her: for any amount of loan $b$, she has to repay more interest in the future. Thus when the interest rate rises, the vertical intercept shifts up, and the horizontal intercept shifts in, as the budget line pivots clockwise around her endowment point.

## Different Interest Rates for Borrowing and Saving

Up to now we’ve been dealing with a single interest rate at which an agent could both borrow and save. In reality, the interest rates people face for saving are often very different from the interest rates charged for borrowing. The reason, in general, is that individual borrowers are a much higher risk than, say, the U.S. Government. Therefore the government can borrow money at a much lower interest rate than an individual.

For example, the following chart from the St. Louis Fed shows the 30-year mortgage rate (a rate at which well-qualified homeowners can borrow money) and the 10-year treasury note rate (a rate which is generally considered a “risk-free return” on savings). As you can see, the mortgage (borrowing) rate is consistently higher than the treasury (saving) rate:

On a more day-to-day basis, it’s not unusual for a savings account in a bank to have an interest rate of nearly zero, while interest on credit card debt can be around 30% — and payday lenders like Cash Call can charge an interest rate that amounts to nearly 90% per year!

How can we picture this in our model of intertemporal consumption? It’s actually quite simple: starting from the endowment, we can can have one interest rate (slope of the budget line) that corresponds to savings, and another that corresponds to borrowing:

Try adjusting the two interest rates (for borrowing and saving). Note that each affects one portion of the budget constraint: the rate for borrowing affects the rate at which one can move to the right from the initial point, and the rate for saving affects the rate at which one can move to the left.

Copyright (c) Christopher Makler / econgraphs.org