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Chapter 20 / Intertemporal Choice

20.2 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 \(c_1 + c_2 = m_1 + m_2\) This is just an endowment budget line \(p_1x_1 + p_2x_2 = p_1e_1 + p_2e_2\) with the variables

if we interpret the variables as follows: \(\begin{aligned} x_1 &= c_1 & \text{ (present consumption)}\\ x_2 &= c_2 & \text{ (future consumption)}\\ e_1 &= m_1 & \text{ (present income)}\\ e_2 &= m_2 & \text{ (future income)} \end{aligned}\) where the price ratio is 1, since a dollar saved today is a dollar consumed tomorrow.

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 \((1+r)c_1 + c_2 = (1+r)m_1 + m_2\) Here we have our endowment budget constraint again, though now the “price” of present consumption is $1 + r$ because spending one dollar today means giving up $1 + r$ dollars in the future.

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.

Effect of a change in the interest rate

How does a change in the interest rate affect the budget line?

Let’s first think about an increase in the interest rate. 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.

On the other hand, a decrease in the interest rate would have the opposite effect: it would make borrowers better off, and savers worse off.

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Next: Preferences over Time
Copyright (c) Christopher Makler / econgraphs.org