Intertemporal Choice

Consuming across time

Now the “two goods” are consumption today (\(c_1\)) and consumption tomorrow (\(c_2\)). The consumer has income \(m_1\) today and \(m_2\) tomorrow, and can borrow or save at interest rate \(r\).

Saving a dollar today returns \(1+r\) dollars tomorrow; borrowing a dollar today costs \(1+r\) tomorrow. So the relative price of consuming today in terms of tomorrow is \((1+r)\). The budget constraint, in present-value form: \[ c_1 + \frac{c_2}{1+r} \;=\; m_1 + \frac{m_2}{1+r}. \] Like the endowment model, this line always passes through the income point \((m_1,m_2)\), and raising \(r\) pivots it around that point (steeper — today’s consumption becomes dearer).

The consumer is a saver (lender) if \(c_1 < m_1\) and a borrower if \(c_1 > m_1\). With Cobb–Douglas preferences \(U=c_1^{a}c_2^{1-a}\) and wealth \(W = m_1 + m_2/(1+r)\): \[ c_1 = a\,W, \qquad c_2 = (1-a)\,W\,(1+r). \] Here \(a\) is effectively patience turned around — a low \(a\) means the consumer weights tomorrow heavily and tends to save.

Try it. Raise the interest rate — the budget line pivots steeper around the income point ■, and saving becomes more attractive. Slide a (weight on today) to turn a saver into a borrower.

What to notice

  • The budget line pivots around the income point ■ as \(r\) changes — borrowing and lending are always done relative to what you earn each period.
  • Higher r steepens the line: a dollar saved today buys more future consumption, so savers gain. A pure saver is made better off by a rate hike; a borrower worse off.
  • Raising a (caring more about today) pushes the choice right, past the income line — the consumer becomes a borrower, pulling future income forward.