Intertemporal Equilibrium and the Interest Rate

Watch a market that trades today's share against tomorrow's share. The interest rate is the price of that swap.

A 🧑 person with plenty today lends the spare share out. A person short today borrows, spends now, and repays tomorrow. The rate settles where the borrowing meets the lending. The formula below is the exchange rate between today's 💰 money and tomorrow's.

For example, one friend receives the full 10,000 won allowance this month. She lends 5,000 won to a friend who is short this month. Next month she receives 5,500 won back, interest included. The lender spends less today and the borrower spends more.

c_1 + \frac{c_2}{1+r} = e_1 + \frac{e_2}{1+r}

1 + r = \frac{E_2}{\delta E_1}

Part of these learning paths

General Equilibrium and Welfare

Examples & Exercises

In a 2-period economy with $\delta = 0.8$ , $e^1 = (7, 3)$ , $E = (10, 10)$ , find the equilibrium interest rate $r$ and explain the meaning of $(1+r)\delta = 1$ .

1 + r = \frac{E_2}{\delta E_1} = 1.25

\delta = 0.8

—

r = 0.25

, flat consumption profile.

Show proof

Credit market clearing gives $1 + r = E_2/(\delta E_1) = 10/8 = 1.25$ , so $r = 0.25$ .
Since $(1+r)\delta = 1$ , the Euler equation $c_2/c_1 = (1+r)\delta$ gives $c_2 = c_1$ : the interest rate exactly offsets impatience, flattening consumption. With $W_1 = 7 + 3/1.25 = 9.4$ , $c^1 = (9.4/1.8,\ 9.4/1.8) = (5.22, 5.22)$ .

In the example above, determine who is the lender and who is the borrower, and verify that the credit market clears.

Consumer 1 lends

(s_1 = 1.78)

; consumer 2 borrows.

Show proof

$s_1 = 7 - 5.22 = 1.78 > 0$ : consumer 1 is the lender. Consumer 2 has $W_2 = 3 + 7/1.25 = 8.6$ , $c^2_1 = 8.6/1.8 = 4.78$ , so $s_2 = 3 - 4.78 = -1.78 < 0$ — the borrower.
$s_1 + s_2 = 1.78 - 1.78 = 0$ : the credit market clears. The period-1-rich consumer lends to the period-2-rich one — the interest rate is the price of that trade.

16. Equilibrium and Time