In micro, splitting consumption between two periods was decided once and finished. In macro, that choice repeats every period, linked to tomorrow again and again.
The saving rate is not a number handed down to anyone. Each household chooses how much to spend today and leave for tomorrow. A higher interest rate or more patience changes what gets left. The formula below adds today's satisfaction to tomorrow's value. Modern macro models run on exactly this rule.
For example, say you get 10,000 won of 💰 allowance. You could spend it all today. But picturing the toy you want tomorrow, you drop some into the 🐷 piggy bank. Weighing tomorrow's fun alongside today's is exactly how the formula thinks.
Two households face the same interest rate r=0.04, the same income y=4 each period, and hold the same assets a=10. Only patience differs: one has β=0.95, the other β=0.80.
How much does each eat today, and how much does each carry into tomorrow?
Rate 0.04, patience 0.95 — the dashed line is the β=0.80 twin.Show proof
Their resources are identical: (1+0.04)⋅10+4=14.4.
Only the split differs. The patient household picks consumption 5.4 and saving 9; the impatient twin picks consumption 10.4 and saving 4.
The saving rate is not a number handed down from above — it is the shadow of patience. The larger β is, the heavier tomorrow's happiness weighs today, so the same budget gets split with less for today and more for tomorrow.
A household with no assets at all (a=0) lives on income y=4 each period.
If the interest rate jumps from r=0.04 to r=0.09, what happens to its first saving (with β=0.95)?
Rate 0.09, patience 0.95 — saving starts even from zero assets.Show proof
At a rate of 0.04 this household eats its whole income of 4 and saves 0. Waiting pays so little that there is no reason to wait.
At a rate of 0.09 the best choice becomes saving 2 and eating 2. With no assets there is no interest income for an income effect to act on — saving is born purely from the substitution effect: the reward for waiting grew.