The Market for Saved Funds

In micro, the question was whether one person saves. In macro, the stage is a market where the whole country pools its saving.

The interest rate is the price of borrowed money. A higher rate coaxes out more saving. It also scares off borrowers who wanted to invest. Where the two meet, as the formulas below show, sits the equilibrium rate. A government deficit eats into the pool, pushes the rate up, and crowds private investment out.

For example, imagine all your classmates pool their 🐷 piggy-bank 💰 money in one box. If many friends want to borrow and money is scarce, the charge goes up. If the teacher borrows big first, less is left for everyone else.

S(r)=S0+srS(r) = S_0 + s r
I(r)=I0−brI(r) = I_0 - b r
r∗=I0−S0+Ds+br^* = \frac{I_0 - S_0 + D}{s + b}

Part of these learning paths

Examples & Exercises

The government runs a deficit D=20D = 20 while base saving stays at S0=40S_0 = 40.

What happens to the interest rate and to private investment?

Saving 4040, deficit 2020 — rate 8%8\%, investment 5252.
Show proof

  1. With no deficit, equilibrium is r∗=(I0−S0)/(s+b)=(100−40)/(4+6)=6r^* = (I_0 - S_0) / (s + b) = (100 - 40) / (4 + 6) = 6 and investment is I=100−6⋅6=64I = 100 - 6 \cdot 6 = 64. The deficit 2020 shifts total saving left by that much, so r∗=(100−40+20)/10=8r^* = (100 - 40 + 20) / 10 = 8.

  2. At the new rate, investment is I=100−6⋅8=52I = 100 - 6 \cdot 8 = 52. The crowded-out amount is given by the formula below — not the whole deficit, only 1212.

    The higher rate coaxes household saving from 6464 up to 7272, covering the other 88.

    Even so, saving = investment shrinks from 6464 to 5252.

    ΔI=−bD/(s+b)=−6⋅20/10=−12\Delta I = -b D / (s + b) = -6 \cdot 20 / 10 = -12

Households tighten their belts and base saving rises from S0=40S_0 = 40 to 6060 (there is no deficit).

What happens to the interest rate and investment?

Saving 6060, deficit 00 — rate 4%4\%, investment 7676.
Show proof

  1. The saving curve shifts right, so equilibrium is r∗=(100−60)/(4+6)=4r^* = (100 - 60) / (4 + 6) = 4 — down 22 points from the old 66. When loanable money is plentiful, its price, the interest rate, falls.

  2. Investment becomes I=100−6⋅4=76I = 100 - 6 \cdot 4 = 76, up 1212 from the old 6464. In this market, extra saving turns into exactly extra investment — the rise in saving, 76−64=1276 - 64 = 12, matches to the last unit.

5. Where the Interest Rate Is Decided