The paradox of thrift at different amounts of extra saving
Equilibrium income and realised saving when households try to save more at every income, with investment fixed.
Extra saving attempted
Equilibrium income
Change in income
Realised saving
0
20,000
0
3,800
100
19,600
−400
3,800
200
19,200
−800
3,800
400
18,400
−1,600
3,800
800
16,800
−3,200
3,800
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What it calculates: Saving (S), Average propensity to save, Consumption, Break-even income.
Updated 22 September 2026 · Transparent assumptions
S = −1,200 + 0.25·Y means dissaving of 1,200 at zero income and 25 cents saved per extra dollar
The savings function is the consumption function seen from the other side. Saving is income minus consumption, so with C = a + bY it is S = Y − a − bY = −a + (1 − b)Y. The intercept is negative — at zero income, autonomous consumption of a is financed by dissaving — and the slope is the marginal propensity to save, s = 1 − b.
S = −a + (1 − b)·Y
At an income of 20,000 ($ billions), saving is −1,200 + 0.25 × 20,000 = 3,800 and the average propensity to save is 3,800 ÷ 20,000 = 0.19, below the MPS of 0.25 because the negative intercept weighs on the average.
Worked example
a = 1,200, s = 0.25, Y = 20,000
S = −1,200 + 0.25 × 20,000
S = 3,800
Saving is zero at 4,800 of income and positive above it
Setting S = 0 gives the break-even income a ÷ s = 1,200 ÷ 0.25 = 4,800, the same point where the consumption function crosses the 45-degree line. Below it the economy dissaves; above it every extra dollar adds 25 cents to saving.
S = 0 → Y = a ÷ s
The APS rises with income along the function: −0.25 at 2,400 of income, zero at 4,800, 0.19 at 20,000 and 0.22 at 40,000, approaching the MPS of 0.25 but never reaching it.
With investment of 3,800, income settles where saving is also 3,800
In a closed economy with no government, output is in equilibrium when the leakage from the spending stream (saving) equals the injection into it (investment). With investment of 3,800, the savings function gives −1,200 + 0.25Y = 3,800, so Y = 20,000 — the same answer the Keynesian cross gives from the spending side.
−a + sY = I → Y = (a + I) ÷ s
If income were higher, saving would exceed investment, spending would fall short of output, unsold goods would pile up and firms would cut production. If income were lower, the reverse. Either way income is pushed toward the point where the two lines cross.
Leakage = injection
−1,200 + 0.25Y = 3,800
0.25Y = 5,000
Y = 20,000
Trying to save 200 more cuts income by 800 and leaves saving at 3,800
Suppose every household decides to save 200 more at every income — the saving line shifts up by 200 to S = −1,000 + 0.25Y. At the old income of 20,000, planned saving is now 4,000 but investment is still 3,800, so spending falls short of output. Income drops until saving is back to 3,800: −1,000 + 0.25Y = 3,800 gives Y = 19,200.
The attempt to save more lowered income by 200 × 4 = 800, the multiplier at work, and total saving did not rise at all. That is Keynes’s paradox of thrift: what is prudent for one household can be self-defeating for all of them together when investment does not respond.
Everyone saves 200 more
S = −1,000 + 0.25Y
−1,000 + 0.25Y = 3,800
Y = 19,200 (down 800)
S = −1,000 + 0.25 × 19,200 = 3,800
If extra saving lowers interest rates and lifts investment, thrift can raise saving after all
The paradox depends on investment staying fixed. In the classical loanable-funds view, a rise in saving lowers the interest rate, which raises investment; if investment rises by as much as saving, income need not fall and the extra saving funds a larger capital stock. At full employment that is the more realistic case.
Which view fits depends on where the economy is. In a deep slump with interest rates near zero, the Keynesian result is the better guide, which is why the paradox of thrift returned to policy debates after 2008 and again in 2020.
The saving line is the quickest route to equilibrium income and the multiplier
Because equilibrium requires saving to equal investment, the savings function gives equilibrium income in one step, and its slope gives the multiplier directly: an extra 1 of investment needs 1 ÷ s of extra income to generate the matching saving. With s = 0.25 that is a multiplier of 4.
k = 1 ÷ s
The consumption function calculator solves the same economy from the spending side; the two pages give identical equilibrium incomes because they are two views of the same accounting identity.
Frequently Asked Questions
What is the savings function?
It is saving as a function of disposable income: S = −a + (1 − b)Y, where a is autonomous consumption and 1 − b is the marginal propensity to save. It is income minus the consumption function.
Why is the intercept of the savings function negative?
At zero income households still consume the autonomous amount a, which they finance by dissaving — drawing on savings or borrowing — so saving at zero income is −a.
What is the paradox of thrift?
If all households try to save more while investment stays fixed, spending and income fall until saving again equals investment. Income ends lower and total saving is unchanged.
How is the savings function related to the consumption function?
They are complements: S = Y − C. The intercepts are −a and a, the slopes 1 − b and b, and both functions give the same break-even and equilibrium incomes.
Sources & References
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