Reading C = a + bY
The linear consumption function has two numbers. The intercept a is autonomous consumption — what households would spend even with no income, paid for by running down savings or borrowing. The slope b is the marginal propensity to consume, the share of each additional dollar of disposable income that is spent.
In the economy used throughout this guide, a = 1,200 and b = 0.75. At a disposable income of 20,000, consumption is 1,200 + 0.75 × 20,000 = 16,200, and saving is whatever is left: 3,800.
Worked example
a = 1,200, b = 0.75, Y = 20,000
C = 1,200 + 0.75 × 20,000 = 16,200
S = 20,000 − 16,200 = 3,800
The saving function is the same line, seen from the other side
Subtract consumption from income and you have the saving function: S = −a + (1 − b)Y. Its intercept is negative — at zero income the economy dissaves the whole of its autonomous consumption — and its slope is the marginal propensity to save, 0.25 here.
Nothing new is being assumed; the two functions are complements, and any question about one can be answered with the other. The savings function calculator and the consumption function calculator give identical break-even and equilibrium incomes for the same economy for exactly that reason.
Break-even income: where the line crosses 45 degrees
Draw consumption against income together with a 45-degree line on which C = Y. The consumption line starts above it (at 1,200) and rises more slowly, so the two cross once. At that income — the break-even income — the economy spends exactly what it earns and saves nothing.
Solving a + bY = Y gives Y = a ÷ (1 − b) = 1,200 ÷ 0.25 = 4,800. Below the break-even income, saving is positive. Below it, consumption exceeds income and saving is negative; above it, saving grows by 0.25 for every extra dollar.
Worked example
Y = 1,200 ÷ (1 − 0.75)
Y = 1,200 ÷ 0.25
Y = 4,800 (saving = 0 here)
Add investment and you have the Keynesian cross
In a closed economy with no government, total planned spending is consumption plus planned investment: AE = a + bY + I. Output is in equilibrium where planned spending equals output — where the AE line crosses the 45-degree line. If output were higher, spending would fall short, unsold goods would pile up, and firms would cut production; if lower, inventories would run down and firms would expand.
With investment of 3,800, the equilibrium is Y = (a + I) ÷ (1 − b) = (1,200 + 3,800) ÷ 0.25 = 20,000. At that income saving is 3,800, exactly equal to investment — the same equilibrium reached from the saving side, where leakages (saving) equal injections (investment).
Worked example
Y = (1,200 + 3,800) ÷ 0.25
Y = 20,000
S = −1,200 + 0.25 × 20,000 = 3,800 = I
The multiplier falls out of the same algebra
The factor 1 ÷ (1 − b) in the equilibrium formula is the spending multiplier. Raise investment by 100 and equilibrium income rises by 100 ÷ 0.25 = 400, because the first 100 of spending becomes income, 75% of which is spent again, and so on down a geometric series that sums to four times the original amount.
That is the whole logic of fiscal stimulus in this model: any autonomous change in spending — investment, government purchases, a shift in a — moves equilibrium output by the multiplier times the change.
Worked example
ΔI = 100
ΔY = 100 ÷ 0.25 = 400
The paradox of thrift, in numbers
Suppose every household decides to save 200 more at every level of income, so the saving line shifts up to S = −1,000 + 0.25Y. At the old income of 20,000, planned saving is now 4,000 while investment is still 3,800: spending falls short of output, and output falls. It keeps falling until saving is back to 3,800, which happens at Y = (1,000 + 3,800) ÷ 0.25 = 19,200.
Income is 800 lower and total saving has not risen at all. Keynes called it the paradox of thrift: what is prudent for one household can be self-defeating for all of them together — when investment does not respond. If extra saving lowers interest rates and raises investment, or if the economy is at full employment, the paradox weakens or disappears.
Worked example
S = −1,000 + 0.25Y, I = 3,800
−1,000 + 0.25Y = 3,800
Y = 19,200 (down 800)
S = 3,800 (unchanged)
Keynes, Kuznets and the consumption puzzle
Keynes proposed the function in 1936 and expected the average propensity to consume to fall as economies grew richer. Cross-section data agreed — richer households spend a smaller share of income — but when Simon Kuznets assembled long-run US data, the average propensity stayed roughly constant over decades.
The resolution came from Milton Friedman's permanent-income hypothesis and Franco Modigliani's life-cycle hypothesis: households spend out of expected lifetime resources, so a short-run consumption function looks Keynesian while the long-run relationship is close to proportional. The straight line in this guide is a short-run tool, and it should not be stretched across decades.
Common mistakes
Treating a change in income as a shift of the consumption function.Income moves you along the line; wealth, interest rates, credit and expectations shift it.Reading the intercept as what poor households actually spend.Autonomous consumption is a feature of the straight line, not a measured floor for any household.Finding equilibrium where consumption equals income.That is the break-even income. Equilibrium output is where consumption plus investment equals income.Assuming the paradox of thrift always holds.It needs fixed investment and idle capacity; with interest-sensitive investment or full employment, extra saving can raise the capital stock instead.Fitting a line through two points and treating it as the economy's behaviour.Two points fix a line exactly and say nothing about how well a line describes the data.