National income equilibrium and the multiplier
Output settles where planned spending equals output. Any change in spending moves national income by a multiple of that change.
The Keynesian income-expenditure model
Keynes argued that, in the short run with spare capacity, output is set by aggregate expenditure (AE): AE = C + I + G + (X − M). Consumption depends on income: C = a + bY, where a is autonomous consumption and b is the marginal propensity to consume (MPC), the share of each extra rupee of income that is spent. The rest is saved: MPS = 1 − MPC. Equilibrium is where Y = AE: firms produce exactly what is planned to be bought, so stocks do not pile up or run down. The same point is where leakages = injections (S + T + M = I + G + X).
Worked example 1: finding equilibrium income
A closed economy has C = 200 + 0.8Y, I = 300 and G = 200 (Rs. billion, no taxes). Find equilibrium income and check it with saving.
- Equilibrium: Y = C + I + G = 200 + 0.8Y + 300 + 200.
- Y − 0.8Y = 700, so 0.2Y = 700 and Y = 700 ÷ 0.2 = 3,500.
- Check: C = 200 + 0.8 × 3,500 = 200 + 2,800 = 3,000; AE = 3,000 + 300 + 200 = 3,500. ✓
- Saving S = Y − C = −200 + 0.2Y = −200 + 700 = 500 = I + G (300 + 200). Leakages = injections. ✓
Equilibrium national income is Rs. 3,500 billion.
How the multiplier works (MPC = 0.8)
100 + 80 + 64 + 51.2 + … = 100 × 1 ÷ (1 − 0.8) = 500.
Worked example 2: the multiplier
In the economy above (MPC = 0.8), investment rises by Rs. 100 billion. (a) Find the multiplier and the change in income. (b) In an open economy where MPC = 0.8, the tax rate t = 0.25 and the marginal propensity to import m = 0.1, what is the multiplier?
- (a) k = 1 ÷ (1 − MPC) = 1 ÷ (1 − 0.8) = 1 ÷ 0.2 = 5.
- ΔY = k × ΔI = 5 × 100 = 500, so Y rises from 3,500 to 4,000.
- Check: 200 + 0.8 × 4,000 + 400 + 200 = 200 + 3,200 + 600 = 4,000. ✓
- (b) Open-economy multiplier k = 1 ÷ [1 − MPC(1 − t) + m] = 1 ÷ [1 − 0.8 × 0.75 + 0.1] = 1 ÷ [1 − 0.6 + 0.1] = 1 ÷ 0.5 = 2.
(a) k = 5; income rises by Rs. 500 billion to 4,000. (b) With taxes and imports, k falls to 2: each round leaks more.
Changes in equilibrium income
| Change | Effect on AE | Effect on equilibrium Y |
|---|---|---|
| Rise in autonomous consumption (confidence, wealth) | Up | Rises by k × the change |
| Rise in investment (lower interest rates, optimism) | Up | Rises by k × the change |
| Rise in government spending | Up | Rises by k × the change |
| Rise in exports (e.g. more tea or apparel orders) | Up | Rises by k × the change |
| Higher taxes, more saving or more imports | Down | Falls |
Limits of the multiplier
- Leakages: high saving, taxes and imports shrink each round of spending.
- Imports: in a small open economy, much extra spending goes on imported goods, so the income created is partly abroad.
- Time lags: the rounds take months, not days.
- Supply limits: near full capacity, extra spending raises prices more than output.
- Crowding out: government borrowing can raise interest rates and reduce private investment.
- The paradox of thrift: if everyone tries to save more at once, income falls and total saving may not rise.
When exports of apparel or tea rise, factory and estate workers earn more and spend in local shops, so incomes rise further: the multiplier. Because Sri Lanka imports much of its fuel, machinery and consumer goods, part of each round leaks abroad, so the multiplier is smaller than in a closed economy.
Short answer: “Define the multiplier.” The ratio of the change in national income to the change in an injection that caused it: k = ΔY ÷ ΔI = 1 ÷ (1 − MPC).
Structured question: “Given C = 200 + 0.8Y, I = 300 and G = 200, find equilibrium income and the effect of a Rs. 100 billion rise in investment.” Y = 700 ÷ 0.2 = 3,500. k = 5, so Y rises by 500 to 4,000.
Essay: “Explain the determination of equilibrium national income and the multiplier effect.” Explain AE and the consumption function, draw the 45° Keynesian cross, show equilibrium and leakages = injections, explain the multiplier process with numbers and its limits.
k = 1 ÷ leakage rate. Bigger MPC → bigger multiplier. Every leak (Saving, Tax, iMports) shrinks it.