The multiplier and equilibrium income
When government spending rises, income rises by more than the amount spent: each extra euro is partly spent again on consumption, which creates more income. How much depends on the marginal propensity to consume.
Formulas and rules
- Goods market: Y = C + I + G with C = C₀ + c·(Y − T)
- Equilibrium income: Y* = (C₀ − c·T + I + G) / (1 − c)
- Spending multiplier: 1 / (1 − c)
- Tax multiplier: −c / (1 − c)
- Balanced budget (G and T rise by the same amount): multiplier 1
Your turn
Three problems on this topic. Type numbers like 12.5 or 1,250 and elasticities with their sign.
Task
In a closed economy without taxes: consumption C = 40 + 0.8·Y, investment I = 40, government spending G = 60 (in € bn). Find equilibrium income, the multiplier and the change in income if government spending rises by € 60 bn.
Equilibrium income Y* in € bn
Multiplier
Change in income ΔY in € bn
Numbers like 12.5 or 1,250; elasticities with their sign (−0.75); percentages as a number (12.5). Round to 2 decimal places.
Worked example
In a closed economy without taxes: consumption C = 170 + 0.8·Y, investment I = 160, government spending G = 280 (in € bn). Find equilibrium income, the multiplier and the change in income if government spending rises by € 50 bn.
Worked solution
- 1.Equilibrium: Y = C + I + G = 170 + 0.8Y + 160 + 280
- 2.0.2·Y = 610 → Y* = 610 / 0.2 = 3,050
- 3.Multiplier = 1 / (1 − c) = 1 / 0.2 = 5
- 4.ΔY = 5 × 50 = 250
Typical mistake: Computing the multiplier as 1/c instead of 1/(1 − c). It depends on the share that is saved: the less of each extra euro is saved, the larger it is.
A closed economy has C = 150 + 0.8·(Y − T), I = 270, G = 300 and a lump-sum tax T = 40 (in € bn). Find equilibrium income and the tax multiplier. By how much does income change if the government raises both spending and taxes by € 70 bn?
Worked solution
- 1.Y = 150 + 0.8(Y − 40) + 270 + 300 → 0.2·Y = 688
- 2.Y* = 688 / 0.2 = 3,440
- 3.Tax multiplier = −c / (1 − c) = −0.8 / 0.2 = −4
- 4.Balanced budget: ΔY = 5 × 70 + (−4) × 70 = 70; the balanced-budget multiplier is 1
Typical mistake: Assuming that the rise in spending and in taxes cancel out. The tax works more weakly because part of it would have been saved; overall income rises by exactly the amount (balanced-budget theorem).
Common questions
Why is the tax multiplier smaller than the spending multiplier?
A tax cut first reaches households, who spend only the share c of it and save the rest. Government spending adds to demand in full straight away.
What does the balanced-budget theorem say?
If the government raises spending and taxes by the same amount, income rises by exactly that amount. The balanced-budget multiplier is 1.