Source: Principles of Macroeconomics, Case/Fair, 8e
Difficulty: Intermediate to Advanced Prerequisites: Part 1 of these Chapter 9 notes (fiscal policy basics, equilibrium with government). You need to be comfortable with MPC, MPS, the consumption function C = a + b(Y - T), and the equilibrium condition Y = C + I + G.
Tags: government spending multiplier, tax multiplier, balanced-budget multiplier, MPC, MPS, expansionary fiscal policy, contractionary fiscal policy, multiplier effect, lump-sum taxes, change in equilibrium output
Part 1 showed how the government enters the model. This section shows what happens when the government changes its spending or taxes. Small changes in G or T produce amplified (multiplied) changes in equilibrium output. Understanding the three multipliers here, the government spending multiplier, the tax multiplier, and the balanced-budget multiplier, is central to exam questions in this chapter. The maths is straightforward once you know MPC and MPS; the conceptual challenge is understanding why the tax multiplier is smaller in absolute value than the spending multiplier.
An increase in government spending raises equilibrium output by more than the initial spending increase (the multiplier effect). A tax cut also raises output, but by a smaller amount per pound of tax cut, because households save part of the extra disposable income. When spending and taxes rise by the same amount, output still rises by exactly that amount (the balanced-budget multiplier equals 1).
Government spending multiplier
The ratio of the change in equilibrium output to the change in government purchases. Formula: 1/MPS. In simple terms, it tells you how many pounds of extra output each pound of extra government spending creates.
Tax multiplier
The ratio of the change in equilibrium output to the change in taxes. Formula: -(MPC/MPS). It is always negative because a tax increase reduces output and a tax cut raises it. The absolute value is always one less than the spending multiplier.
Balanced-budget multiplier
The multiplier that applies when government spending and taxes change by the same amount simultaneously. It always equals 1, regardless of the MPC. If G and T both rise by $100, output rises by exactly $100.
Marginal propensity to consume (MPC)
The fraction of each additional pound of disposable income that households spend on consumption. MPC + MPS = 1.
Marginal propensity to save (MPS)
The fraction of each additional pound of disposable income that households save. MPS = 1 - MPC.
Expansionary fiscal policy
Policy aimed at increasing output and reducing unemployment: increase government purchases and/or decrease taxes.
Contractionary fiscal policy
Policy aimed at reducing output (to curb inflation, for instance): decrease government purchases and/or increase taxes.
Autonomous spending
Spending that does not depend on the level of income. Planned investment (I) and planned government spending (G) are autonomous in this model. Saving is not autonomous; it depends on income.
When the government increases purchases, the initial spending becomes someone's income. That person spends a fraction (MPC) of it, which becomes another person's income, and so on. The total effect on output is larger than the initial change.
Government spending multiplier = 1 / MPS
Equivalently, since MPS = 1 - MPC:
Government spending multiplier = 1 / (1 - MPC)
Worked examples:
MPS = 0.2 → multiplier = 1/0.2 = 5
MPS = 0.1 → multiplier = 1/0.1 = 10
MPC = 0.6 → MPS = 0.4 → multiplier = 1/0.4 = 2.5
If the multiplier is 5 and government purchases increase by $100 billion, output increases by 5 x $100 = $500 billion.
If the multiplier is 4 and government spending decreases by $50 billion, output decreases by 4 x $50 = $200 billion.
At the original equilibrium, output equals planned expenditure. When G rises by, say, $100 billion, planned expenditure immediately exceeds output by $100 billion. Firms see inventories falling unexpectedly and respond by increasing production. That extra production is extra income, part of which is consumed, triggering further rounds of spending and production. The process converges to a new equilibrium where output has risen by the full multiplier amount.
A tax change works differently from a spending change. When the government cuts taxes by $1, households receive an extra $1 of disposable income, but they only spend MPC of it (saving the rest). So the initial boost to spending is smaller than a $1 increase in G. This is why the tax multiplier is smaller in absolute value.
Tax multiplier = -(MPC / MPS)
The negative sign reflects the inverse relationship: a tax increase reduces output, a tax cut raises it.
Worked examples:
MPC = 0.9 → tax multiplier = -(0.9/0.1) = -9
MPC = 0.5 → tax multiplier = -(0.5/0.5) = -1
MPC = 0.55 → tax multiplier = -(0.55/0.45) = -1.22
MPS = 0.3 → MPC = 0.7 → tax multiplier = -(0.7/0.3) = -2.33
MPS = 0.25 → MPC = 0.75 → tax multiplier = -(0.75/0.25) = -3
If the tax multiplier is -6 and taxes are reduced by $100 billion, output increases by (-6) x (-$100) = $600 billion.
If the tax multiplier is -5 and taxes are increased by $10 billion, output falls by (-5) x ($10) = $50 billion.
The tax multiplier is always exactly one less than the spending multiplier (in absolute value):
Tax multiplier = -(spending multiplier - 1)
Or equivalently: spending multiplier = |tax multiplier| + 1.
Worked examples:
Spending multiplier = 10 → tax multiplier = -9
Tax multiplier = -6.66 → spending multiplier = 6.66 + 1 = 7.66
A decrease in lump-sum taxes shifts the consumption function upward (parallel shift). It does not change the slope, because the MPC out of income has not changed. An increase in lump-sum taxes shifts it downward.
When both G and T change, calculate each effect separately and add them:
Change in Y from spending change = spending multiplier x change in G
Change in Y from tax change = tax multiplier x change in T
Total change in Y = sum of both
Worked example (MPS = 0.5, so spending multiplier = 2, tax multiplier = -1):
G increases by $400 and T decreases by $400.
From G: 2 x $400 = +$800
From T: (-1) x (-$400) = +$400
Total: +$1,200
Worked example (MPC = 0.8, so spending multiplier = 5, tax multiplier = -4):
G increases by $100 and T decreases by $100.
From G: 5 x $100 = +$500
From T: (-4) x (-$100) = +$400
Total: +$900
When the government increases both spending and taxes by the same amount, the net effect on output is an increase equal to that amount. The multiplier is exactly 1, regardless of the MPC.
Balanced-budget multiplier = 1
This works because the spending multiplier is always exactly 1 larger than the absolute value of the tax multiplier. The extra round of spending from G more than offsets the drag from higher taxes.
Worked examples:
G and T both increase by $700 → output increases by $700.
G and T both decrease by $800 → output decreases by $800.
G and T both increase by $300 (with MPC = 0.8) → output increases by $300. The MPC does not matter here.
These are a common exam format. You are told the desired change in output and the MPC, and you must recommend the right policy.
To increase output by $300 billion with MPC = 0.8:
Spending multiplier = 1/0.2 = 5. Required increase in G = $300/5 = $60 billion.
Alternatively, tax multiplier = -4. Required tax cut = $300/4 = $75 billion.
Alternatively, using the balanced-budget multiplier: increase G and T both by $300 billion.
To decrease output by $100 billion with MPC = 0.9:
Spending multiplier = 1/0.1 = 10. Required decrease in G = $100/10 = $10 billion.
To reduce output by $10 billion with MPC = 0.6:
Spending multiplier = 1/0.4 = 2.5. Required decrease in G = $10/2.5 = $4 billion.
When policy changes but output has not yet adjusted:
At the old equilibrium, if G increases, planned expenditure now exceeds output. There is an unplanned fall in inventories, which signals firms to produce more.
At the old equilibrium, if G decreases, output exceeds planned expenditure. There is an unplanned rise in inventories, which signals firms to produce less.
Using Table 9.3 as an example (MPC = 0.8, MPS = 0.2, so spending multiplier = 5):
Equilibrium output = $1,400 million.
If G increases by $50 million, new equilibrium = 1,400 + (5 x 50) = $1,650 million.
If taxes rise from $100 to $120 million (increase of $20), tax multiplier = -4. Change in Y = -4 x 20 = -80. New equilibrium = 1,400 - 80 = $1,320 million.
Using Table 9.4 (MPC = 0.9, MPS = 0.1, so spending multiplier = 10, tax multiplier = -9):
Equilibrium = $3,400 billion.
G increases to $400 (increase of $200). New equilibrium = 3,400 + (10 x 200) = $5,400 billion.
G decreases by $100. New equilibrium = 3,400 - (10 x 100) = $2,400 billion.
Taxes reduced from $100 to $50 (decrease of $50). Change = -9 x (-50) = +450. New equilibrium = 3,400 + 450 = $3,850 billion.
From Figure 9.5 (equilibrium at Y = 3,000, AE intercept = 600):
Slope of AE line = (3,000 - 600) / 3,000 = 0.8. So MPC = 0.8, MPS = 0.2.
Spending multiplier = 1/0.2 = 5. Tax multiplier = -(0.8/0.2) = -4.
If G increases by $100: new equilibrium = 3,000 + 500 = $3,500.
If G decreases by $200: new equilibrium = 3,000 - 1,000 = $2,000.
If T increases by $50: change = -4 x 50 = -200. New equilibrium = 3,000 - 200 = $2,800.
Balanced-budget increase of $100 (G up $100, T up $100): output rises by $100 to $3,100.
Formula | Meaning |
|---|---|
Government spending multiplier = 1/MPS | Effect of a $1 change in G on Y |
Tax multiplier = -(MPC/MPS) | Effect of a $1 change in T on Y |
Balanced-budget multiplier = 1 | Effect when G and T change by the same amount |
Change in Y = multiplier x change in G (or T) | How to calculate the new equilibrium |
Tax multiplier = -(spending multiplier - 1) | Relationship between the two multipliers |
When governments announce stimulus packages during recessions, the multiplier is the reason a relatively modest spending increase can have a larger impact on GDP. Conversely, austerity measures (spending cuts or tax increases) reduce GDP by more than the cut itself, which is why deficit reduction during a downturn can be self-defeating. The balanced-budget multiplier explains why a government can boost output even while keeping its budget balanced, by raising both spending and taxes simultaneously.
Students often think a $1 tax cut has the same effect on output as a $1 increase in government spending. It does not. The tax cut is weaker because households save part of the extra income before spending kicks in.
The balanced-budget multiplier is always 1. Students sometimes try to calculate it using the MPC and get confused. The MPC does not matter for this result.
The tax multiplier is negative, which means a tax increase reduces output. Students sometimes drop the negative sign and get the direction wrong.
"Autonomous" does not mean "unimportant." It means "not dependent on income." Investment and government spending are autonomous in this model; saving is not.
You will almost certainly face a policy-adviser question. Know how to work backwards from a desired output change to find the required change in G or T.
Be able to calculate both multipliers from MPC or MPS, and know the relationship between them.
Table and diagram questions will ask you to find the new equilibrium after a policy change. Always compute the multiplier first, then apply it to the change.
The balanced-budget multiplier = 1 is a favourite exam item. If G and T change by the same amount, output changes by that amount, full stop.
True or false: As the MPC decreases, the government spending multiplier increases. False. A lower MPC means a higher MPS, and since the multiplier = 1/MPS, a higher MPS means a smaller multiplier.
True or false: A tax cut of $10 billion will have less effect on the economy than an increase in government purchases of $10 billion. True. The tax multiplier is smaller in absolute value than the spending multiplier.
True or false: If the government increases taxes by $1 billion and increases spending by $1 billion, equilibrium output increases by $1 billion. True. That is the balanced-budget multiplier at work.
Fill in the blank: If MPC = 0.75, the government spending multiplier is ____ and the tax multiplier is ____. Spending multiplier = 1/0.25 = 4. Tax multiplier = -(0.75/0.25) = -3.
Q: What is the formula for the government spending multiplier?
A: 1/MPS (equivalently, 1/(1 - MPC)).
Q: If the MPC is 0.8, what is the tax multiplier?
A: -(0.8/0.2) = -4.
Q: The government wants to increase output by $300 billion. The MPC is 0.8. By how much should government purchases increase?
A: Spending multiplier = 5. Required increase = $300/5 = $60 billion.
Q: If government spending increases by $100 billion and taxes increase by $100 billion, what happens to equilibrium output?
A: It increases by $100 billion (balanced-budget multiplier = 1).
Q: If the government spending multiplier is 10, what is the tax multiplier?
A: -(10 - 1) = -9.
Q: If the tax multiplier is -6 and taxes are reduced by $100 billion, what happens to output?
A: Output increases by $600 billion.
Q: At the original equilibrium, government spending increases by $100 billion. What happens to inventories at the old output level?
A: There is an unplanned fall in inventories because planned expenditure now exceeds output.
The multiplier concept here is the fiscal-policy version of the expenditure multiplier from Chapter 8. In later chapters, the model becomes more complex: the money market (Chapter 11-12) introduces interest rate effects that can crowd out some of the fiscal stimulus, and the open economy adds net exports as another leakage. The automatic stabilisers discussed in Part 3 of these notes are a direct application of the multiplier working in reverse, dampening the economy's swings without deliberate policy action.
government spending multiplier, tax multiplier, balanced-budget multiplier, 1/MPS, MPC/MPS, fiscal stimulus, fiscal contraction, expansionary fiscal policy, contractionary fiscal policy, multiplier effect, lump-sum tax, autonomous spending, policy adviser problem, change in equilibrium output, Case Fair Chapter 9, Principles of Macroeconomics Chapter 9, ECO 2013