Fiscal policy can also change taxes. A tax cut works through the same rounds as the The Spending Multiplier, with one difference at the start.
Key idea
$$\text{tax multiplier} = -\dfrac{MPC}{MPS}$$ A tax cut of $\Delta T$ raises real GDP by $\dfrac{MPC}{MPS} \times \Delta T$ (with the price level unchanged); a tax increase lowers it. Transfer payments work like a tax cut. A change in $G$ matched by an equal change in taxes has a balanced-budget multiplier of 1.
Why it works
When the government buys \$100 of goods, the first round is \$100 of spending. When it cuts taxes by \$100, households save part before spending anything: with $MPC = 0.8$, the first round is only \$80. After that, the rounds are the same, each 0.8 times the last. So the total is $80 \times 5 = 400$, which is $\dfrac{0.8}{0.2} \times 100$. The tax multiplier (4 here) is exactly one less than the spending multiplier (5). The minus sign says that raising taxes lowers GDP.
Worked examples
Worked example: A tax increase
The MPC is 0.75. Taxes rise by \$40 billion. With the price level unchanged, how does real GDP change?
- Tax multiplier $= -\dfrac{0.75}{0.25} = -3$.
- $\Delta GDP = -3 \times 40 = -120$ billion dollars.
Answer: Falls by \$120 billion
Worked example: Spending versus tax cut
The MPC is 0.9. Which raises GDP more: \$10 billion more in government purchases or a \$10 billion tax cut? By how much?
- Spending: $\dfrac{1}{0.1} \times 10 = 100$.
- Tax cut: $\dfrac{0.9}{0.1} \times 10 = 90$.
- Purchases raise GDP \$10 billion more: the first round the government spends directly.
Answer: Purchases, by \$10 billion
Worked example: Balanced budget
The government raises purchases by \$50 billion and raises taxes by \$50 billion. MPC = 0.8.
- Purchases: $+5 \times 50 = +250$. Taxes: $-4 \times 50 = -200$.
- Net: $+50$ billion, the size of the spending change.
Answer: GDP rises \$50 billion
Common mistake
Using the spending multiplier for a tax change, forgetting that part of a tax cut is saved before any spending happens, or missing the sign: a tax increase lowers GDP. Check: the tax multiplier's size is one less than the spending multiplier's, and its sign is opposite to the tax change.