Spending Multiplier vs Tax Multiplier

Why the tax multiplier is always smaller

Both multipliers run the same re-spending chain, they just start it at different points. A dollar of government purchases enters aggregate demand in full, because the government buys a dollar of goods or services with it. A dollar of tax cut goes to households first, and households save the MPS share, so only MPC dollars reach the spending stream in round one. From round two onward the two paths are identical, which is why the tax multiplier equals the spending multiplier times MPC, carrying a negative sign. With an MPC of 0.8 the spending multiplier is 5 and the tax multiplier is -4, and for any MPC strictly between 0 and 1 the gap between them is exactly 1 in this simple model. You can see the gap another way: a $50 billion tax cut injects only $40 billion in round one, and the missing $10 billion, multiplied by 5, is the entire $50 billion by which the tax cut falls short of $50 billion of purchases.

Getting the sign right in a calculation

The tax multiplier is negative, so the direction of the answer has to come from the change in taxes rather than from your intuition about the policy. Write it as change in real GDP equals the tax multiplier times the change in taxes. A tax cut is a negative change in taxes, and a negative multiplier times a negative change gives a positive change in GDP. Take an MPC of 0.75, so the spending multiplier is 4 and the tax multiplier is -3. A $200 billion tax cut raises real GDP by -3 times -$200 billion, or $600 billion, while $200 billion of extra government purchases raises it by $800 billion. Dropping the negative sign gives 3 times -$200 billion, a $600 billion contraction, which reverses the whole answer. The same sign logic explains the balanced budget multiplier of 1: raise both G and T by $200 billion and you get $800 billion minus $600 billion, a net $200 billion.

What both multipliers assume

These formulas come from a stripped-down Keynesian model, and AP questions expect you to know where it stops working. Both assume spare capacity, so extra spending raises real output instead of only the price level; along a steep or vertical aggregate supply curve the real effect shrinks toward zero. Both ignore crowding out, the rise in interest rates from government borrowing that can offset private investment. Both also treat saving as the only leakage from the income stream, so once you allow for income taxes and spending on imports, each round of re-spending loses more than MPS and both multipliers get smaller. The tax multiplier formula also assumes a lump-sum tax change rather than a change in the tax rate, since a rate change alters the size of the leakage itself. You can run the arithmetic at /calculate/spending-multiplier and /calculate/tax-multiplier.

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