Suppose the government buys \$100 billion more of goods. That \$100 billion becomes someone's income, and they spend part of it, set by the MPC. That spending becomes someone else's income, and so on. Adding these rounds needs a little decimal arithmetic.
Key idea
$$\text{spending multiplier} = \dfrac{1}{1 - MPC} = \dfrac{1}{MPS}$$ With the price level unchanged, a change in autonomous spending ($G$, $I$, exports, or consumption not caused by income) changes real GDP by the multiplier times that change.
Why it works
With $MPC = 0.8$, a first \$100 of new spending creates \$100 of income. Then \$80 is spent, then \$64, then \$51.20, and so on: each round is 0.8 times the last. The total is a geometric series: $$100 + 100(0.8) + 100(0.8)^2 + \cdots = \dfrac{100}{1 - 0.8} = 500.$$ The rounds shrink because at each step the MPS share is saved and leaves the chain. A bigger MPC means less leaks out each round, so the multiplier is bigger.
Worked examples
Worked example: Finding the change in GDP
The MPC is 0.75 and investment rises by \$40 billion. With the price level unchanged, how much does real GDP rise?
- Multiplier $= \dfrac{1}{1 - 0.75} = \dfrac{1}{0.25} = 4$.
- $\Delta GDP = 4 \times 40 = 160$ billion dollars.
Answer: \$160 billion
Worked example: Working backward
A \$50 billion rise in government purchases raises real GDP by \$250 billion. What is the MPC?
- Multiplier $= \dfrac{250}{50} = 5$.
- $\dfrac{1}{MPS} = 5$, so $MPS = 0.2$ and $MPC = 0.8$.
Answer: 0.8
Worked example: It works in reverse
The MPS is 0.25. Business investment falls by \$20 billion. With the price level unchanged, how does real GDP change?
- Multiplier $= \dfrac{1}{0.25} = 4$.
- $\Delta GDP = 4 \times (-20) = -80$ billion dollars. Lost spending is lost income, which cuts spending again in every round.
Answer: Falls by \$80 billion
This simple multiplier assumes prices stay put and that only saving leaks out. In real economies taxes and imports also leak out of each round, and prices may rise, so measured multipliers are smaller. How large they are in practice is one of the questions economists debate.
Common mistake
Using the MPC itself as the multiplier, or dividing 1 by the MPC instead of by the MPS. With $MPC = 0.8$ the multiplier is 5, not 0.8 and not 1.25. Check: the multiplier is always at least 1, and it grows as the MPC grows.