The multiplied total already includes the initial injection.
In a simple fixed-price model, a sustained autonomous injection creates successive income rounds. If r is the constant share of each additional income amount respent on domestic output and 0 <= r < 1, the multiplier is k = 1/(1 - r). With marginal saving, net tax and import shares defined consistently from the same income change, k = 1/(MPS + MPT + MPM). In the closed two-sector case this reduces to 1/(1 - MPC). The model assumes spare capacity, stable behaviour and no offsetting autonomous changes; it does not guarantee a measured real-GDP effect when prices, interest rates, imports or expectations respond differently.
- Domestic-spending form
- The multiplier k is the total income increase per unit of extra autonomous spending. If a constant share r of each extra income unit is spent on domestic output, k = 1/(1 - r), with 0 <= r < 1.
- Withdrawal form
- In the simple open model, k = 1/(MPS + MPT + MPM), using saving, net tax and import shares of the same income change. These are the shares not passed into the next domestic spending round.
- Income change
- Multiply the autonomous spending increase by k once. The resulting total includes the first round as well as later income-induced rounds; fixed prices, spare capacity and stable behaviour are assumed.
Apply the idea
Closed two-sector case
Without taxes and imports, r = MPC and k = 1/(1 - MPC) = 1/MPS.
Convergence
For 0 <= r < 1, each round shrinks and the running total approaches a finite value. If r is one or more, this shrinking-round result does not apply; the formula cannot supply the same finite positive multiplier.
Limits
Stable marginal behaviour, spare capacity, fixed prices and no offsetting autonomous changes are assumptions. The formula is not an exact empirical forecast when they fail.
| Round | Additional income | Cumulative additional income |
|---|---|---|
| Initial | 10 | 10 |
| Second | 6 | 16 |
| Third | 3.6 | 19.6 |
| Fourth | 2.16 | 21.76 |
| All rounds in the model | Progressively smaller later additions | 25 |
Worked example: Add the shrinking income rounds
An additional autonomous purchase of domestic output is 10 units per period. Each extra income unit produces 0.6 units of further domestic spending; the remaining 0.4 leaks into saving, net tax or imports. Prices are fixed and there is enough capacity to supply all the additional output.
- The initial income round is 10. Respending 0.6 gives 10 x 0.6 = 6 next, then 6 x 0.6 = 3.6 and 3.6 x 0.6 = 2.16. The first four rounds total 21.76; later rounds add progressively smaller amounts.
- The full model multiplier is 1/(1 - 0.6) = 2.5. Total equilibrium income rises by 2.5 x 10 = 25 units per period.
- The additional induced income beyond the initial 10 is 15, not 25. Do not add the initial injection again to the calculated total.
- If the marginal withdrawal share rises to 0.5, k falls to 2 and the same injection raises model income by 20. These are conditional model results, separate from a price-adjusting AD/AS simulation.
Watch out for this
An injection of 10 with multiplier 2.5 raises income by 35 because 25 must be added to the initial 10.
The multiplier result 25 already includes the initial 10. Subsequent induced rounds contribute the remaining 15.
Check your understanding
With MPS = 0.2, MPT = 0.1 and MPM = 0.1 on a consistent income basis, what is k?
- 2.5.
- 5.
- 0.4.