Multiplier rounds and leakage conventions
| English | 中文 | Pinyin |
|---|---|---|
| autonomous injection | 自主注入 | zì zhǔ zhù rù |
| marginal propensity to withdraw | 边际漏出倾向 | biān jì lòu chū qīng xiàng |
A decision you can investigate
- An autonomous purchase creates income for someone else, who spends part of it. Subsequent recipients repeat the process while some income leaves each domestic spending round.
- The multiplier adds rounds; it does not mean every original currency unit is counted repeatedly without a matching transaction.
Build the explanation
- The multiplier k is the ratio ΔY/ΔJ of the total income change to an initial autonomous injection 自主注入 in a stated model. In a closed fixed-price model without tax or imports and with constant MPC, k=1/(1−MPC). MPC is a marginal response, not an average consumption ratio. Larger saving leakage reduces later spending rounds and k.
- For the fuller withdrawal model, the marginal propensity to withdraw 边际漏出倾向 is MPW=MPS+MPT+MPM and k=1/MPW, with all marginal propensities measured consistently against the same additional income. Taxes, saving and imports withdraw spending from domestic rounds. Imported consumption can be inside total C, so simply using 1/(1−MPC) with an inclusive consumption propensity would miss import leakage. Name the convention and tax-income base before substituting.
Work through the evidence
- In the no-tax/no-import case, MPC=0.75 gives k=1/(1−0.75)=4. Injection 20 produces successive income rounds 20,15,11.25,8.4375; their sum 54.6875 is only the first four rounds. The infinite geometric sum is 80 under the stated constant-response assumptions.
- In a separate open-economy example, each extra 100 income generates tax 10, saving 20 and consumption 70, of which 20 is imported and 50 domestic. Against the same gross-income base, MPT=0.1, MPS=0.2, MPM=0.2, so MPW=0.5 and k=2. Injection 20 gives total income 40 in that model. Total-consumption MPC 0.7 cannot be used alone to give 1/0.3 here because it includes imported purchases and tax also withdraws income.
What is the closed-model multiplier with MPC 0.75?
1/(1−0.75)=4.
What is the withdrawal-based multiplier in the open example?
1/(0.2+0.1+0.2)=2.
A multiplier of 4 guarantees that any injection 20 raises real output 80 immediately.
That response requires the model’s capacity, price, timing and behavioural assumptions.
Test the limits
- If a propensity is measured against disposable income rather than gross additional income, convert the base before combining it with tax propensities. For example, a disposable-income MPC 0.8 with proportional tax 0.25 generates consumption 0.8×0.75=0.6 per extra gross-income unit before its import content is removed.
- Constant propensities, spare capacity, unchanged prices, no offsetting spending and sufficient time are strong assumptions. At capacity, higher demand can mainly raise prices; imports, interest responses, financing and confidence can weaken the real-output effect. The multiplier magnifies the AD impact of an injection in the model; it does not automatically shift LRAS or certify that a policy benefits every group.
Why does inclusive consumption MPC 0.7 not suffice for the open example?
Use consistent withdrawal shares and avoid losing tax/import leakage.
Apply and explain your answer
- Why is 54.6875 not the full income response in the first model?
- It includes only four rounds; further positive but shrinking rounds sum to the remaining 25.3125 under the model.
Match the terms to their meanings.
Use each term for its stated economic relationship.
Use the terms precisely
- marginal propensity to withdraw: The share of additional income withdrawn through saving, taxation and imports on a consistent income base.
- autonomous injection: An initial spending increase treated as independent of the induced income changes in the stated model.
In the no-tax/no-import case, MPC=0.75 gives k=1/(1−0.75)=4. Injection 20 produces successive income rounds 20,15,11.25,8.4375; their sum 54.6875 is only the first four rounds. The infinite geometric sum is 80 under the stated constant-response assumptions. In a separate open-economy example, each extra 100 income generates tax 10, saving 20 and consumption 70, of which 20 is imported and 50 domestic. Against the same gross-income base, MPT=0.1, MPS=0.2, MPM=0.2, so MPW=0.5 and k=2. Injection 20 gives total income 40 in that model. Total-consumption MPC 0.7 cannot be used alone to give 1/0.3 here because it includes imported purchases and tax also withdraws income.
If a propensity is measured against disposable income rather than gross additional income, convert the base before combining it with tax propensities. For example, a disposable-income MPC 0.8 with proportional tax 0.25 generates consumption 0.8×0.75=0.6 per extra gross-income unit before its import content is removed. Constant propensities, spare capacity, unchanged prices, no offsetting spending and sufficient time are strong assumptions. At capacity, higher demand can mainly raise prices; imports, interest responses, financing and confidence can weaken the real-output effect. The multiplier magnifies the AD impact of an injection in the model; it does not automatically shift LRAS or certify that a policy benefits every group.
The multiplier k is the ratio ΔY/ΔJ of the total income change to an initial autonomous injection in a stated model. In a closed fixed-price model without tax or imports and with constant MPC, k=1/(1−MPC). MPC is a marginal response, not an average consumption ratio. Larger saving leakage reduces later spending rounds and k.