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Lesson M04.L04: The Multiplier Effect: Derivation and Examples

Module: Macroeconomics in the Short-Run: The Basic Keynesian Model Level: intro Duration: 30 minutes Learning Objective: Derive the spending multiplier algebraically as 1/(1-MPC); explain the round-by-round amplification; apply to fiscal policy examples including Australian stimulus packages. Data as of: 2024 Provenance: OpenStax Macro 3e | Australian Treasury | MIT OCW 14.02

Explanation

One of Keynes's most important insights is the multiplier effect: an initial change in autonomous spending (investment, government spending, or exports) causes a larger final change in equilibrium output. This happens because spending in one person's hands becomes income for another, who then spends a fraction of it, and so on.

Derivation of the spending multiplier:

Start from the equilibrium condition: Y = PE = Autonomous Spending (A) + MPC ร— Y

Rearranging: Y โˆ’ MPC ร— Y = A Y(1 โˆ’ MPC) = A Y = A ร— [1 รท (1 โˆ’ MPC)]

The term in brackets is the spending multiplier (k):

k = 1 รท (1 โˆ’ MPC)

So a change in autonomous spending ฮ”A causes a change in equilibrium output:

ฮ”Y = k ร— ฮ”A

Round-by-round logic: If the government spends $1 billion and MPC = 0.80: - Round 1: Government spends $1,000M โ†’ income rises $1,000M - Round 2: Recipients spend 80% โ†’ $800M more income - Round 3: 80% of $800M โ†’ $640M more income - Round 4: $512M โ€ฆ and so on

Total = $1,000M ร— (1 + 0.8 + 0.64 + 0.51 + โ€ฆ) = $1,000M ร— [1/(1โˆ’0.8)] = \(1,000M ร— 5 = **\)5,000M**

The higher the MPC, the larger the multiplier โ€” more of each round of income is re-spent rather than saved.

Australian example โ€” 2008โ€“09 GFC stimulus: The Rudd Government deployed approximately $42 billion in stimulus (the Nation Building and Economic Stimulus Plan). This included $900 cash payments to households and infrastructure spending. Treasury estimated the short-run fiscal multiplier at around 1.4โ€“1.7, implying total GDP impact of $60โ€“70 billion โ€” Australia avoided recession (one of the few OECD countries to do so).

Worked Example

Scenario: The Australian government increases infrastructure spending by $20 billion. MPC = 0.75.

Step 1 โ€” Calculate the multiplier: k = 1 รท (1 โˆ’ 0.75) = 1 รท 0.25 = 4

Step 2 โ€” Calculate the total change in equilibrium output: ฮ”Y = k ร— ฮ”G = 4 ร— \(20B = **\)80 billion increase in GDP**

Step 3 โ€” Trace the first three rounds: | Round | New Spending | New Income | |---|---|---| | 1 (govt) | $20.00B | $20.00B | | 2 (recipients spend 75%) | $15.00B | $15.00B | | 3 | $11.25B | $11.25B | | 4 | $8.44B | \(8.44B | | โ€ฆ (continues) | โ€ฆ | โ€ฆ | | **Total** | **\)80B | $80B** |

Step 4 โ€” Check with formula: Sum of infinite geometric series = 20 ร— [1/(1โˆ’0.75)] = 20 ร— 4 = $80B โœ“

Step 5 โ€” What if MPC were 0.60 instead? k = 1 รท (1 โˆ’ 0.60) = 1 รท 0.40 = 2.5 ฮ”Y = 2.5 ร— \(20B = **\)50B** โ€” lower multiplier, smaller total impact.

Common Misconception

Misconception: The multiplier means government spending is "free" โ€” each dollar spent creates more than a dollar in GDP, so the policy pays for itself.

Correction: The multiplier amplifies the output effect, not the fiscal cost. If the multiplier is 4, a $20B spending increase raises GDP by $80B โ€” but the government still had to spend (and borrow) $20B. The multiplier does not mean the spending is costless; it means the economic activity generated is larger than the initial outlay. Moreover, real-world multipliers are often below the simple textbook value because of taxes (which reduce the income households can respend), imports (which leak spending abroad), crowding-out of private investment, and changes in confidence. Australian empirical estimates cluster around 0.8โ€“1.7, not 4.

Practice Prompts

  1. MPC is 0.90. What is the spending multiplier? โ†’ Answer: k = 1 รท (1 โˆ’ 0.90) = 1 รท 0.10 = 10. A high MPC means a very large multiplier โ€” most of each round of income is re-spent.

  2. The Reserve Bank of Australia effectively stimulates $5 billion of additional private investment (by cutting interest rates). MPC = 0.80. What is the total increase in equilibrium output? โ†’ Answer: k = 1 รท (1 โˆ’ 0.80) = 5. ฮ”Y = 5 ร— \(5B = **\)25 billion**.

  3. During the 2008โ€“09 GFC, the Rudd Government sent $900 cash payments to approximately 8.7 million eligible Australians (total โ‰ˆ $7.8B). Using a multiplier of 1.5, estimate the total GDP impact. โ†’ Answer: ฮ”Y = 1.5 ร— \(7.8B = **\)11.7 billion** additional GDP. (Note: the real multiplier for lump-sum transfers is typically lower than for government purchases because some recipients save the payment โ€” consistent with the observed spending behaviour at the time.)

Further Resources