Demonstration 1 of 4
Temporary bonus or permanent raise
How much of an income increase is spent now, and why does persistence matter?
A forward-looking household spreads resources evenly over its horizon. A one-period bonus raises resources by the bonus only, so each period gets a quarter of it. A permanent raise lifts every period's income, so consumption rises one for one. A binding need with no borrowing makes current spending follow current cash instead.
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C_t is consumption in period t. Income is 40,000 dollars a period for four periods, the interest rate is zero and the household starts with no assets. The increase arrives in period 1 only, or in every period. The need is an essential period-1 spending level with no borrowing allowed.
Predict first. Does a one-time 20,000 dollar bonus raise period-1 consumption by more or less than 20,000?
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Constructed example: the chapter's hypothetical household (40,000 a period, a 20,000 bonus, the permanent contrast and the 50,000 need); increases of 10,000 and 40,000 are added for comparison.
Calculated values
- Resources
- $180,000
- Smoothed level
- $45,000
- Period-1 consumption
- $45,000
- Change in period-1 consumption
- $5,000
- Saved in period 1
- $15,000
- Later consumption per period
- $45,000.00
With a one-period bonus of $20,000, resources are $180,000 and the smoothed level is 180,000 / 4 = 45,000 dollars. Consumption is $45,000 in every period. Period-1 consumption rises by $5,000 and the household saves $15,000 of period-1 income.
Worked steps
- Resources = 60,000 + 40,000 + 40,000 + 40,000 = 180,000
- Smoothed level = 180,000 / 4 = 45,000
- Change in C1 = 45,000 - 40,000 = 5,000
- Saved in period 1 = 60,000 - 45,000 = 15,000
Use the idea
Before forecasting spending from a tax rebate or a raise, ask how long households expect it to last and whether they can borrow.
Where the conclusion applies
Four periods, zero interest, perfect foresight about income, a wish for equal consumption and free saving. Uncertainty, precautionary motives and costly credit change the split.
Check your understanding: If the 20,000 increase is permanent, what is period-1 consumption?
Chapter 53 source: section "Permanent-income hypothesis".
Demonstration 2 of 4
A tax cut paid back with interest
Does a debt-financed tax cut change spending if the household sees the later tax coming?
The cut and the later tax have the same present value, so the household's wealth is unchanged and it saves the cut. A household that wanted to borrow uses the cut as a loan instead, which changes the timing of consumption.
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The government cuts year-1 taxes by 1,000 dollars and repays the bond in year 2 at interest rate r. Desired borrowing is the amount the household already wanted to move from year 2 into year 1 but could not borrow.
Predict first. Does a rational household with no borrowing limit spend any of the cut?
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Constructed example: the chapter's hypothetical 1,000 dollar cut at 5 percent with desired borrowing 0 and 400; rates of 0 and 10 percent and desired borrowing of 800 are added for comparison.
Calculated values
- Future tax
- $1,050.00
- Present value of future tax
- $1,000.00
- Year-1 saving
- $1,000.00
- Year-1 consumption change
- $0.00
- Saving with interest
- $1,050.00
- Remaining year-2 burden
- $0.00
- Present value of remaining burden
- $0.00
The 1,000 dollar cut is repaid with a year-2 tax of 1.05 x 1,000 = 1,050.00 dollars, worth $1,000.00 today. Saving $1,000 grows to $1,050.00, leaving $0.00. The household saves the whole cut, the deposit exactly covers the later tax, and consumption is unchanged in both years: Ricardian equivalence holds.
Worked steps
- Future tax = 1.05 x 1,000 = 1,050.00
- Present value = 1,050.00 / 1.05 = 1,000.00
- Saved = 1,000 - 0 = 1,000
- Saving grows to 1.05 x 1,000 = 1,050.00
- Remaining burden = 1,050.00 - 1,050.00 = 0.00
Use the idea
When judging a temporary tax cut, ask who pays the later tax and how many recipients are short of cash today.
Where the conclusion applies
Lump-sum taxes, one household that pays the later tax, saving at the government's interest rate and unchanged government purchases.
Check your understanding: With the 400 dollar constraint at 5 percent, what second-year tax burden is left after savings?
Chapter 53 source: section "Ricardian equivalence".
Demonstration 3 of 4
Everyone saves, nobody saves more
If all households try to save more, does realized saving rise?
Lower spending is lower income for sellers. Output falls by the cut times the multiplier until saving out of the smaller income again equals fixed investment.
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Y is output and income, C = a + cY is consumption with autonomous part a and marginal propensity to consume c, and planned investment I is fixed at 80. Realized saving is S = Y - C.
Predict first. When households cut autonomous consumption from 20 to 10 to save more, does realized saving rise?
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Constructed example: the chapter's hypothetical economy (a = 20 then 10, c = 0.8, I = 80); a = 15 and c = 0.6 and 0.75 are added for comparison.
Calculated values
- Output Y
- 500.00
- Consumption C
- 420.00
- Realized saving S
- 80.00
- Multiplier
- 5.00
- Output loss versus a = 20
- 0.00
- Planned saving at the old income
- 80.00
Equilibrium output is (20 + 80) / (1 - 0.80) = 500.00, with C = 420.00 and S = 500.00 - 420.00 = 80.00. The multiplier is 1 / 0.20 = 5.00. This is the starting economy: realized saving equals investment, 80.
Worked steps
- Y = (20 + 80) / (1 - 0.80) = 100 / 0.20 = 500.00
- C = 20 + 0.80 x 500.00 = 420.00
- S = 500.00 - 420.00 = 80.00
- Loss = 500.00 - 500.00 = 0.00
Use the idea
In a slump with idle capacity, expect a general rise in thrift to lower output unless investment rises to absorb the saving.
Where the conclusion applies
Fixed prices, a closed economy with no government, a constant marginal propensity to consume and investment that does not respond to income or interest rates.
Check your understanding: With a = 10 and c = 0.8, what is the output loss?
Chapter 53 source: section "Paradox of thrift".
Demonstration 4 of 4
Spend and tax the same amount
Why does output rise by exactly the extra purchases when taxes rise just as much?
The purchase enters demand in full, but the tax lowers consumption only by c times the tax. The difference, (1 - c) times the change, is multiplied by 1 / (1 - c), which returns exactly the original change.
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c is the marginal propensity to consume. Government purchases and lump-sum taxes both rise by the same amount, in million dollars, with fixed prices, fixed investment and no imports.
Predict first. Does a higher marginal propensity to consume make the balanced-budget multiplier larger than 1?
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Constructed example: the chapter's hypothetical economy (c = 0.75, 40 million); propensities 0.5 and 0.9 and changes of 20 and 80 million are added for comparison.
Calculated values
- First round
- 10.00
- Purchases effect
- 160.00
- Tax effect
- -120.00
- Net change in output
- 40.00
- Balanced-budget multiplier
- 1.00
- Running total after 15 rounds
- 39.47
Purchases add 40 million and the tax removes 0.75 x 40 = 30.00 of consumption, so the first round is 10.00. The rounds sum to 10.00 / (1 - 0.75) = 40.00. Separately, the purchases effect is 160.00 and the tax effect -120.00, netting to 40.00: the multiplier is 1 at this marginal propensity to consume, as at every other.
Worked steps
- First round = 40 - 0.75 x 40 = 10.00
- Rounds: 10.00, 7.5000, 5.6250, ...
- Sum = 10.00 / (1 - 0.75) = 40.00
- Purchases effect = 40 / 0.25 = 160.00
- Tax effect = -0.75 x 40 / 0.25 = -120.00
- Net = 160.00 + (-120.00) = 40.00
Use the idea
A spending program funded by an equal current tax still adds demand in a slack economy, by about its own size under these assumptions.
Where the conclusion applies
Fixed prices, one marginal propensity to consume for everyone, lump-sum taxes, fixed investment, no imports and no monetary offset.
Check your understanding: At a propensity of 0.9 and 40 million, what are the two separate effects?
Chapter 53 source: section "Balanced-budget multiplier".