Taxes, Disposable Income, and Consumption

Why Trace a Tax Change Through Consumption?

A tax change does not enter planned expenditure in the same way as a change in government purchases. A lump-sum tax affects households’ disposable income; households then adjust consumption according to how much they spend from each additional dollar of disposable income. The size and direction of the consumption response therefore depend on both the tax change and the marginal propensity to consume.

The key is to keep the steps separate. First calculate the change in disposable income. Then apply the consumption response. Finally, if the question asks for planned expenditure at a given output, include the resulting consumption change while holding other expenditure components fixed. This direct calculation does not, by itself, determine how output subsequently responds.

Disposable Income and the Consumption Function

Let $Y$ denote current income or output, and let $T$ denote a household’s lump-sum tax payment. A lump-sum tax is a fixed dollar amount in this model: the tax bill does not automatically rise or fall with income. Disposable income is the income remaining after this tax.

Definition: Disposable income is $Y_d=Y-T$. In the linear consumption function, $C=\bar{C}+cY_d$, $C$ is planned consumption, $\bar{C}$ is autonomous consumption, and $c$ is the marginal propensity to consume (MPC). The model assumes $0<c<1$, and holds $\bar{C}$ and $c$ fixed while the change is analyzed.

The MPC is the change in planned consumption associated with a one-unit change in disposable income, with the other features of the consumption function held fixed. For example, if $c=0.75$, households increase consumption by 75 cents for each additional dollar of disposable income in this model. The remaining 25 cents is not consumed. The MPC is not the fraction of total income paid in taxes, nor does it mean that all consumption equals $cY_d$: the function also includes autonomous consumption $\bar{C}$.

Combining the definitions gives a consumption function written directly in terms of income and taxes:

C=\bar{C}+c(Y-T)

This expression makes the tax channel explicit. At a fixed level of income, a higher tax reduces disposable income and consumption. A tax cut has the opposite effect. The response is less than the dollar amount of the tax change when $0<c<1$, because households consume only a share of the change in disposable income.

Deriving the Direct Consumption Response

Compare an initial situation with a new one. Keep the autonomous consumption term and MPC unchanged. Subtract the initial consumption equation from the new one. Since the consumption function is linear, the difference in consumption equals the MPC times the difference in disposable income.

\Delta C=c\Delta Y_d

Disposable income is $Y-T$, so its change is the change in income minus the change in taxes. Substitution gives the general change formula.

\Delta Y_d=\Delta Y-\Delta T \qquad\Longrightarrow\qquad \Delta C=c(\Delta Y-\Delta T)

This formula applies to changes between two situations under the stated fixed-MPC consumption model. In particular, to isolate the direct effect of a tax change, hold income fixed. Then $\Delta Y=0$, and the consumption change is the negative of the MPC multiplied by the tax change.

\Delta C=-c\Delta T \qquad\text{when }\Delta Y=0

The sign convention matters: $\Delta T$ means new taxes minus initial taxes. A tax increase has $\Delta T>0$, so consumption falls; a tax cut has $\Delta T<0$, so consumption rises. This relationship describes the consumption response at a given income. It does not calculate a later change in income or the total effect after expenditure and output respond.

From Consumption to Planned Expenditure

In a simple closed-economy expenditure model, planned expenditure is consumption plus planned investment and government purchases:

$$ E=C+I+G $$

Here $I$ and $G$ are treated as fixed for the direct comparison. At a given level of income, a lump-sum tax change affects $E$ through its effect on $C$. Thus the change in planned expenditure is the same as the direct change in consumption. This does not say that actual output must immediately equal planned expenditure: if planned spending differs from output, inventories can change. Nor does it include any subsequent output response.

The formula $E=C+I+G$ is the simplified model used here. If the model includes other planned expenditure components, such as net exports, they must also be included in $E$. The direct tax effect remains the consumption change when those other components are held fixed.

Worked Example: A Tax Cut at Fixed Income

A household has income of $6,000, pays a lump-sum tax of $800, and follows the consumption function $C=300+0.75Y_d$. Planned investment is $900 and government purchases are $1,200. Calculate initial disposable income, consumption, and planned expenditure. Then calculate each again after the tax is cut by $200, holding income and the other expenditure components fixed.

Method. Calculate disposable income first, then use the consumption function, and finally add planned investment and government purchases. The MPC is $c=0.75$; it applies to disposable income, not directly to the tax bill as a dollar-for-dollar consumption change.

Initially, disposable income is $Y_d=Y-T=\$6{,}000-\$800=\$5{,}200$. Consumption is $C=\$300+0.75(\$5{,}200)=\$300+\$3{,}900=\$4{,}200$. Initial planned expenditure is $E=\$4{,}200+\$900+\$1{,}200=\$6{,}300$.

A $200 tax cut changes the tax bill to $T=\$800-\$200=\$600$. The new disposable income is $Y_d=\$6{,}000-\$600=\$5{,}400$. New consumption is $C=\$300+0.75(\$5{,}400)=\$300+\$4{,}050=\$4{,}350$. New planned expenditure is $E=\$4{,}350+\$900+\$1{,}200=\$6{,}450$.

Check. Disposable income rises by $200, and consumption rises by $0.75(\$200)=\$150$. Planned expenditure also rises by $150 because investment and government purchases are unchanged. The level calculations and the change calculation agree.

Worked Example: Income and Taxes Change Together

All amounts are in thousands of dollars. Initially, income is 50, the lump-sum tax is 8, and consumption follows $C=4+0.60Y_d$. Income then rises to 54 while the tax rises to 10. Planned investment remains 12 and government purchases remain 10. Find the change in disposable income, consumption, and planned expenditure.

Method. Use $\Delta Y_d=\Delta Y-\Delta T$, then multiply the disposable-income change by the MPC. Do not attribute the entire consumption change to the tax change, because income also changes.

Income rises by $\Delta Y=54-50=4$, while taxes rise by $\Delta T=10-8=2$. Therefore, disposable income changes by $\Delta Y_d=4-2=2$. Initially, $Y_d=50-8=42$, so consumption is $C=4+0.60(42)=4+25.2=29.2$. In the new situation, $Y_d=54-10=44$, so consumption is $C=4+0.60(44)=4+26.4=30.4$. The consumption increase is $30.4-29.2=1.2$.

Initial planned expenditure is $E=29.2+12+10=51.2$. New planned expenditure is $E=30.4+12+10=52.4$. It rises by $52.4-51.2=1.2$.

Check. The income increase contributes $0.60(4)=2.4$ to consumption, while the tax increase contributes $-0.60(2)=-1.2$. Their net effect is $2.4-1.2=1.2$, matching the level calculation. All amounts are in thousands of dollars.

Worked Example: A Tax Cut Shifts the Consumption Schedule

All amounts are in thousands of dollars. A closed-economy model has $C=5+0.80(Y-T)$, planned investment of 6, and government purchases of 4. Compare planned expenditure at incomes of 20 and 40 when the lump-sum tax is 3, and then when it is cut to 1. Show whether the MPC changes.

Method. Evaluate disposable income and consumption at each income and tax combination, then add the fixed expenditure components. The tax change shifts consumption at a given income; it does not alter the MPC under this model.

With $T=3$ and $Y=20$, disposable income is $20-3=17$, consumption is $5+0.80(17)=5+13.6=18.6$, and planned expenditure is $18.6+6+4=28.6$. With $T=3$ and $Y=40$, disposable income is $40-3=37$, consumption is $5+0.80(37)=5+29.6=34.6$, and planned expenditure is $34.6+6+4=44.6$.

After the tax cut to $T=1$, at $Y=20$, disposable income is $20-1=19$, consumption is $5+0.80(19)=5+15.2=20.2$, and planned expenditure is $20.2+6+4=30.2$. At $Y=40$, disposable income is $40-1=39$, consumption is $5+0.80(39)=5+31.2=36.2$, and planned expenditure is $36.2+6+4=46.2$.

At either income, the tax cut raises disposable income by 2 and consumption and planned expenditure by $0.80(2)=1.6$. Across the two income levels, planned expenditure rises by $44.6-28.6=16$ before the tax cut and by $46.2-30.2=16$ after it, while income rises by 20. The slope is $16/20=0.80$ in both cases, confirming that the tax cut changes the schedule’s position, not its slope.

Check. The fixed tax cut produces the same direct consumption increase at both income levels because the MPC is unchanged. The calculated shift of 1.6 matches the tax-change calculation.

Common Mistakes and the Model’s Limits

Treating a tax change as a dollar-for-dollar consumption change. If the MPC is 0.70 and taxes rise by $100 at fixed income, consumption falls by $0.70(\$100)=\$70$, not $100.
Reversing the sign. With $\Delta T=T_{\text{new}}-T_{\text{initial}}$, a tax increase is positive and lowers consumption: $\Delta C=-c\Delta T$. A tax cut is negative and raises consumption.
Applying the MPC to total income instead of the change in disposable income. When income and taxes both change, first calculate $\Delta Y_d=\Delta Y-\Delta T$. Then calculate $\Delta C=c\Delta Y_d$.
Calling the direct expenditure shift the total output response. The direct calculation holds income fixed. If output changes later, that can change disposable income and consumption too. The direct tax-to-consumption calculation alone does not quantify those subsequent responses.

The model is a useful way to trace the immediate channel from a lump-sum tax to household spending, but its conclusions depend on its assumptions. It treats the tax as independent of income, keeps the MPC and autonomous consumption fixed, and holds other planned expenditure components fixed for the direct comparison. A tax system whose tax payment varies with income, or a situation where the MPC changes, requires a different specification.

Key takeaway: At fixed income, a lump-sum tax change alters disposable income by the negative of the tax change. Consumption responds by the MPC times that disposable-income change, so $\Delta C=-c\Delta T$. Planned expenditure changes by the same amount when all other components are fixed.

Check Your Understanding

For each problem, apply the stated consumption model and track the tax change’s direct effect carefully.

A household has fixed income and MPC 0.70. Its lump-sum tax rises from $900 to $1,100. By how much does consumption change? (Answer: consumption decreases by $140.)
A household’s income rises by $500 while its lump-sum tax falls by $120. If the MPC is 0.65, what is the change in disposable income and consumption? (Answer: disposable income increases by $620 and consumption increases by $403.)
All amounts are in thousands of dollars. A household has $C=2+0.75(Y-T)$, income 30, and tax 6. The tax is cut by 2; planned investment is 4 and government purchases are 5. What are consumption and planned expenditure after the cut? (Answer: consumption is 21.5 and planned expenditure is 30.5 thousand dollars.)
Income falls by $100 while a lump-sum tax is cut by $250. If the MPC is 0.80, what is the resulting consumption change? (Answer: consumption increases by $120.)
Two households each receive a $100 lump-sum tax cut at fixed income. Their MPCs are 0.40 and 0.90. Assuming both consumption functions remain linear, what is their combined consumption increase? (Answer: $130.)

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