Proportional Income Taxes and the Spending Multiplier

Why Proportional Taxes Change the Spending Multiplier

When government purchases rise, the initial spending becomes income for someone else. Some of that additional income is then spent, creating further rounds of expenditure. With a proportional income tax, however, an increase in income also raises taxes. Households receive only part of each additional dollar as disposable income, so their induced consumption is smaller than it would be without the tax.

This tutorial derives the government-purchases multiplier in a basic Keynesian expenditure model. It focuses on how a proportional tax changes the response of consumption to output; it does not model changes in interest rates, prices, imports, or productive capacity.

Definition: A proportional income-tax rule makes tax revenue depend on income according to $T=T_0+tY$. Here $T$ is total tax revenue, $T_0$ is a fixed tax component, $Y$ is output or income, and $t$ is the proportional tax rate, with $0\leq t<1$. Each additional dollar of income raises taxes by $t$ dollars. If $T_0=0$, the rule is simply $T=tY$.

Disposable income is income remaining after taxes: $Y_d=Y-T$. Let consumption be $C=a+cY_d$, where $a$ is autonomous consumption and $c$ is the marginal propensity to consume out of disposable income. In the model, $0\leq c\leq1$: households consume $c$ of each additional dollar of disposable income. The rate $t$ applies to income, while $c$ applies to disposable income; confusing these two rates leads to an incorrect multiplier.

Deriving the Multiplier

Suppose planned expenditure is consumption plus autonomous investment and government purchases: $E=C+I+G$. Treat $I$, $G$, $T_0$, and $a$ as fixed while deriving the response to a change in $G$. Equilibrium in the Keynesian cross occurs where output equals planned expenditure, $Y=E$. Substituting the tax rule into disposable income gives:

$$ Y_d=Y-(T_0+tY)=(1-t)Y-T_0 $$

Therefore consumption is:

C=a+c\big((1-t)Y-T_0\big) =\big(a-cT_0\big)+c(1-t)Y

The coefficient on $Y$, $c(1-t)$, is the induced increase in consumption for each additional dollar of output. One dollar of extra income leaves $1-t$ dollars after proportional taxes, and households consume the fraction $c$ of that disposable-income increase. Hence the induced consumption response is $c(1-t)$, not $c$.

Write all expenditure not induced by current output as $A=a-cT_0+I+G$. Then equilibrium is $Y=A+c(1-t)Y$. Rearranging gives the equilibrium level and the multiplier for a change in government purchases:

Y\big[1-c(1-t)\big]=A \qquad\Longrightarrow\qquad Y=\frac{A}{1-c(1-t)}

k_G=\frac{\Delta Y}{\Delta G} =\frac{1}{1-c(1-t)}

The multiplier formula applies when the model is stable, so that the induced-expenditure coefficient is less than one: $c(1-t)<1$. Under the usual assumptions $0\leq c\leq1$ and $0\leq t<1$, this condition holds except for the limiting case $c=1$ and $t=0$. In that case, each expenditure round would be matched by another equally large round, and the finite geometric-series interpretation would fail.

Key result: In the basic expenditure model with fixed prices and autonomous investment, the government-purchases multiplier under a proportional income-tax rate $t$ is $k_G=1/[1-c(1-t)]$, provided $c(1-t)<1$. A higher $t$ reduces the induced consumption response and, for $c>0$, lowers the multiplier.

Why the Formula Holds: The Spending Rounds

The same result can be understood by following an initial increase in government purchases through successive rounds. An initial $\Delta G$ raises income by $\Delta G$. The associated disposable-income increase is $(1-t)\Delta G$, of which households spend $c(1-t)\Delta G$. That consumption becomes income in the next round, and the same response applies again.

\Delta Y =\Delta G+c(1-t)\Delta G+[c(1-t)]^2\Delta G+\cdots

Let $r=c(1-t)$. If $0\leq r<1$, the geometric series $1+r+r^2+\cdots$ converges to $1/(1-r)$. Factoring out $\Delta G$ therefore gives:

\Delta Y=\Delta G\frac{1}{1-r} =\frac{\Delta G}{1-c(1-t)}

This also explains why the fixed tax component $T_0$ does not appear in the spending multiplier: it changes the initial level of disposable income and thus the autonomous component of expenditure, but it does not change the response of taxes to an additional dollar of income. The proportional rate $t$, by contrast, changes that response.

Applying the Multiplier

Worked Example: A Purchases Increase with a Proportional Tax

In a hypothetical economy, the marginal propensity to consume out of disposable income is $c=0.80$, and taxes equal $25\%$ of income, so $t=0.25$. The government increases purchases by $\$40$ billion. Using the basic expenditure model, find the multiplier and the resulting change in equilibrium output.

Method and assumptions. Use $k_G=1/[1-c(1-t)]$. Treat prices and autonomous private expenditure as fixed, and assume the tax rule remains unchanged during the adjustment.

First, find the share of an additional dollar of output that remains after proportional taxes:

$$ 1-t=1-0.25=0.75 $$

The induced consumption response to one additional dollar of output is:

$$ c(1-t)=0.80(0.75)=0.60 $$

The expenditure rounds converge because $0.60<1$. The multiplier is:

k_G=\frac{1}{1-0.60}=\frac{1}{0.40}=2.5

Finally, multiply the purchases increase by the multiplier:

\Delta Y=k_G\Delta G=2.5(\$40\text{ billion}) =\$100\text{ billion}

Check. The first round adds $\$40$ billion; the next adds $0.60(\$40)=\$24$ billion; the following round adds $0.60(\$24)=\$14.4$ billion. The rounds diminish, as required, and their total is $\$40/(1-0.60)=\$100$ billion. The common mistake avoided is using $c=0.80$ as the induced response without accounting for the tax on additional income.

Worked Example: Finding Equilibrium and a Spending Change

A hypothetical economy has consumption $C=120+0.75(Y-T)$, taxes $T=20+0.20Y$, investment $I=80$, and government purchases $G=100$, with all amounts in billions of dollars. Find equilibrium output. Then calculate the change in equilibrium output if government purchases increase by $\$12$ billion, with the tax rule and other autonomous spending unchanged.

Method and assumptions. Substitute the tax rule into consumption, form planned expenditure $E=C+I+G$, and solve $Y=E$. For the spending change, use the multiplier implied by the same consumption and tax rates.

Start with disposable income:

$$ Y_d=Y-(20+0.20Y)=0.80Y-20 $$

Substitute into the consumption function:

$$ C=120+0.75(0.80Y-20) =120+0.60Y-15 =105+0.60Y $$

Planned expenditure is then:

$$ E=(105+0.60Y)+80+100=285+0.60Y $$

At equilibrium, $Y=E$, so:

Y=285+0.60Y \qquad\Longrightarrow\qquad 0.40Y=285 \qquad\Longrightarrow\qquad Y=712.5

Equilibrium output is therefore $\$712.5$ billion. For the spending increase, the induced response is $c(1-t)=0.75(0.80)=0.60$, and the multiplier is $1/(1-0.60)=2.5$. Thus:

\Delta Y=2.5(\$12\text{ billion})=\$30\text{ billion}

Check. After the change, the equilibrium equation becomes $Y=297+0.60Y$, because autonomous expenditure rises from 285 to 297. Solving gives $0.40Y=297$, or $Y=742.5$ billion, which is $\$30$ billion above the original equilibrium. This check confirms that the multiplier change and the direct equilibrium calculation agree.

Worked Example: Comparing Two Proportional Tax Rates

A hypothetical economy has $c=0.80$. Compare the output response to a $\$28$ billion increase in government purchases when the proportional income-tax rate is first $t=0.10$ and then $t=0.30$. Assume the rate is fixed within each case and all other model conditions are unchanged.

Method and assumptions. Calculate the induced consumption response and multiplier separately for each tax rate. This isolates the effect of the tax rate rather than treating the two cases as one simultaneous policy change.

At $t=0.10$, the after-tax share of additional income is $1-0.10=0.90$. The induced consumption response is $0.80(0.90)=0.72$, so:

k_G=\frac{1}{1-0.72}=\frac{1}{0.28} =3.571428\ldots

The output change is $3.571428\ldots(\$28\text{ billion})=\$100$ billion. At $t=0.30$, the after-tax share is $1-0.30=0.70$, and induced consumption is $0.80(0.70)=0.56$. Therefore:

k_G=\frac{1}{1-0.56}=\frac{1}{0.44} =2.272727\ldots

The resulting output change is $2.272727\ldots(\$28\text{ billion})=\$63.63636\ldots$ billion, or $\$63.64$ billion to the nearest hundredth of a billion. Raising the tax rate from $0.10$ to $0.30$ lowers the multiplier and the response to the same purchases increase.

Check. The induced response falls from $0.72$ to $0.56$, so the expenditure rounds shrink more quickly in the second case. The common mistake avoided is subtracting the tax rate directly from the MPC: the relevant coefficient is $c(1-t)$, not $c-t$.

Common Mistakes and Model Limits

The multiplier is a result of a particular short-run model, not a universal forecast of the effect of government purchases. It assumes that the price level, tax rule, and other autonomous expenditure are fixed; that output adjusts to planned expenditure; and that the induced-consumption coefficient is stable. If interest rates, prices, imports, or production constraints respond, the simple formula may not describe the full effect. Those factors are outside this model.

A frequent exam error is to apply the no-tax multiplier $1/(1-c)$ even though taxes rise with income. For example, with $c=0.80$ and $t=0.25$, the incorrect calculation is:

k_G^{\text{wrong}}=\frac{1}{1-0.80}=5

That calculation assumes all additional income is disposable. The correction is to account for the fraction $1-t=0.75$ retained after the proportional tax:

k_G^{\text{correct}} =\frac{1}{1-0.80(0.75)} =\frac{1}{0.40} =2.5

Another trap is to treat $T_0$ as if it were part of the proportional rate. A fixed tax affects the intercept of consumption and therefore the equilibrium level of output, but it does not alter $c(1-t)$ or the purchases multiplier when $T_0$ is held fixed. Always identify which part of the tax rule changes with income before choosing the multiplier.

Key takeaway: A proportional tax reduces the share of additional income available for consumption. First find the induced response $c(1-t)$; then calculate the government-purchases multiplier as $1/[1-c(1-t)]$. The formula requires a stable expenditure process and fixed model conditions.

Check Your Understanding

For each problem, use the basic expenditure model and state or apply its assumptions. Amounts are in dollars unless another unit is given.

An economy has $c=0.70$ and a proportional tax rate of $t=0.20$. Find the induced consumption response to one additional dollar of output and the government-purchases multiplier. (Answer: induced response $=0.56$; $k_G=25/11\approx2.2727$.)
With $c=0.80$ and $t=0.25$, government purchases rise by $\$36$ million. Find the multiplier and the change in equilibrium output. (Answer: $k_G=2.5$; $\Delta Y=\$90$ million.)
Consumption is $C=90+0.80(Y-T)$, taxes are $T=10+0.25Y$, investment is $70$, and government purchases are $100$. Find equilibrium output, then find the output change if purchases increase by $20$, with other terms unchanged. (Answer: initial $Y=630$; $\Delta Y=50$.)
For $c=0.75$, compare the purchases multiplier at $t=0.20$ and $t=0.40$. For a $\$40$ million purchases increase, calculate the output response in each case and the difference between them. (Answer: multipliers $2.5$ and $20/11\approx1.8182$; output responses $\$100$ million and approximately $\$72.73$ million; difference approximately $\$27.27$ million.)
Consumption is $C=60+0.80(Y-T)$, taxes are $T=15+0.30Y$, investment is $90$, and government purchases are $80$. Find equilibrium output and the output increase from an additional $\$11$ of purchases. (Answer: initial $Y=495.45$; $\Delta Y=25$.)

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