Mixed-Leakage Multiplier Problems

Why Mixed-Leakage Problems Need Careful Accounting

A change in government purchases adds directly to planned expenditure. A tax change works differently: it affects disposable income, and only the resulting change in consumption enters expenditure. Meanwhile, higher output can generate both additional proportional tax payments and additional imports. A reliable calculation must keep these channels separate.

The central task is to identify which quantities change and whether the tax change is a lump-sum tax change or a change in the proportional tax rate. At a fixed tax rate, the multiplier gives a direct formula for combining purchases and lump-sum taxes. A change in the tax rate changes the multiplier itself, so a finite rate change generally requires solving the initial and final equilibria.

Definition: In the simple expenditure model, $c$ is the marginal propensity to consume out of disposable income, $t$ is the proportional income-tax rate, and $m$ is the marginal propensity to import out of income. The parameter $c$ satisfies $0 \leq c \leq 1$; the rates $t$ and $m$ are nonnegative. A lump-sum tax $T_0$ does not vary with income. A change in $T_0$ is distinct from a change in $t$.

The Equilibrium Model and the Multiplier

Assume a short-run expenditure model in which the price level is fixed, output adjusts to planned expenditure, and investment and exports do not respond to output. Let consumption depend on disposable income, with proportional taxes and a lump-sum tax, and let imports include an income-dependent component:

C=C_0+c\bigl[(1-t)Y-T_0\bigr], \qquad M=M_0+mY.

Here $Y$ is output, $C_0$ is autonomous consumption, $M_0$ is autonomous imports, and $I$, $G$, and $X$ denote investment, government purchases, and exports. Goods-market equilibrium requires output to equal planned expenditure:

$$ Y=C+I+G+X-M. $$

Substituting the consumption and import functions gives:

Y=C_0+c\bigl[(1-t)Y-T_0\bigr]+I+G+X-M_0-mY.

The terms that vary with output are $c(1-t)Y$, which is induced consumption, and $-mY$, which is the induced import leakage. Collecting them on the left gives:

\bigl[1-c(1-t)+m\bigr]Y=C_0-cT_0+I+G+X-M_0.

Define the mixed-leakage denominator $D=1-c(1-t)+m$. For a stable equilibrium in this model, $D>0$. Holding $c$, $t$, and $m$ fixed, the change in equilibrium output caused by a change in government purchases and a change in the lump-sum tax is:

\Delta Y=\frac{\Delta G-c\Delta T_0}{1-c(1-t)+m}.

The sign convention is important: a tax increase has $\Delta T_0>0$, while a tax cut has $\Delta T_0<0$. The formula applies to changes in these autonomous policy amounts while the proportional tax rate and other model parameters remain fixed.

Why the Formula Holds

At a fixed tax rate, the change in consumption has two parts. The lump-sum tax change directly alters disposable income by $-\Delta T_0$; the output change alters disposable income by $(1-t)\Delta Y$. Thus:

\Delta C=c\bigl[(1-t)\Delta Y-\Delta T_0\bigr].

Imports change by $\Delta M=m\Delta Y$. Subtracting the initial equilibrium condition from the final one, with investment and exports unchanged, yields:

\Delta Y=\Delta C+\Delta G-\Delta M =c\bigl[(1-t)\Delta Y-\Delta T_0\bigr]+\Delta G-m\Delta Y.

Move the terms containing $\Delta Y$ to the left. The result is $D\Delta Y=\Delta G-c\Delta T_0$, which gives the stated formula when $D>0$. This also explains the leakages in the denominator: part of each additional dollar of output is not spent on domestic output because it is saved, paid as proportional tax, or spent on imports. The tax effect in the numerator is multiplied by $c$, since a dollar of lump-sum tax change does not change consumption dollar for dollar unless $c=1$.

Theorem: With fixed $c$, $t$, and $m$, the government-purchase multiplier is $1/D$, and the lump-sum tax multiplier is $-c/D$, where $D=1-c(1-t)+m>0$. A purchase change and a lump-sum tax change can be combined in the numerator; do not apply either multiplier while also counting the same induced response separately.

Worked Applications

Worked Example: A Purchase Increase and a Lump-Sum Tax Increase

A hypothetical economy has $c=0.80$, $t=0.25$, and $m=0.10$. Government purchases rise by $\$30$ billion, while lump-sum taxes rise by $\$10$ billion. Find the change in equilibrium output, in billions of dollars. Assume prices, investment, exports, and all model parameters other than the stated policy amounts are unchanged.

Method. Use the combined-change formula because the proportional tax rate stays fixed. First calculate the denominator:

$$ D=1-c(1-t)+m =1-0.80(1-0.25)+0.10 =1-0.80(0.75)+0.10 =1-0.60+0.10 =0.50. $$

The numerator is the direct purchase increase less the consumption reduction from the lump-sum tax increase:

\Delta G-c\Delta T_0 =30-0.80(10) =30-8 =22.

Therefore:

\Delta Y=\frac{22}{0.50}=44.

Equilibrium output rises by $\$44$ billion. As a check, the purchase multiplier is $1/0.50=2$, giving a purchase effect of $2(30)=60$. The tax multiplier is $-0.80/0.50=-1.6$, giving a tax effect of $-1.6(10)=-16$. Together, $60-16=44$, matching the combined calculation. The common mistake is to subtract the full $\$10$ billion tax increase from purchases; only $c\Delta T_0=\$8$ billion is the initial consumption effect.

Worked Example: A Lump-Sum Tax Cut With Three Induced Leakages

In a hypothetical economy, $c=0.75$, $t=0.20$, and $m=0.05$. The government cuts lump-sum taxes by $\$18$ billion and makes no change to purchases. Calculate the output change, then verify it by tracing the changes in consumption and imports. Report amounts in billions of dollars.

Method. A tax cut means $\Delta T_0=-18$, not $+18$. The denominator is:

$$ D=1-0.75(1-0.20)+0.05 =1-0.75(0.80)+0.05 =1-0.60+0.05 =0.45. $$

With $\Delta G=0$, the output change is:

\Delta Y=\frac{0-0.75(-18)}{0.45} =\frac{13.5}{0.45} =30.

So output rises by $\$30$ billion. To check the flow of expenditure, higher output raises disposable income by $(1-t)\Delta Y=0.80(30)=\$24$ billion. The tax cut adds another $\$18$ billion to disposable income, so total disposable income rises by $24+18=\$42$ billion. Consumption rises by $c(42)=0.75(42)=\$31.5$ billion. Imports rise by $m\Delta Y=0.05(30)=\$1.5$ billion. With purchases unchanged, the net expenditure increase is $31.5-1.5=\$30$ billion, exactly the output change. This check avoids treating the tax cut as an equal-sized direct addition to expenditure.

Worked Example: A Purchase Change Alongside a Tax-Rate Change

A hypothetical economy has $c=0.80$ and $m=0.10$. Initially, the proportional tax rate is $t_0=0.20$, and total autonomous net expenditure—the right-hand side of the equilibrium equation after income-dependent terms are collected—is $\$460$ billion. The tax rate then rises to $t_1=0.25$, while a government-purchase increase raises autonomous net expenditure by $\$50$ billion. Find the initial and final equilibrium outputs and the output change.

Method. Since the tax rate changes, calculate each equilibrium using its own denominator rather than applying the fixed-rate multiplier to both policy changes. Initially:

D_0=1-0.80(1-0.20)+0.10 =1-0.64+0.10 =0.46, \qquad Y_0=\frac{460}{0.46}=1000.

The final autonomous net expenditure is $460+50=\$510$ billion. At the new tax rate:

D_1=1-0.80(1-0.25)+0.10 =1-0.60+0.10 =0.50, \qquad Y_1=\frac{510}{0.50}=1020.

Thus output rises by $Y_1-Y_0=1020-1000=\$20$ billion. The purchase increase raises output, while the higher tax rate reduces induced consumption; the final equilibrium shows their combined effect. The common mistake is to use the initial multiplier for the whole change without accounting for the fact that $t$ changed. Reporting both equilibria makes the comparison explicit.

Tax-Rate Changes Versus Lump-Sum Tax Changes

The fixed-rate formula cannot be used by simply substituting a change in the tax rate for $\Delta T_0$. A rate change affects the income-dependent term $c(1-t)Y$, so it changes the denominator. For a finite change in the rate, calculate the initial and final equilibria using their respective rates:

Y_0=\frac{K_0}{1-c(1-t_0)+m}, \qquad Y_1=\frac{K_1}{1-c(1-t_1)+m}, \qquad \Delta Y=Y_1-Y_0,

where $K_0$ and $K_1$ are the initial and final autonomous net expenditure amounts, including any changes in government purchases or lump-sum taxes as appropriate. If only the tax rate changes, $K_1=K_0$. If purchases also change, include that change in $K_1$. This equilibrium comparison is exact for the stated linear model.

Classify each policy change.
Separate government purchases, lump-sum taxes, and proportional tax rates. Record whether each amount rises or falls.

Calculate the relevant leakage denominator.
For a fixed tax rate, use $D=1-c(1-t)+m$. If the rate changes, calculate a separate $D$ for each equilibrium.

Apply the correct equation.
At a fixed rate, use $\Delta Y=(\Delta G-c\Delta T_0)/D$. For a rate change, solve the initial and final equilibria and subtract.

Check signs and units.
A purchase increase and a lump-sum tax cut raise planned expenditure; a purchase decrease and a tax increase reduce it. Keep the output units consistent with all policy amounts.

Common Mistakes and a Practical Check

Mixed-leakage calculations are most reliable when the policy shock is written in words before numbers are substituted. The denominator represents the output response after induced leakages; the numerator represents the initial autonomous change in planned expenditure. Mixing those roles commonly produces an incorrect multiplier or sign.

Using the wrong tax sign. If taxes are cut by $\$q$, write $\Delta T_0=-q$. Then $-c\Delta T_0$ is positive.
Applying the tax multiplier to a tax rate. The fixed-rate lump-sum tax multiplier applies to $\Delta T_0$, not to $\Delta t$. A rate change alters the denominator.
Leaving imports out of the denominator. With income-dependent imports, include $+m$ in $D$. Omitting it overstates the response of domestic output.
Subtracting the entire tax change from purchases. The immediate consumption response to a lump-sum tax change is $c\Delta T_0$, not $\Delta T_0$ unless $c=1$.
Counting induced effects twice. Once the multiplier includes proportional taxes and imports, do not add separate induced-tax or import adjustments to the calculated $\Delta Y$.

A useful final check is to return to the equilibrium-change equation. At a fixed rate, confirm that $D\Delta Y=\Delta G-c\Delta T_0$. For a rate change, confirm that each output satisfies its own equilibrium equation with the corresponding denominator. These checks test both the arithmetic and the model setup.

Key takeaway: At a fixed proportional tax rate, combine purchase and lump-sum tax changes in the numerator and include proportional taxes and income-dependent imports in the denominator. When the tax rate itself changes, solve the initial and final equilibria separately.

Check Your Understanding

For each problem, use the stated simple expenditure model and keep the units consistent. Answers are provided at the end of each question.

An economy has $c=0.60$, $t=0.25$, and $m=0.05$. Government purchases rise by $\$24$ billion and lump-sum taxes rise by $\$8$ billion. What is the equilibrium output change? (Answer: output rises by $\$32$ billion.)
With $c=0.75$, $t=0.20$, and $m=0.05$, the government cuts lump-sum taxes by $\$13.5$ billion and leaves purchases unchanged. Calculate the output change. (Answer: output rises by $\$22.5$ billion.)
Given $c=0.70$, $t=0.20$, and $m=0.10$, purchases rise by $\$18$ billion and lump-sum taxes rise by $\$9$ billion. Find the net change in equilibrium output. (Answer: output rises by approximately $\$21.6667$ billion.)
An economy has $c=0.75$, $m=0.05$, and initial autonomous net expenditure of $\$450$ billion. Its tax rate rises from $0.20$ to $0.30$, while purchases rise enough to increase autonomous net expenditure by $\$22.5$ billion. Find the initial output, final output, and output change. (Answer: initial output is $\$1{,}000$ billion, final output is $\$900$ billion, and output falls by $\$100$ billion.)
At fixed $c=0.80$, $t=0.25$, and $m=0.15$, a lump-sum tax cut of $\$11$ billion occurs. What government-purchase increase, together with that tax cut, is needed to raise output by $\$40$ billion? (Answer: purchases must rise by $\$13.2$ billion.)

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