The Balanced-Budget Multiplier

Why the Balanced-Budget Multiplier Matters

A government can raise purchases and raise taxes by the same amount. In that case, the change in purchases is matched by the change in tax revenue, so the policy change is balanced in the budget sense. But the effect on output is not zero: government purchases enter planned expenditure dollar for dollar, whereas a dollar of lump-sum tax reduces consumption only by the fraction households would have spent. The remaining fraction would have been saved.

The balanced-budget multiplier measures the change in equilibrium output when government purchases and lump-sum taxes increase by equal amounts. In the simple closed-economy model, it equals one. This result is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

The Model and Its Assumptions

Use the simple closed-economy Keynesian expenditure model. Output $Y$ is in equilibrium when it equals planned expenditure: consumption $C$, investment $I$, and government purchases $G$. Consumption depends linearly on disposable income $Y-T$, where $T$ is a lump-sum tax. The marginal propensity to consume is $c$, with $0<c<1$.

Model assumptions: Planned expenditure is $C+I+G$, with $C=C_0+c(Y-T)$. The consumption intercept $C_0$, $c$, investment $I$, and the price level are fixed. Taxes are lump-sum: $T$ does not vary with income. The economy is closed, so imports are absent. There is no capacity constraint or policy response that changes these relationships during the calculation.

The equilibrium condition is therefore $Y=C+I+G$. The result in this tutorial depends on the assumptions above. It does not automatically apply if taxes vary with income, imports or other leakages are introduced, or investment and consumption respond to other changing conditions.

Deriving the Balanced-Budget Multiplier

First compare two equilibria, before and after a policy change. Write $\Delta Y$, $\Delta G$, and $\Delta T$ for the changes in output, government purchases, and lump-sum taxes. Since $C_0$ and $c$ stay fixed, the change in consumption is $c(\Delta Y-\Delta T)$: disposable income changes by $\Delta Y-\Delta T$, and consumption changes by $c$ times that amount.

\Delta Y=c(\Delta Y-\Delta T)+\Delta I+\Delta G

In this model investment is fixed, so $\Delta I=0$. Rearranging gives the general response to changes in purchases and lump-sum taxes:

(1-c)\Delta Y=\Delta G-c\Delta T \qquad\Longrightarrow\qquad \Delta Y=\frac{1}{1-c}\Delta G-\frac{c}{1-c}\Delta T

Now impose the balanced-budget change, $\Delta G=\Delta T$. Substitution yields:

\Delta Y=\frac{1-c}{1-c}\Delta G=\Delta G \qquad\Longrightarrow\qquad \frac{\Delta Y}{\Delta G}=1

Theorem: In the simple closed-economy model with a fixed marginal propensity to consume and lump-sum taxes, equal changes in government purchases and taxes produce an equal change in equilibrium output. Thus the balanced-budget multiplier is one.

This result can also be understood through the expenditure rounds. If purchases and taxes both rise by $B$, the immediate change in planned expenditure at the original output is $B-cB=(1-c)B$: purchases add $B$, while the tax increase reduces consumption by $cB$. Subsequent rounds of consumption spending generate a total response equal to the initial change multiplied by $1/(1-c)$. Therefore, $(1-c)B/(1-c)=B$. The tax increase does not cancel the purchases increase dollar for dollar in expenditure because households would have consumed only the fraction $c$ of that tax amount.

Worked Calculations

Worked Example: Finding the Initial and New Equilibrium

A hypothetical closed economy has $C=70+0.70(Y-T)$, $I=80$, $G=130$, and $T=100$, all amounts in millions of dollars. Find initial equilibrium output. Then suppose purchases and lump-sum taxes each increase by $24 million. Find the new output and verify the result.

Method. Substitute the consumption function into $Y=C+I+G$ to find the initial equilibrium. Then apply the balanced-budget result and check the new equilibrium directly.

Initially, the equilibrium equation is $Y=70+0.70(Y-100)+80+130$. Collecting the terms that do not contain $Y$ gives $Y=210+0.70Y$, so $0.30Y=210$ and $Y=210/0.30=700$. Initial output is $700 million.

At this output, disposable income is $700-100=600$. Consumption is $70+0.70(600)=70+420=490$. Planned expenditure is $490+80+130=700$, confirming the initial equilibrium.

The two policy changes are $\Delta G=\Delta T=24$. Because the balanced-budget multiplier is one, $\Delta Y=24$, so new output is $700+24=$$724 million. New purchases are $130+24=154$, and new taxes are $100+24=124$.

At the proposed new output, disposable income is $724-124=600$, so consumption remains $70+0.70(600)=490$. Planned expenditure is $490+80+154=724$, exactly equal to output. The result checks. This calculation avoids the mistake of treating the $24 million tax increase as a $24 million reduction in consumption: consumption falls only by the marginal propensity to consume times the tax change, and here the rise in output offsets that change in disposable income.

Worked Example: Using the Expenditure-Rounds Interpretation

A hypothetical economy has $C=40+0.80(Y-T)$, $I=90$, $G=120$, and $T=150$, in millions of dollars. Government purchases and lump-sum taxes are each raised by $35 million. Calculate initial and new output, and show how the first-round expenditure change relates to the total change.

Method. Find the initial equilibrium from the expenditure equation. For the policy, calculate the immediate change in planned expenditure at the original output, then use the balanced-budget multiplier and verify the new equilibrium.

Initially, $Y=40+0.80(Y-150)+90+120$. The constant terms sum to $40-120+90+120=130$, so $Y=130+0.80Y$. Thus $0.20Y=130$, giving initial output $Y=650$ million.

At $Y=650$, disposable income is $650-150=500$. Consumption is $40+0.80(500)=440$, and planned expenditure is $440+90+120=650$. The starting output is an equilibrium.

At the original output, the purchases increase adds $35 million to planned expenditure, while the tax increase reduces consumption by $0.80(35)=28$ million. The immediate net increase is $35-28=7$ million. Since $1/(1-0.80)=5$, the rounds of expenditure give a total output change of $5(7)=35$ million. Equivalently, the balanced-budget multiplier directly gives $\Delta Y=\Delta G=35$. New output is $650+35=$$685 million.

New taxes are $150+35=185$, and new purchases are $120+35=155$. At output $685$, disposable income is $685-185=500$; consumption remains $40+0.80(500)=440$. Planned expenditure is $440+90+155=685$, confirming the answer. The first-round increase is only $7 million, not $35 million; the remaining increase comes through subsequent expenditure rounds.

Worked Example: Choosing the Balanced Policy Change to Meet an Output Goal

A hypothetical economy has $C=30+0.60(Y-T)$, $I=70$, $G=90$, and $T=50$, in millions of dollars. A policy is intended to raise output by $28 million using equal increases in purchases and lump-sum taxes. Find initial output, the required change in each policy variable, and the new equilibrium output.

Method. Solve the initial equilibrium, then use the unit balanced-budget multiplier to translate the target output change into the required equal policy changes. Substitute into the new equilibrium to check.

Initially, $Y=30+0.60(Y-50)+70+90$. The constant terms are $30-30+70+90=160$, giving $Y=160+0.60Y$. Thus $0.40Y=160$ and initial output is $400$ million.

Since the balanced-budget multiplier is one, an output increase of $28 million requires $\Delta G=28$ million and $\Delta T=28$ million. New output is $400+28=428$ million. New purchases are $90+28=118$, and new taxes are $50+28=78$.

At $Y=428$, disposable income is $428-78=350$. Consumption is $30+0.60(350)=30+210=240$. Planned expenditure is $240+70+118=428$, equal to output. The initial output, policy changes, and new equilibrium are consistent.

Common Exam Trap and Limits of the Result

A frequent error is to apply only the government-purchases multiplier, $1/(1-c)$, to the increase in purchases, while ignoring the simultaneous tax increase. For example, with $c=0.80$ and equal increases of $35 million, that incorrect calculation gives $35/(1-0.80)=175$ million. It treats the purchases increase as if taxes did not change.

The correction is to include both changes:

\Delta Y=\frac{35-0.80(35)}{1-0.80} =\frac{7}{0.20} =35\text{ million}

Another way to see the error is that equal changes in purchases and lump-sum taxes imply $\Delta G=\Delta T$, not $\Delta T=0$. The tax increase lowers consumption at any given output, but its effect is only $c\Delta T$. In the specified model, the full multiplier expression then reduces to $\Delta Y=\Delta G$.

The result concerns a change in the budget: the increase in purchases is matched by an equal increase in tax revenue. It does not say that the government's total budget must have been balanced before the policy or will be balanced afterward. Nor should the unit multiplier be carried over unchanged to a model with income-dependent taxes, imports, changing investment, or other features excluded here. State the model and check whether the changes are equal before using the result.

Key takeaway: With fixed investment, a constant marginal propensity to consume, and lump-sum taxes in the simple closed-economy model, $\Delta G=\Delta T$ implies $\Delta Y=\Delta G$. Verify that the tax change is included and that the model assumptions fit the problem.

Check Your Understanding

For each problem, use the stated simple closed-economy model and treat taxes as lump-sum. Amounts are in millions of dollars unless stated otherwise.

An economy has $c=0.60$. Purchases and taxes each rise by $20 million. What is the change in equilibrium output? (Answer: output rises by $20 million.)
An economy has $C=50+0.75(Y-T)$, $I=60$, $G=100$, and $T=80$. Find initial output. Then find new output if purchases and taxes each rise by $18 million. (Answer: initial output is $600 million; new output is $618 million.)
With a fixed marginal propensity to consume of $0.80$, a balanced increase in purchases and lump-sum taxes is intended to raise output by $45 million. What increase is required in each? (Answer: purchases and taxes must each rise by $45 million.)
Output is initially $500 million, and $c=0.40$. Purchases and taxes each rise by $30 million. Calculate the first-round increase in planned expenditure at the original output and the total change in equilibrium output. (Answer: first-round planned expenditure rises by $18 million; equilibrium output rises by $30 million.)
In the simple model, $c=0.70$, purchases rise by $50 million, and lump-sum taxes rise by $40 million. Is this a balanced-budget change, and what is the resulting change in output? (Answer: it is not balanced because the changes differ by $10 million; output rises by $(50-0.70(40))/(1-0.70)=\$73.33$ million.)

What You'll Learn