Why the Tax Multiplier Is Smaller Than the Spending Multiplier

Why Equal-Sized Fiscal Changes Have Different Effects

An increase in government purchases enters planned expenditure directly: the government buys the full amount of the additional goods and services. A lump-sum tax reduction works less directly. It raises households’ disposable income, but households spend only part of that increase and save the rest. That difference in the first round of spending explains why, under the basic Keynesian-cross model, an equal-sized tax change has a smaller effect on equilibrium output than a government-purchases change.

The comparison is useful when interpreting fiscal-policy scenarios: the key is not only the dollar size of a policy change, but also how that change first affects planned expenditure. The results below are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

The Model and Its Multipliers

Use the simple short-run expenditure model with a fixed price level and idle productive capacity. Let $Y$ denote real output, $C$ consumption, $I_0$ autonomous investment, $G$ government purchases, and $T$ lump-sum taxes. Consumption is $C=C_0+c(Y-T)$, where $C_0$ is autonomous consumption and $c$ is the marginal propensity to consume (MPC). Assume $0<c<1$. Investment and all other autonomous components are held constant when comparing policy changes. There are no proportional income taxes, imports, interest-rate responses, or supply constraints in this model.

Equilibrium output equals planned expenditure. The policy multipliers describe the change in equilibrium output caused by a one-unit change in a fiscal variable, holding the other exogenous components fixed.

Definition: The government-purchases multiplier is the change in equilibrium output divided by the change in government purchases, with lump-sum taxes and other autonomous components held fixed. The lump-sum tax multiplier is the change in equilibrium output divided by the change in lump-sum taxes, with government purchases and other autonomous components held fixed. A positive tax change means a tax increase; a negative tax change means a tax cut.

The equilibrium condition is $Y=C+I_0+G$. Substituting the consumption function and comparing two equilibria gives a compact way to derive both multipliers.

\begin{aligned} \Delta Y &= c(\Delta Y-\Delta T)+\Delta G,\\ (1-c)\Delta Y &= \Delta G-c\Delta T,\\ \Delta Y &= \frac{1}{1-c}\Delta G-\frac{c}{1-c}\Delta T. \end{aligned}

The equation gives the government-purchases multiplier as $1/(1-c)$ and the lump-sum tax multiplier as $-c/(1-c)$. The negative sign on the tax multiplier captures direction: a tax increase reduces output, while a tax cut raises it. For $0<c<1$, the absolute tax multiplier is $c/(1-c)$, which is $c$ times the government-purchases multiplier. Because $c<1$, it is strictly smaller.

Key result: In this model, an equal-sized increase in government purchases raises equilibrium output more than an equal-sized lump-sum tax cut. The magnitude of the tax multiplier is exactly the MPC share of the government-purchases multiplier: $$ \left|\frac{-c}{1-c}\right|=c\left(\frac{1}{1-c}\right)<\frac{1}{1-c}. $$

Why the Tax Multiplier Is Smaller

A purchases increase of one dollar adds one dollar to planned expenditure in the first round. By contrast, a one-dollar tax cut raises disposable income by one dollar but initially raises consumption by only $c$ dollars. The remaining $1-c$ dollars are not spent on consumption in this model. Later spending rounds amplify both initial changes, but they do not erase the difference between the initial injections.

The same idea can be seen by writing the total response as a sequence of expenditure rounds. For $0<c<1$, successive rounds form a convergent geometric series.

\begin{aligned} \text{Purchases increase of } \Delta G:\quad &\Delta G+c\Delta G+c^2\Delta G+\cdots =\frac{\Delta G}{1-c},\\ \text{Tax cut of } \Delta T<0:\quad &c(-\Delta T)+c^2(-\Delta T)+c^3(-\Delta T)+\cdots =\frac{c(-\Delta T)}{1-c}. \end{aligned}

In the tax-cut sequence, the first term is $c(-\Delta T)$, not the full tax cut. This is the central reason the tax multiplier is smaller in magnitude. The result depends on the stated model assumptions; it should not be carried over unchanged to a model with proportional taxes, imports, interest-sensitive spending, or a binding supply constraint.

Worked Example: Comparing Equal-Sized Policy Changes

In a hypothetical economy, the MPC is $0.75$. Compare the output effect of a government-purchases increase of $40 billion with the effect of a lump-sum tax cut of $40 billion. Assume the simple expenditure model above and hold other autonomous components fixed.

Method. Calculate each multiplier from the MPC, then apply the signed policy change. The common mistake to avoid is treating a tax cut as if the full amount were an immediate addition to consumption.

The government-purchases multiplier is $1/(1-c)=1/(1-0.75)=1/0.25=4$. Thus the purchases change gives $\Delta Y=4(40)=160$, or an output increase of $160 billion.

The tax multiplier is $-c/(1-c)=-0.75/0.25=-3$. A $40 billion tax cut means $\Delta T=-40$, so $\Delta Y=(-3)(-40)=120$, or an output increase of $120 billion.

Check. The tax-cut effect is $0.75$ of the purchases effect: $0.75(160)=120$. It is smaller by $40 billion. The first-round consumption increase from the tax cut is only $0.75(40)=30$ billion, rather than $40 billion.

Using the Multipliers With Signed Changes

A reliable calculation keeps the sign of each policy change. In the equation $\Delta Y=[1/(1-c)]\Delta G-[c/(1-c)]\Delta T$, an increase in $G$ is positive, an increase in $T$ is positive, and a tax cut is a negative $\Delta T$. This convention automatically produces the correct direction for each output response.

For example, an increase in taxes contributes a negative term to $\Delta Y$, while a tax cut contributes a positive term. A policy may also involve changes in both purchases and taxes; in that case, calculate each contribution with its own signed change and add them. This calculation does not make the two multipliers equal: it simply combines their separate effects under the model.

Worked Example: Tracing the Rounds After a Tax Increase

A hypothetical economy has an MPC of $0.60$. Compare a $25 billion increase in lump-sum taxes with a $25 billion increase in government purchases. Assume all other autonomous expenditure is unchanged.

Method. Apply each signed change to its multiplier, then check the result using the first-round expenditure sequence. This avoids confusing the initial consumption response to taxes with the total output response.

The purchases multiplier is $1/(1-0.60)=1/0.40=2.5$. Therefore, $\Delta Y=2.5(25)=62.5$, an output increase of $62.5 billion.

The tax multiplier is $-0.60/0.40=-1.5$. For the tax increase, $\Delta T=25$, so $\Delta Y=(-1.5)(25)=-37.5$. Output falls by $37.5 billion.

Round-by-round check for the tax increase. The $25 billion tax increase first reduces disposable income by $25 billion. Consumption initially falls by $0.60(25)=15$ billion. The next round falls by $0.60(15)=9$ billion, followed by $0.60(9)=5.4$ billion. The total fall is $15+9+5.4+\cdots=15/(1-0.60)=37.5$ billion, matching the multiplier calculation.

Round-by-round check for purchases. The initial increase is $25 billion, followed by $0.60(25)=15$, then $0.60(15)=9$, and so on. The total is $25+15+9+\cdots=25/(1-0.60)=62.5$ billion. Both checks agree with the calculated signs and magnitudes.

Finding a Policy Change for a Target Output Effect

The same formulas can be rearranged to find the policy change needed for a specified output response. For a desired increase in output, a tax cut must be larger in dollar terms than a purchases increase because each dollar of tax reduction initially generates only $c$ dollars of consumption. Always keep track of whether the requested value is a tax increase or a tax cut.

Worked Example: Solving for the Required Tax Cut

In a hypothetical economy, $c=0.80$. Policymakers want equilibrium output to rise by $90 billion using either a government-purchases increase alone or a lump-sum tax cut alone. Find each required policy change.

Method. Calculate both multipliers and divide the target output change by the relevant multiplier. The common mistake to avoid is comparing the policy amounts without first distinguishing a tax increase from a tax cut.

The purchases multiplier is $1/(1-0.80)=1/0.20=5$. The required purchases increase is therefore $\Delta G=90/5=18$, or $18 billion.

The tax multiplier is $-0.80/0.20=-4$. Set $90=(-4)\Delta T$. Then $\Delta T=90/(-4)=-22.5$. The negative sign means a tax cut of $22.5 billion.

Check. The tax-cut contribution is $(-4)(-22.5)=90$ billion, and the purchases contribution is $5(18)=90$ billion. The required tax cut is larger than the purchases increase, as expected.

Common Exam Trap: Using the Wrong Tax Effect

A frequent error is to use the government-purchases multiplier for a tax change, or to omit the MPC from the numerator of the tax multiplier. Consider $c=0.70$ and a $10 billion tax cut. The incorrect calculation might use $1/(1-c)=1/0.30=3.333$, giving an output increase of about $33.33 billion. That applies the purchases multiplier to a tax cut and overstates the effect.

The correct tax multiplier is $-0.70/0.30=-2.333\ldots$. Since the tax change is $\Delta T=-10$, the output change is $(-2.333\ldots)(-10)=23.333\ldots$ billion. Equivalently, the first-round consumption increase is only $0.70(10)=7$ billion, and later rounds multiply that initial $7 billion by $1/(1-0.70)$.

Use the right numerator. The purchases multiplier has numerator 1; the tax multiplier has numerator $-c$.
Use the signed tax change. A tax cut has $\Delta T<0$, so multiplying by the negative tax multiplier produces a positive output change.
Compare like with like. The “smaller” claim refers to the absolute output effect of equal-sized changes, one in purchases and one in lump-sum taxes, under the same model assumptions.

Key takeaway: Government purchases enter planned expenditure dollar for dollar, while a lump-sum tax change affects consumption only through the MPC. Consequently, for $0<c<1$, the absolute lump-sum tax multiplier is $c$ times the government-purchases multiplier.

Check Your Understanding

For each problem, use the simple expenditure model with no proportional taxes, imports, interest-rate response, or binding supply constraint. Treat all amounts as dollars in the stated units.

A hypothetical economy has $c=0.70$. Compare a $18 billion increase in government purchases with a $18 billion tax cut. Find both output changes. (Answer: purchases increase output by $60 billion; the tax cut increases output by $42 billion.)
In an economy with $c=0.65$, calculate the output effect of a $12 billion lump-sum tax increase and a separate $12 billion government-purchases increase. Give each result to two decimal places. (Answer: tax increase, $-$22.29$ billion; purchases increase, $+$34.29$ billion.)
With $c=0.80$, find the tax cut and, separately, the purchases increase that would each raise output by $72 billion. (Answer: tax cut of $18 billion; purchases increase of $14.4 billion.)
A hypothetical economy has $c=0.40$ and receives a $30 billion lump-sum tax cut. Calculate the output increase and compare it with the effect of a separate $30 billion increase in government purchases. (Answer: tax cut, $+$20$ billion; purchases increase, $+$50$ billion.)
In a model using only these two multipliers, the government-purchases multiplier is $3.2$. Infer the MPC, calculate the lump-sum tax multiplier, and find the output effect of a $50 billion tax cut. (Answer: $c=0.6875$; tax multiplier $=-2.2$; output increases by $110 billion.)

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