Deriving the Government-Purchases Multiplier

Why Government Purchases Affect Equilibrium Output

In the Keynesian-cross model, equilibrium output is the level at which firms' production matches planned expenditure. Government purchases are one component of planned expenditure. If purchases rise, planned expenditure rises directly; equilibrium output then changes as the model returns to equality between output and planned expenditure.

The government-purchases multiplier measures the change in equilibrium output associated with a one-unit change in government purchases, when the model's other determinants are held fixed. This tutorial derives that relationship from the equilibrium-output equation. The result is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Model and assumptions: Consider a closed economy with no exports or imports. Planned expenditure is consumption plus autonomous investment plus government purchases. Consumption is linear in disposable income, and taxes are fixed lump-sum taxes. Prices are fixed, output adjusts to planned expenditure, and the marginal propensity to consume is constant with $0 \leq c < 1$. For the government-purchases multiplier, consumption parameters, taxes, investment, and all other autonomous spending remain unchanged as government purchases change. The model does not include a monetary-policy response or a change in interest rates that crowds out private spending.

Let $Y$ denote equilibrium output, $C_0$ autonomous consumption, $c$ the marginal propensity to consume (MPC), $T$ lump-sum taxes, $I$ autonomous investment, and $G$ government purchases. In this model, disposable income is $Y-T$. The consumption function is $C=C_0+c(Y-T)$, and planned expenditure is $C+I+G$.

Deriving the Multiplier from Equilibrium

At equilibrium, output equals planned expenditure. Substituting the consumption function into that condition gives an equation in $Y$. Collecting the terms involving output reveals how a change in purchases affects equilibrium output.

\begin{aligned} Y &= C + I + G \\ &= C_0 + c(Y-T) + I + G \\ &= C_0 + cY - cT + I + G \\ (1-c)Y &= C_0 - cT + I + G \\ Y &= \frac{C_0 - cT + I + G}{1-c} \end{aligned}

The condition $c<1$ ensures that $1-c$ is positive, so this equation has a finite solution. To isolate the effect of government purchases, compare two equilibria that differ only in $G$. Since $C_0$, $c$, $T$, and $I$ do not change, subtracting the initial equilibrium equation from the new one leaves the change in output on one side and the change in purchases on the other.

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

Result: In the closed-economy model with fixed lump-sum taxes and the other assumptions stated above, the government-purchases multiplier is $1/(1-c)$. Thus, for a change in purchases with all other determinants fixed, $\Delta Y = \Delta G/(1-c)$. The multiplier is positive and at least 1 when $0 \leq c < 1$. It is not a universal estimate of the effect of government purchases outside this model.

The algebra explains the result: in the equilibrium equation, an extra unit of $G$ raises the right-hand side by one unit. But equilibrium output also appears on the right through consumption, as $cY$. Moving that output-dependent term to the left leaves the coefficient $1-c$, so the output change needed to restore equality is $\Delta G/(1-c)$. The multiplier is therefore the reciprocal of the share of an additional unit of output not spent on consumption in this model.

The fixed-tax condition matters. The expression $1/(1-c)$ is the response to a change in $G$ while $T$ is unchanged. A simultaneous change in taxes, investment, or another component of planned expenditure is a different comparison. Likewise, if taxes depend on income, imports are included, or prices and interest rates respond, the equilibrium equation and multiplier may differ.

Applying the Equilibrium Equation

The equation can be used in two related ways. If the change in government purchases is known, multiply it by $1/(1-c)$ to find the change in equilibrium output. If the question instead gives an output target, rearrange the same relationship to find the required purchases change. When solving for an initial or final output level, use the full equilibrium equation, not only the change formula.

Worked Example: Find Initial and New Equilibrium Output

A hypothetical closed economy has consumption $C=80+0.75(Y-40)$, autonomous investment $I=100$, and government purchases $G=120$. All amounts are in millions of dollars. Prices are fixed, taxes are lump-sum and remain at 40, and the other model assumptions hold. Find initial equilibrium output and the new output if government purchases rise by 20 million dollars.

Method. First solve the equilibrium equation for the initial output. Then apply the purchases multiplier to the change in $G$. The MPC is $c=0.75$, so $1-c=0.25$.

Substitute the data into the equilibrium equation. The net autonomous-spending term is $C_0-cT+I+G=80-(0.75)(40)+100+120=80-30+100+120=270$. Therefore, initial output is $Y=270/0.25=1{,}080$ million dollars.

The multiplier is $1/(1-0.75)=1/0.25=4$. With $\Delta G=20$ million dollars, $\Delta Y=4(20)=80$ million dollars. New equilibrium output is $1{,}080+80=1{,}160$ million dollars.

Check. At the new output, disposable income is $1{,}160-40=1{,}120$. Consumption is $80+0.75(1{,}120)=80+840=920$. Planned expenditure is $920+100+140=1{,}160$, equal to output. The calculation is consistent with equilibrium.

Common mistake avoided: Using the initial level of $G$ as though it were the change would calculate the effect of 120 million dollars rather than the 20 million dollar increase. The multiplier applies to $\Delta G$, not to the level of government purchases.

Worked Example: Infer the MPC from Two Equilibria

In a hypothetical economy, government purchases rise from 60 to 78 billion dollars while lump-sum taxes, autonomous consumption, and autonomous investment remain fixed. Equilibrium output rises from 720 to 792 billion dollars. Under the stated closed-economy model, infer the multiplier and MPC, then verify the final equilibrium using consumption $C=50+c(Y-40)$ and autonomous investment $I=100$.

Method. Calculate the observed changes first. Their ratio gives the model's multiplier; the multiplier equation can then be rearranged to find $c$.

The purchases change is $\Delta G=78-60=18$ billion dollars. The output change is $\Delta Y=792-720=72$ billion dollars. Thus, the multiplier is $72/18=4$. Since $4=1/(1-c)$, we have $1-c=1/4=0.25$, so $c=0.75$.

To verify the initial equilibrium, calculate planned expenditure at $Y=720$: disposable income is $720-40=680$, consumption is $50+0.75(680)=50+510=560$, and planned expenditure is $560+100+60=720$.

At the new output, disposable income is $792-40=752$, consumption is $50+0.75(752)=50+564=614$, and planned expenditure is $614+100+78=792$. Both output levels satisfy the equilibrium condition.

Common mistake avoided: The 18-billion-dollar purchases increase is not itself the multiplier. The multiplier is the ratio of the output change to the purchases change, $72/18=4$.

Worked Example: Find Purchases Needed for an Output Target

A hypothetical closed economy has $C=40+0.80(Y-50)$, autonomous investment $I=100$, and government purchases $G=90$, all measured in billions of dollars. Lump-sum taxes remain fixed at 50 billion dollars. Find initial equilibrium output and the increase in government purchases needed to raise equilibrium output by 60 billion dollars.

Method. Solve the full equilibrium equation for the initial level, then rearrange the multiplier relationship to solve for the required change in purchases.

The net autonomous-spending term is $40-(0.80)(50)+100+90=40-40+100+90=190$. Since $1-c=1-0.80=0.20$, initial equilibrium output is $190/0.20=950$ billion dollars.

The multiplier is $1/0.20=5$. For a target increase of $\Delta Y=60$ billion dollars, $\Delta G=\Delta Y/5=60/5=12$ billion dollars. The new level of purchases is $90+12=102$ billion dollars, and the target output is $950+60=1{,}010$ billion dollars.

Check. At the new purchases level, the net autonomous-spending term is $40-40+100+102=202$. Dividing by $1-c=0.20$ gives $202/0.20=1{,}010$ billion dollars, the target output.

Common mistake avoided: Dividing by the multiplier is appropriate when solving for $\Delta G$ from a target $\Delta Y$. Multiplying the target output change by 5 would reverse the relationship.

Interpretation and Common Pitfalls

The multiplier depends on the MPC: as $c$ rises toward 1, the denominator $1-c$ becomes smaller and the model's multiplier becomes larger. For example, if $c=0.80$, the multiplier is $1/(1-0.80)=5$; if $c=0.50$, it is $1/(1-0.50)=2$. These are implications of the model, not claims that an observed policy change must produce those exact output responses.

A frequent error is to use $c$ itself as the multiplier. Suppose $c=0.80$ and government purchases rise by 10 million dollars. The incorrect calculation $0.80(10)=8$ million dollars treats the MPC as the output response per unit of purchases. In fact, $1-c=0.20$, so the multiplier is $1/0.20=5$, and the model predicts $\Delta Y=5(10)=50$ million dollars. The MPC is a parameter in consumption; it is not the government-purchases multiplier.

Another error is to apply the result while allowing other determinants to change. The subtraction that gives $(1-c)\Delta Y=\Delta G$ depends on the other terms being unchanged. If taxes change at the same time, for example, the comparison is not a pure government-purchases change. Keep the policy change and the model's fixed conditions clear before using the formula.

Identify the model inputs.
Check that the economy is treated as closed, taxes are fixed lump-sum taxes, and the MPC satisfies $0 \leq c < 1$.

Separate levels from changes.
Use the full equilibrium equation to find an output level; use the multiplier with $\Delta G$ to find a change in output.

Check the result.
Substitute the resulting output into planned expenditure and verify that the two are equal.

Key takeaway: With fixed lump-sum taxes and all other determinants held constant, the equilibrium-output equation implies $\Delta Y/\Delta G=1/(1-c)$. State the model conditions, distinguish changes from levels, and check that the result satisfies equilibrium.

Check Your Understanding

For each problem, use the closed-economy model with fixed lump-sum taxes and the other assumptions stated in the question.

The MPC is 0.65 and government purchases rise by 20 million dollars. Find the multiplier and the predicted change in equilibrium output. (Answer: multiplier $20/7 \approx 2.857$; output rises by $\$57.14$ million.)
Purchases rise by 18 billion dollars and equilibrium output rises by 72 billion dollars, with all other determinants fixed. Find the implied multiplier and MPC. (Answer: multiplier $4$; MPC $0.75$.)
A hypothetical economy has $C=45+0.75(Y-30)$, $I=55$, and $G=80$, with all values in billions of dollars. Find initial equilibrium output, then find output after purchases rise by 5 billion dollars. (Answer: initial output $\$630$ billion; new output $\$650$ billion.)
Equilibrium output is initially 900 billion dollars. Purchases increase by 25 billion dollars, and the MPC is 0.80. Assuming no other determinant changes, find the new output and the multiplier. (Answer: multiplier $5$; new output $\$1{,}025$ billion.)
A hypothetical economy has an MPC of 0.70. Policymakers seek an equilibrium-output increase of 36 billion dollars through government purchases alone. Find the required purchases increase. (Answer: $\$10.8$ billion.)

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