Worked Government-Purchases Shocks

Why Work Through a Government-Purchases Shock?

A multiplier formula is useful only if it is applied to the right quantity. A government-purchases shock is a change in purchases; the multiplier converts that change into a change in equilibrium output. It does not, by itself, give the new output level. A numerical solution therefore has two distinct jobs: find the starting equilibrium, then calculate how the shock changes it.

This tutorial applies the simple Keynesian-cross model to those calculations. It focuses on autonomous consumption, planned investment, and government purchases, with no taxes or imports in the model. The examples show how to solve for initial output, handle increases and decreases in purchases, and verify the result. The treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Model and assumptions: Planned expenditure is $E=C+I+G$, where $C=C_0+cY$, $I$ is fixed planned investment, and $G$ is government purchases. Here $C_0$, $I$, and $G$ are autonomous amounts, $Y$ is output, and $c$ is the marginal propensity to consume, with $0<c<1$. Equilibrium output satisfies $Y=E$. The model holds prices and $c$ fixed and excludes taxes, imports, interest-rate responses, and capacity constraints.

From the Initial Equilibrium to the Shock

In equilibrium, output equals planned expenditure. Substituting the consumption function into that condition gives $Y=C_0+cY+I+G$. The initial equilibrium can therefore be found by collecting the terms involving $Y$ on one side. Once that level is known, a government-purchases change can be evaluated separately.

$$ Y=C_0+cY+I+G $$

Y^*=\frac{C_0+I+G}{1-c}

The asterisk marks an equilibrium level. If purchases change by $\Delta G$, while $C_0$, $I$, and $c$ remain unchanged, the change in equilibrium output is

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

A positive $\Delta G$ raises equilibrium output in this model; a negative $\Delta G$ lowers it. The new equilibrium level is the initial level plus the change: $Y_1^*=Y_0^*+\Delta Y$. Keeping these quantities separate prevents a common error: reporting the change as if it were the final output level.

Why the Multiplier Applies to the Change

The change formula follows directly from comparing two equilibria under the same model. Before the shock, output satisfies $Y_0^*=C_0+cY_0^*+I+G_0$. After the shock, it satisfies $Y_1^*=C_0+cY_1^*+I+G_1$. Subtract the first equilibrium equation from the second. The unchanged autonomous terms cancel, leaving $\Delta Y=c\Delta Y+\Delta G$. Rearranging gives $(1-c)\Delta Y=\Delta G$, so dividing by $1-c$ yields the multiplier formula. This also explains why the shock is multiplied: the initial change in purchases generates further rounds of consumption as output and income rise.

\begin{aligned} Y_1^*-Y_0^*&=c(Y_1^*-Y_0^*)+(G_1-G_0)\\ \Delta Y&=c\Delta Y+\Delta G\\ (1-c)\Delta Y&=\Delta G\\ \Delta Y&=\frac{\Delta G}{1-c} \end{aligned}

The multiplier is $1/(1-c)$, which is greater than 1 when $0<c<1$. The formula relies on the model’s assumptions: if the marginal propensity to consume changes between equilibria, for example, the subtraction above does not produce the same constant-$c$ multiplier. Likewise, this simple calculation is not a complete prediction of actual output when excluded responses—such as changing prices or interest rates—matter.

A Reliable Calculation Sequence

For a numerical question, identify the starting values and the size and direction of the government-purchases change before calculating. Then solve the initial equilibrium and the change separately. The steps below keep the units and the two output quantities explicit.

Record the model inputs.
Identify $C_0$, $c$, initial $I$, initial $G_0$, and the signed change $\Delta G$. Keep all dollar amounts in the same units.

Solve initial equilibrium output.
Calculate $Y_0^*=(C_0+I+G_0)/(1-c)$. This is the output level before the shock.

Calculate the output change.
Find the multiplier $1/(1-c)$, then multiply it by the signed $\Delta G$.

Find and check the new level.
Set $Y_1^*=Y_0^*+\Delta Y$. Check by substituting $Y_1^*$ into $C_0+cY+I+G_1$; it should equal $Y_1^*$.

Worked Applications

Each example follows the same sequence but changes the direction or structure of the shock. The checks use the post-shock values of government purchases and consumption.

Worked Example: An Increase in Government Purchases

A simplified economy has planned consumption $C=120+0.75Y$, planned investment $I=80$, and initial government purchases $G_0=100$, all measured in millions of dollars. Government purchases rise by $\$20$ million. Find initial equilibrium output, the change in output, and the new equilibrium output.

Method. Solve the initial equilibrium first, then apply the multiplier to the signed purchases change.

1. Find initial equilibrium. The autonomous expenditure total is $C_0+I+G_0=120+80+100=300$ million. The marginal propensity to consume is $c=0.75$, so $1-c=1-0.75=0.25$. Therefore,

Y_0^*=\frac{300}{0.25}=1{,}200\text{ million dollars}.

2. Find the multiplier and output change. The multiplier is $1/(1-c)=1/0.25=4$. The shock is $\Delta G=+20$ million, so $\Delta Y=4(20)=+80$ million.

3. Find the new output level. $Y_1^*=1{,}200+80=1{,}280$ million dollars.

4. Check the new equilibrium. New government purchases are $G_1=100+20=120$ million. At $Y=1{,}280$, consumption is $C=120+0.75(1{,}280)=120+960=1{,}080$ million. Planned expenditure is $1{,}080+80+120=1{,}280$ million, equal to output.

Common mistake avoided: Adding only the initial $\$20$ million of purchases would give a change of $\$20$ million, not the model’s $\$80$ million change. The induced consumption rounds account for the difference.

Worked Example: A Cut in Government Purchases

In another economy, planned consumption is $C=90+0.60Y$, planned investment is $I=150$, and initial government purchases are $G_0=210$, all in millions of dollars. Purchases are reduced by $\$40$ million. Calculate initial equilibrium output, the output change, and the new equilibrium.

Method. Treat the cut as a negative change, rather than using the absolute size of the cut as if it were an increase.

1. Find initial equilibrium. Total autonomous expenditure is $90+150+210=450$ million. The unspent fraction of an additional dollar of income is $1-c=1-0.60=0.40$. Thus,

Y_0^*=\frac{450}{0.40}=1{,}125\text{ million dollars}.

2. Find the output change. The multiplier is $1/0.40=2.5$. Since purchases fall, $\Delta G=-40$ million. Therefore, $\Delta Y=2.5(-40)=-100$ million.

3. Find the new output level. $Y_1^*=1{,}125-100=1{,}025$ million dollars. The new purchases level is $G_1=210-40=170$ million.

4. Check the result. At the new output, consumption is $C=90+0.60(1{,}025)=90+615=705$ million. Planned expenditure is $705+150+170=1{,}025$ million, equal to the new output.

Common mistake avoided: Using $+40$ instead of $-40$ would incorrectly predict an increase of $100$ million. The sign of the shock must carry through the multiplier calculation.

Worked Example: Several Changes and a Final Equilibrium Check

A regional economy is represented by $C=180+0.75Y$, fixed planned investment $I=170$, and initial government purchases $G_0=250$, in millions of dollars. Purchases first rise by $\$30$ million and are then reduced by $\$10$ million relative to that higher level. Find the initial equilibrium, the net output change, the final output, and verify the result.

Method. Combine the successive purchases changes to find the final change relative to the initial level; then use that net shock.

1. Find initial equilibrium. Initial autonomous expenditure is $180+170+250=600$ million. Since $1-c=1-0.75=0.25$,

Y_0^*=\frac{600}{0.25}=2{,}400\text{ million dollars}.

2. Find the net purchases change. The first change is $+30$ million and the second is $-10$ million. Their sum is $\Delta G=30-10=+20$ million. Final purchases are $G_1=250+20=270$ million.

3. Find the net output change and final level. The multiplier is $1/0.25=4$. Hence $\Delta Y=4(20)=80$ million and $Y_1^*=2{,}400+80=2{,}480$ million.

4. Check the final equilibrium. Consumption at $Y=2{,}480$ is $C=180+0.75(2{,}480)=180+1{,}860=2{,}040$ million. Planned expenditure is $2{,}040+170+270=2{,}480$ million, matching output.

Common mistake avoided: Applying the multiplier to the initial purchases level, or to the first increase alone, would not answer the question about the final change relative to the initial equilibrium. The net shock is $20$ million, not $30$ million.

Common Errors and Model Limits

The multiplier is a relationship between a change in government purchases and a change in equilibrium output. It is not a multiplier on the whole initial output level. In the first example, multiplying the initial output of $1{,}200$ million by 4 would produce an irrelevant figure; the multiplier applies to the $\$20$ million shock, giving an $\$80$ million output change.

A second frequent error is to stop after finding $\Delta Y$. If a question asks for the new equilibrium output, add the signed change to initial output. Conversely, if it asks only for the effect of the shock, report the change rather than substituting the new level. Always label the result.

Finally, do not use this formula without checking the model. It assumes a constant $c$ between the two equilibria and holds other autonomous expenditure fixed. It also omits taxes, imports, changes in prices, and interest-rate responses. If a problem supplies a different model or changes additional variables, the simple government-purchases multiplier alone is not sufficient.

Key takeaway: In the simple Keynesian-cross model, solve the initial level with $Y_0^*=(C_0+I+G_0)/(1-c)$, calculate the shock’s effect with $\Delta Y=\Delta G/(1-c)$, and report the new level as $Y_0^*+\Delta Y$. Verify the result by checking that planned expenditure equals output.

Check Your Understanding

For each problem, keep the shock’s sign and units explicit. Assume the simple Keynesian-cross model and its assumptions unless the question states otherwise.

An economy has $C=75+0.50Y$, $I=125$, and $G_0=100$, in millions of dollars. Purchases increase by $\$15$ million. Find initial equilibrium output, the output change, and the new equilibrium. (Answer: $Y_0^*=\$600$ million; $\Delta Y=+\$30$ million; $Y_1^*=\$630$ million.)
Planned consumption is $C=110+0.80Y$, planned investment is $90$, and initial government purchases are $150$, all in millions. Purchases are cut by $10$. Find the initial and new equilibrium output. (Answer: $Y_0^*=\$1{,}750$ million; $\Delta Y=-\$50$ million; $Y_1^*=\$1{,}700$ million.)
In an economy with $C=200+0.60Y$, $I=180$, and $G_0=120$, all in millions, government purchases first rise by $25$ and then fall by $5$ from the raised level. Find the net shock and final equilibrium output. (Answer: net $\Delta G=+\$20$ million; $Y_0^*=\$1{,}250$ million; $\Delta Y=+\$50$ million; $Y_1^*=\$1{,}300$ million.)
An economy has $C=140+0.75Y$, $I=160$, and $G_0=200$, in millions of dollars. Government purchases fall to $170$. Calculate the multiplier, initial equilibrium, output change, and new equilibrium. (Answer: multiplier $4$; $Y_0^*=\$2{,}000$ million; $\Delta Y=-\$120$ million; $Y_1^*=\$1{,}880$ million.)
For $C=50+0.50Y$, $I=70$, and $G_0=80$, all in millions, a policy raises purchases by $10$. A candidate reports the output change as $10$ and the new output as the initial output plus $10$. Give the correct change and new equilibrium, and identify the error. (Answer: $Y_0^*=\$400$ million; multiplier $2$; $\Delta Y=+\$20$ million; $Y_1^*=\$420$ million; the candidate omitted the induced expenditure rounds.)

What You'll Learn