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## AP®︎/College Macroeconomics

### Course: AP®︎/College Macroeconomics>Unit 3

Lesson 2: Multipliers

# Lesson summary: The expenditure and tax multipliers

In this lesson summary we cover the key takeaways and terminology related to spending multipliers and tax multipliers. Topics include how to calculate the expenditure multiplier and the tax multiplier.

## Lesson overview

How can an additional $\mathrm{}1$ in spending lead to more than $\mathrm{}1$ additional GDP? Because of a thing called the multiplier effect. A dollar spent by one person is income for a different person. But if the person who receives that dollar is going to spend some of it and save some of it, then this sparks a process that magnifies (or “multiplies”) that first change.
The fraction of how much is spent will determine the ultimate increase that the first dollar will cause—the higher the fraction that is spent, the bigger the multiplier. That means that if there is a change in autonomous spending, such as the government decides to increase government spending, the final impact on real GDP will depend on the fraction that is spent instead of saved.

## Key Terms

Termdefinition
multiplier effectwhen a change in spending leads to a much larger change in real GDP than the initial change; for example, if a government spends $\mathrm{}100$, and that change in spending leads to real GDP increasing by $\mathrm{}400$, then the multiplier effect has multiplied the initial impact four times.
change in autonomous spendingchanges in spending that happen in response to something besides an increase in income; for example, if the government decides to spend money on building a new bridge because they want to build a bridge (not because they have extra income lying around) or one firm decides to build a $\mathrm{}10$ million fidget spinner factory. The key thing that makes a change autonomous is that it is not happening in response to an increase in income,
marginal propensity to consume ($MPC$)the proportion of any additional income that is spent; for example, if your $MPC$ is $0.75$ that means for every $\mathrm{}1$ more income you get, you will save $25$ cents and spend $75$ cents.
marginal propensity to save ($MPS$)the proportion of any additional income that is saved; note that $MPC+MPS=1$.
expenditure multiplierthe magnitude of how much real GDP will change in response to an autonomous change in aggregate spending; for example, if the expenditure multiplier is $5$, then $\mathrm{}100$ in government spending results in a total increase in real GDP of $\mathrm{}500$. This means that if a country has an output gap of $\mathrm{}500$, it doesn’t have to increase government spending by $\mathrm{}500$ to close that gap.
tax multiplierthe ratio of the total change in real GDP caused by a change in taxes; for example, if the tax multiplier is $-4$, then a $\mathrm{}100$ tax increase will decrease real GDP by $\mathrm{}400$. For example, if the government has an output gap of $\mathrm{}400$ million and the tax multiplier is $-4$, then the government can close that gap by decreasing taxes by only $\mathrm{}100$ million.

## Key Takeaways

### A $\mathrm{}1$‍  increase in autonomous spending means more than a $\mathrm{}1$‍  increase in real GDP

Governments usually spend money if they have a reason, or an objective to spend it. They don’t spend more money just because the economy is doing well and national income is increasing (in fact, we will learn later that it is usually the opposite). This kind of an increase in spending called an autonomous increase in government spending.
A change in autonomous spending will lead to a much larger final change in real GDP because of the multiplier effect. That spending will have a much larger final impact on real GDP. For example, if the government buys apples from Jack, Jack then uses that money to buy latte’s from Jill, and Jill buys a computer from Pedro. The final impact of the government’s purchase of apples will be bigger than just that purchase because we add up the apples, lattes, and computers in real GDP.

### The expenditure multiplier

The expenditure multiplier shows what impact a change in autonomous spending will have on total spending and aggregate demand in the economy. To find the expenditure multiplier, divide the final change in real GDP by the change in autonomous spending.
For example, if the government spends to send the first gerbil into space, and this increase in government spending results in a increase in real GDP, then the multiplier is $4$.
How does that happen? The government buys a rocket from Rocket’s R Us for , which gets counted in government spending. Rocket’s R Us uses this to pay their employees. Those employees will save some of that , but spend the rest on t-shirts. The t-shirt sellers then spend the income earned selling t-shirt on kayaking lessons, and so on.
We can see the impact of this multiplier effect on in the table below:
Roundamount spent = additional income x $MPC$cumulative impact on real GDP
$1$ Government buys rockets$\mathrm{}1,000,000$$\mathrm{}1,000,000$
$2$ Employees get $\mathrm{}1,000,000$ in income and buy t-shirts$\mathrm{}750,000$$\mathrm{}1,750,000$
3 The t-shirt sellers use t-shirt revenue to buy kayaking lessons$\mathrm{}562,500$$\mathrm{}2,312,500$
$4$ The kayaking instructors use the income to buy gym membership$\mathrm{}421,875$$\mathrm{}2,734,375$
$5$ The gym owners use income to buy space heaters$\mathrm{}316,406.25$$\mathrm{}3,050,781.25$
***
Eventual effect$\mathrm{}4,000,000$
But wait, there’s more! The expenditure multiplier can also tell us how much more or less spending is needed to close an output gap. For example, if we know the multiplier is $5$ and there is a positive output gap, only more spending is needed to close it.

### The tax multiplier

A change in taxes also results in a multiplier effect. The tax multiplier tells you just how big of a change you will see in real GDP as a result of a change in taxes. For example, imagine the government gives out a total of in tax refunds. As a result, there is a increase in real GDP. Therefore, the tax multiplier is $-3$. The tax multiplier is always one less in magnitude than the expenditure multiplier, and it is always a negative number.
We can see how this plays out in the table shown below. Suppose of spending on gerbil rockets, the government gave out dollars in tax rebates to the employees at Rockets R Us.
Roundamount spent= additional income x $MPC$cumulative impact on real GDP
$1$ Employees get in tax rebates and buy t-shirts$\mathrm{}750,000$$\mathrm{}750,000$
2 t-shirt sellers use t-shirt revenue to buy kayaking lessons$\mathrm{}562,500$$\mathrm{}1,312,500$
$3$ kayaking instructors use income to buy gym membership$\mathrm{}421,875$$\mathrm{}1,734,375$
$4$ gym owners use income to buy space heaters$\mathrm{}316,406.25$$\mathrm{}2,050,781.25$
***
Eventual effect$\mathrm{}3,000,000$
Why is it less? Notice that there is a round missing here: the initial in tax rebates themselves are not counted. The impact of the tax is indirect, not direct. That missing initial amount is why the tax multiplier is always 1 less than the expenditure multiplier.

### The $MPC$‍  and multipliers

The more that is spent out each dollar, the bigger the multiplier will be. Why? Because if money is spent, that money becomes income to another agent who then spends it, and that continues with a chain reaction of spending.
The tax multiplier tells us the final increase in real GDP that will occur as the result of a change in taxes. Interestingly, the tax multiplier is always smaller than the expenditure multiplier by exactly $1$. So if the expenditures multiplier is $4$, the tax multiplier is $3$ and if the expenditures multiplier is $10$, the tax multiplier is $9$.

### The $MPC$‍  and the $MPS$‍

The marginal propensity to consume is the change in spending that occurs when income changes, divided by that change in disposable income. If someone spends $\mathrm{}75$ when they have $\mathrm{}100$ more in income, the $MPC$ is $0.75$.
There are only two things you can do with money: spend it or save it. That means whatever proportion not spent must be saved. Economists call this the marginal propensity to save ($MPS$). So if the $MPC$ is $0.75$, the $MPS$ is $0.25$. The sum of $MPC$ and $MPS$ is always $1$.

## Key Equations

### Marginal propensity to consume ($MPC$‍ )

$MPC=\frac{\text{change in spending}}{\text{change in income}}$

### Marginal propensity to save MPS

$MPS=1-MPC$

### Expenditure multiplier

$\begin{array}{rl}\text{expenditure multiplier}& =\frac{1}{1-MPC}\end{array}$

### Tax multiplier

$\text{tax multiplier}=\frac{-MPC}{MPS}$

### Final impact on GDP

$\text{final impact on GDP}=\text{multiplier}×\text{autonomous change}$
For example, if the the tax multiplier is $-3$, and taxes increase by $\mathrm{}400$, then:
$\begin{array}{rl}\text{final impact on GDP}& =-3×\text{400}\\ \\ & =-\mathrm{}1,200\end{array}$
So an increase in taxes of $\mathrm{}400$ will decrease GDP by $\mathrm{}1200$.

## Common misperceptions

• Some people think that a $\mathrm{}100$ increase in government spending should have the same impact as a $\mathrm{}100$ decrease in taxes. But spending has a bigger impact than changes in taxes. A change in taxes has a smaller impact on GDP than a change in spending because of the first step in the expansion process. The initial change in autonomous spending doesn’t get saved in the government spending expansion process, but it does get saved in the tax change process.
• A common misstep is to forget that the spending multiplier and the tax multiplier have the same sign. The tax multiplier is negative, the expenditure multiplier is positive. This is because an increase in aggregate expenditures will increase real GDP, and an increase in taxes will decrease real GDP.
• You won’t be able to use a calculator on the exam. Most test writers know this and take this into account. One of the ways a teacher or test grader might help you out is to use MPCs and MPSs that have multipliers that are easy to calculate. Some commonly used MPCs and their corresponding multipliers are given in the table below. However, don’t just memorize these! You should always show all of your work on free-response questions.
Value of $MPC$Expenditure multiplierTax multiplier
$0.9$$10$$-9$
$0.8$$5$$-4$
$0.7$$4$$-3$
$0.5$$2$$-1$

## Discussion questions

• A $\mathrm{}300$ increase in autonomous expenditures has lead to a $\mathrm{}1,500$ increase in real GDP. Based on this information, what is the marginal propensity to consume?
• Assume that the government of Libbyland has a balanced budget. However, the government in Libbyland implements a tax cut to households. If the marginal propensity to save in Libbyland is $0.25$, what will be the final impact this tax cut will have on real GDP? Show all work.
• Jacksonia has a recessionary gap of . What additional information will we need to know in order to find the amount of autonomous spending that would close that gap?

## Want to join the conversation?

• I'm confused about GDP decrease and taxes. At one point, the article says that taxes will decrease GDP by some negative factor (the tax multiplier)--and this makes sense. But the articles later uses an example wherein people receive a tax refund, which I imagine will increase GDP--but the tax multiplier is still negative. How can the tax multiplier always be negative if it applies to tax refunds/cuts, which I'm assuming will increase GDP?
• The last example illustrates this. Say the multiplier is -3. If you tax someone $100, this results in:$100 x -3 = -$300 to the GDP. A refund would be a negative tax. Therefore if you tax someone -$100, this results in: -$100 x -3 =$300 to the GDP.
• also, shouldn't an MPC of 0.7 give you an expenditure multiplier of 3.333 instead of 4? to get 4 you should have an MPC of 0.75 right?
• I love Walter White
• I'm uncertain about the MPC and tax multiplier effects.

Assume the government taxes $100 away, and simultaneously gives a$100 refund. The tax multiplier, with an MPC of 0.9, is -9; the expenditure multiplier is 10. So GDP increases by $100. Notice that the net change in taxes is$0.

If the government reduces taxes by $100, then that's$900 of additional GDP; but if the government makes a $100 payment, that's$1,000 more GDP.

All of this, of course, has some issues: the population isn't flat, and there's an income effect, so a $100 tax increase on income over$10M and a $100 tax refund to people with incomes below$0.01M will have a net-positive effect because MPC is higher at lower incomes.

Mathematically, a transfer is the same as a tax cut; but in practice it's not. How are the above things used?
Suppose GDP=$1000. After taxing$100 we will decrease GDP by $900 (because 1000+(-9)*100=100), meaning that now GDP will be GDP=$100. Suppose after that the government will change its mind and make $100 tax rebates. In this case GDP will increase by$900, totaling $1000 (because 100+(-9)*(-100)=1000). We use (-100) instead of positive 100 because giving rebats is practically the same thing as taxing people with negative sums of money. For the same reason we use the tax multiplier when giving people rebates instead of the expenditures multiplier. As you can see, we are back to where we started, our GDP is the same as before taxation. (3 votes) • If a positive output gap means that the actual output > potential output, why would governments want to close that gap? (3 votes) • It would mean that inflation is high, and that means prices are higher than they should be. By slowing the economy back down and reducing the output gap, prices would drop again. (2 votes) • There's an error above: MPC = 0.7 => MPS = 1-MPC = 0.3 => Exp.Mult. = 1/0.3 = 3.3 (not 4) => Tax Mult. = -2.3 (not -3) (3 votes) • I have a question. When we are using the multiplier factors (expenditure or taxes); we use them to determine the effect of additional income on the Real GDP of the country. Based on the MPC and MPS, we could determine the effect of this additional income on the economic cycle. The question is: why we could not use these methods to (predict) or (estimate) the overall number of the GDP of a nation for referencing? in another way; what are the pros and cons if we assumed the additional income is the total income? and is it a valid method, even it is not highly accurate, but just using it for prediction and referencing? (3 votes) • In the second discussion question on Libbyland, is the change in GDP really only$300. I feel like it should be $3,000,00. I'm confused. (3 votes) • is the answer to the third Discussion Question (Jacksonia), simply that we need the MPC (or MPS)? (1 vote) • Since you can find the expenditure multiplier from the MPS or MPC, your answer is correct. However, it is ultimately the multiplier that is needed to know the amount that will fill the gap. (4 votes) • shouldn't an initial$1000 expenditure with an MPC of 0.9 end in a $10000 GDP increase instead of$9000? because 1000×(1÷(1-0.9))=‬ 10000