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### Kurs: Makroekonomia - program rozszerzony>Rozdział 3

Lekcja 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.

## Podsumowanie lekcji

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.

## Kluczowe terminy

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.

## Do zapamiętania — najważniejsze zagadnienia

### 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 $1$.

## Najważniejsze równania

### 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 $\mathrm{}1200$.

## Częste błędy

• 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$

## Pytania do dyskusji

• 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?

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