Fiscal Multiplier

Fiscal Multiplier Effect Graph

The multiplier effect is illustrated in Figure 1 below.

The rise in government purchases of precisely $20 billion moves the aggregate-demand curve from AD₁ to AD₂ at the beginning, which is a rightward shift of exactly $20 billion. However, the aggregate demand curve shifts even more to AD₃ when consumers and firms react by increasing their spending.

Fiscal Multiplier Formula

The government can influence the economy by either changing its spending, changing how much to collect in taxes, or doing both. As such, there are three fiscal multipliers for which we have to consider: spending multiplier, tax multiplier, and balanced budget multiplier.

Expenditure multiplier formula

The expenditure multiplier helps us determine how much an increase in one autonomous variable, such as an increase in government spending, changes a country's real GDP. When there is an increase in government purchases of goods and services, it will cause additional purchases to happen as a result of more money flowing to the hands of businesses and individuals. This, in turn, will cause further increases in real GDP.

Let's consider a simple case to find out about the effect of an increase in government spending on the economy. To do that, assume that the entire money that the government spends goes to households. Individuals can consume parts of this money and save the rest. Therefore, we have to take into account how much households are going to spend. The marginal propensity to consume (MPC) describes the portion of disposable income that an individual spends. This will help us find out how much an increase in government spending changes the real GDP.

The formula for the expenditure multiplier is:

If individuals spend 80% percent of their disposable income and save the remaining, the expenditure multiplier is: $\frac{1}{1-0.8}=\frac{1}{0.2}=5$

This means that for a one-dollar increase in government spending, the real GDP will increase by five dollars. If the government spends $10 billion, the real GDP will increase by $50 billion.

Tax multiplier formula

Tax multiplier refers to the change in real GDP due to a change in taxes. The tax multiplier is quite helpful in helping governments design the right tax rate regime. They do this by analyzing how much different tax rates would affect the real GDP so that they can change it according to the desired real GDP level.

However, one important thing to note is that aggregate demand changes are not proportional to the increase or decrease in taxes. That is because we don't always consume our entire disposable income; rather, we save some of it. So, if the taxes were to decrease, and as a result, we'd have more disposable income, it doesn't mean that we will spend it all. Hence, the change in aggregate demand will be less than the change in taxes.

The equation for the tax multiplier is as follows:

Where MPC is the marginal propensity to consume.

Note that the denominator (1 - MPC) is equal to the marginal propensity to save (MPS), which describes the portion of disposable income that an individual saves.

A change in government transfers works in a similar way. You can think of an increase in transfer payments as a decrease in tax collection. It increases people's disposable incomes just like a decrease in taxes would do.

Balanced budget multiplier formula

We discussed the spending multiplier and the tax multiplier. What happens when the government increases taxes to fund some of its current government expenditures? In such a case, both multipliers will play a role in the aggregate demand. The impact of that is analyzed using the balanced budget multiplier.

The balanced budget multiplier is found by summing up the spending and tax multiplier.

$\frac{1}{1-MPC}+\frac{-MPC}{1-MPC}=\frac{1-MPC}{1-MPC}=1$

As the balanced budget multiplier equals one, it means that whenever the government increases its spending by 1 dollar by funding it through tax increases, it increases aggregate spending by 1 dollar.

Fiscal Multiplier Derivation

Using some basic algebra, we can derive the formula for the size of the multiplier effect that takes place when an increase in government purchases causes an increase in disposable income, which in turn increases consumer spending. This formula considers that an increase in government purchases causes an increase in consumer spending. The marginal propensity to spend, known as MPC, is crucial in this formula since it indicates the proportion of a household's additional disposable income spent rather than saved.

Take, for instance, the case when the marginal tendency to spend is three-quarters. This indicates that a family will pay $0.75 (three-quarters of a dollar) and put $0.25 away for savings for every additional dollar that the household earns.

Consider the case when the U.S government spends $20 billion in purchases. Suppose the MPC is at three-quarters, when the employees and business owners receive $20 billion from the government contract, they will raise their consumer spending by three-quarters of $20 billion, which is $15 billion.

Following the impacts in sequential order allows us to calculate the influence of a shift in government spending on total market demand.

The rise in consumer spending of $15 billion causes a rise in income for the employees and owners of the businesses that manufacture the items that consumers buy.

The money that the business owners and employees spend will cause another increase in consumption as other people are receiving more income too.

To derive the formula will have to add all these effects together.

$$\begin{gathered} Changeingovernmentpurchases=\$20billion \\ Firstchangeinconsumption=MPCx\$20billion \\ Secondchangeinconsumption=MP{C}^{2}x\$20billion \\ Thirdchangeinconsumption=MP{C}^{3}x\$20billion \\ ......... \\ Totalchangeindemand=(1+MPC+MP{C}^2+MP{C}^3+...)x\$20billion. \end{gathered}$$

The multiplier can then be written as:

$Multiplier=1+MPC+MP{C}^2+MP{C}^3+...$

This provides an infinite geometric series, and from math, we know that:

$1+x+x^2+x^3+...=1/(1–x).$

Hence multiplier is:

$Multiplier=\frac{1}{(1-MPC)}$

Fiscal Multiplier Example

Assume that the U.S government wants to increase taxes by $20 billion dollars. The marginal propensity to consume (MPC) is 0.6 and the marginal propensity to save (MPS) is 0.4. What does it mean for the real GDP, and by how much will it change?

Let's consider another example. Let's assume that the government wants to increase its spending on public hospitals. For this, the government has prepared a budget of $25 billion. The marginal propensity to consume is 0.8. By how much would the GDP increase if the government was to spend $25 billion on hospitals?