Multiplier

Multiplier Formula

To use the multiplier formulas, we have to calculate the marginal propensity to consume (MPC) and the marginal propensity to save (MPS) first, since they feature heavily in the multiplier equations.

MPC and MPS formula

If consumer spending increases because the consumer has more disposable income, we calculate the MPC by dividing the change in consumer spending by the change in disposable income. It will look something like this:

\(\frac{\Delta \text {Consumer Spending}}{\Delta \text{Disposable Income}}=MPC \)

If we assume that people do not spend all of their disposable income, then where does the rest of the income go? It goes into savings. This is where the MPS comes in since it accounts for the amount of disposable income that the MPC does not. The formula for the MPS looks like this:

\(1-MPC=MPS\)

Expenditure Multiplier Formula

Now we are ready to calculate the expenditure multiplier. Instead of calculating each round of spending individually and adding them together until we arrive at the total increase of real GDP that the initial change in aggregate spending caused, we use this formula:

\(\frac{1}{1-MPC}=\text{Expenditure Multiplier}\)

Tax Multiplier Formula

Taxes have an inverse relationship with consumer spending. The MPC is in the place of the 1 in the numerator because people do not spend the entire equivalent of their tax cut, just like they do not spend all their disposable income. They only spend in proportion to their MPC and save the rest, unlike in the expenditure formula where $1 in spending increases real GDP and disposable income by $1. The tax multiplier is negative because of the inverse relationship where an increase in taxes causes a decrease in spending. The tax multiplier formula helps us calculate the effect of a tax policy on GDP.

\(\frac{-MPC}{(1-MPC)}=\text{Tax Multiplier}\)

Multiplier Theory in Economics

The multiplier theory refers to when an economic factor increases, it generates a higher total of other economic variables than the increase of the initial factor. When there is an autonomous change in aggregate spending more money is spent in the economy. People will earn this money in the form of wages and profits. They will then save a portion of this money and put the rest back into the economy by doing things like paying rent, buying groceries, or paying someone to babysit.

Now the money increases someone else's disposable income, a portion of which they will save and a portion of which they will spend. Each round of spending increases the real GDP. As the money cycles through the economy, a portion of it is saved and a portion is spent, which means the amount that is reinvested each round is shrinking. Eventually, the amount of money reinvested in the economy will equal 0.

Here is a visual representation of the rounds of spending:

The initial increase in investment spending on new solar farms is $500 million. The increase in disposable income is $32 million and consumer spending increased by $24 million.

$24 million divided by $32 million gives us a MPC = 0.75.

Table 1. The multiplier effect - StudySmarter

If we were to plug in all of the values manually we would eventually discover that the total increase in real GDP is $2,000 million, which is $2 billion. Using the formula it would look like this:

$\frac{1}{(1-0.75)}\times \$500million=totalincreaseinGDP\frac{1}{0.25}\times \$500million=4\times \$500million=\$2billion$

Even though the initial increase in investment was only $500 million, the total increase in real GDP was $2 billion. The increase in one economic factor generated a higher total of other economic variables.

The more likely people are to spend or the higher the MPC, the higher the multiplier is. When the multiplier is high, there is a larger increase in the effect of the initial autonomous change in aggregate spending. If the multiplier is low, and peoples’ MPS is high, then there will be a smaller effect.

So far we have been under the assumption that there are no government taxes or spending. The tax multiplier is similar to the expenditure multiplier in that the effects are multiplied through the rounds of spending. It differs in that the relationship between taxes and consumer spending is inverse.

As governments increase taxes and disposable income decreases, consumer spending falls. As each $1 is taxed, disposable income decreases by less than $1. Consumer spending increases in proportion to the MPC in the case of a tax cut or the MPS in the case of a tax increase. This is why the government spending and expenditure multiplier have a greater effect than the tax multiplier. This leads to less output in each round of spending, resulting in less total real GDP.

Economic impact of the multiplier

The economic impact of the multiplier is economic growth due to injections into the economy in the form of spending and investments. As these injections flow through the economy, they contribute to a nation's GDP by stimulating production, consumption, investment, and expenditure at various stages.

The multiplier effect benefits the economy because a small increase in expenditure, investment, or tax cut, has a magnified effect on the economy. Of course, the size of the effect depends on society’s marginal propensity to consume (MPC) and marginal propensity to save (MPS).

If the MPC is high and people spend more of their income, injecting it back into the economy, the multiplier effect will be stronger and therefore the effect on the total real GDP will be greater. When society’s MPS is high, they save more, the multiplier effect is weaker, and the total real GDP effect will be smaller.

Multiplier in Four Sector Economy

The four sector economy is made up of households, firms, the government, and the foreign sector. As seen in Figure 1, money flows through these four sectors through government spending and investing, taxes, private income, and spending, as well as imports and exports in a circular flow.

Leakages consist of taxes, savings, and imports because the money spent on those does not continue to cycle in the economy. Injections are exports, investments, and government spending because they increase the supply of money flowing through the economy.

Figure 1. Four sector economy circular flow diagram

The multiplier effect can be applied to several components. Firms and households account for the autonomous change in aggregate supply. For whatever reason firms and households decide they want to invest in improving their landscaping, so there is an injection of funds into the economy to pay for landscape design, purchasing soil and gravel, installing sprinklers, and gardener wages. The impact on the real GDP of these rounds of spending is explained by the expenditure multiplier. The government can also provide the initial increase in funds in the form of government spending and tax policy which both have their own multiplier effects.